Topologizing Homeomorphism Groups

This paper surveys topologies, called admissible group topologies, of the full group of self-homeomorphismsH(X) of a Tychonoff space X, which yield continuity of both the group operations and at the same time provide continuity of the evaluation function or, in other words, make the evaluation function a group action of H(X) on X. By means of a compact extension procedure, beyond local compactness and in two essentially different cases of rim-compactness, we show that the complete upper-semilattice


Introduction
The "incipit" of the homeomorphism group theory resides in the early seminal work of Birkoff [1].With an apparent simplicity joined with an impressive bright proof strategy, Birkoff positively answered to the query: there exists a topology on the full self-homeomorphism group H() of a compact metric space  which makes it into a topological group and a subspace of the Hilbert cube?The area, originating from [1], has initially evolved relaxing the compactness condition by passing from the class of compact metric spaces, as in Birkoff, to the class of  2 locally compact spaces, as in Arens [2].In [2] Arens focused on those topologies which yield continuity of both the group operations, product and inverse function, and also, at the same time, provide continuity of the evaluation function  : (, ) ∈ H() ×  → () ∈  and posed the problem of the existence for noncompact spaces  of the least element in the upper-semilattice (ordered by the usual inclusion) L  () of all topologies with these two features, that he called admissible group topologies.Of course, there are many different ways to topologize H().For instance, it can be endowed with the subspace topology induced by any of all known function space topologies.Nevertheless, following Birkoff and Arens, we also focused our investigation on topologies which make H() a topological group and the evaluation function a group action of H() on  and, rather obviously, looked at uniform topologies.In fact, uniform topologies make continuous the evaluation function.Furthermore, they make continuous both product and inverse function at (, ) and at , respectively, where  is the identity function of .Being well aware that if  is compact  2 , then the compact-open topology on H(), which is also the uniform topology derived from the unique totally bounded uniformity on , is an admissible group topology, we searched the admissible group topologies on H() by means of a compact extension procedure.Whenever  is Tychonoff, since any self-homeomorphism of  continuously extends to (), the Stone-Cech compactification of , then H() embeds as a subgroup in H().Analogously, whenever  is locally compact  2 , H() embeds as a subgroup in H( ∞ ), where  ∞ is the one-point compactification of .Thereby, the relativization to H() of the compact-open topology on H() and that on H( ∞ ) are both admissible group topologies.Accordingly, the previous significant examples strongly suggest investigating those uniform topologies on H() derived from totally bounded uniformities on  whose uniform completion is a  2 -compactification of  to which any self-homeomorphism of  continuously extends.We say that a  2 -compactification () of  has the lifting property if every self-homeomorphism of  continuously extends to ().Whenever () is a  2 -compactification of  with the lifting property, the homeomorphism group H() embeds as subgroup in H(()) equipped with the compact-open topology.Thus, the induced topology, that is, the topology of uniform convergence determined by the unique totally bounded uniformity associated with (), is an admissible group topology.Furthermore, the compact extension procedure appears as a powerful method to prove the existence of a least admissible group topology.The problem of the existence of a least element in L  () for non-compact space  goes back to Arens [2], who proved that, if  is locally compact  2 , then the -topology, which is generated by the collection of all sets of the type: where  is closed,  is open in  and  or  −  is compact, is the least admissible group topology.He also proved that, with the additional property of local connectedness for , the -topology agrees with the compact-open topology.In the direction of extending Arens' result beyond the class of locally compact spaces, it comes as very natural idea to weaken local compactness into rim-compactness, since, to a rim-compact  2 space  is attached the Freudenthal compactification () [3][4][5], to which any selfhomeomorphism continuously extends [6].A space  is rimcompact if and only if any of its points admits arbitrarily small neighborhoods with compact boundaries.The group topology   induced by () on H() has a simple description as the set-open topology admitting like subbasic open sets all sets [, ], as in (1), but where now  is a closed set with compact boundary in  and again  is open in .
However, rim-compactness by itself is not enough to assure the admissible group topology   determined by the Freudenthal compactification to be the least element in L  ().
As for the space of natural numbers N, for instance, the Freudenthal compactification (N) induces on H(N) the closed-open topology which differs from the compact-open topology which in the case is the -topology.Nevertheless, we performed the result in two substantially different cases of rim-compactness: the former one, where  is rim-compact,  2 , and locally connected, [7]; the latter one, in the first step, where  is the rational number space Q equipped with the Euclidean topology and, next, where  is a product of  2 zero-dimensional spaces each satisfying the property: any two nonempty clopen subspaces are homeomorphic, [8].In the former, whenever  is a locally connected, rim-compact  2 space, we construct in two steps a  2 -compactification of , (), in which ()− zero-dimensionally embeds and to which any self-homeomorphism of  continuously extends.
In the first step,  comes densely embedded into the disjoint union of the Freudenthal compactifications of its components, (), which is a locally compact  2 space to which any self-homeomorphism of  continuously extends.In the second step, in turn () comes embedded in its one-point compactification (), and, as a matter of fact, where c⋅c   → stands for continuous convergence, unfortunately is not topological [9,10].Therefore, in the beginning one has no clear indication and fluctuates between arguments promoting existence or nonexistence in L  (Q) of a least element.What Arens wrote seems to contain a subliminal message of nonexistence.On the contrary, checking in details his construction or completing in their minimal group topologies the uniform topologies induced by non-Archimedean metric compactifications of Q anytime one runs into the closed-open topology which is induced by the Stone-Cech compactification which in the rational case is also the Freudenthal compactification [11].Two arguments seem to promote the existence.On one side, the fine or Whitney topology on H(Q) determines an admissible group topology on H(Q) strictly finer than the closed-open topology: so, the closedopen topology is not too fine.On the other side, the Stone-Cech compactification is the only one  2 -compactification of Q with the lifting property: so, the closed-open topology seems enough fine.In conclusion, H(Q), even though it admits no least admissible topology [2], it still supports the clopen-open topology as the least admissible group topology.This issue is essentially achieved by the property: any two non-empty clopen subspaces of Q are homeomorphic, as it is derived from the topological characterization of Q.Therefore, following the rational trace, we focus just on the class of zero-dimensional spaces satisfying the property: any two non-empty clopen subspaces are homeomorphic and their products.All zero-dimensional spaces of diversity one [12] and all compact zero-dimensional spaces of diversity two [13] are of this kind.Among them we recognize as leaders the rationals, the irrationals, the Baire spaces, and the Cantor discontinuum.In all previous results the least element in L  () is achieved as a uniform topology that can be viewed also as a set-open topology.Accordingly, in the approach to the zero-dimensional case we explored the class of bases of clopen sets in  to select the ones that determine a clopenopen topology that is an admissible group topology induced by a  2 -compactification of  with the lifting property.The bases of clopen sets of  closed under complements and invariant under homeomorphisms of  emerge as the right tool: they make the match.We show that if  = ∏ ∈   is a product of zero-dimensional spaces each of which satisfies the property: any two non-empty clopen subspaces are homeomorphic, then L  () is a complete lattice.Besides, its least element is a clopen-open topology with the left, the right, and the two-sided uniformities all non-Archimedean, thus zerodimensional [8,11,14].
As rim-compactness is a weak and peripherical compactness property, one might think any further relaxation as impossible.But, we show that rim-compactness for  is not a necessary condition for the existence of the least admissible group topology on H().More precisely, we show that the full group of self-homeomorphisms of the product space R × Q, where R and Q are the sets of the real and rational numbers, respectively, both carrying the Euclidean topology, admits a least admissible group topology even though notoriously R×Q is not rim-compact, [15].To achieve this result we carry on another efficient way to produce admissible group topologies in substitution of, or in parallel with, the compact extension procedure.By exploring the literature on the different ways to control efficiently closeness between self-homeomorphisms of a Tychonoff space, we arrive at several different remarkable ideas: drawn by covers yielding the open-cover topology [16]; by uniformities yielding uniform topologies [16][17][18]; or, in the metric case, also by the compatible metrics yielding the limitation topology [1,19] and by the continuous functions to the positive real numbers yielding the fine or Whitney topology [20].Namely, as for the metric setting, three of the examined methods collapse in just one.As a matter of fact, in the metric setting there are only two substantially different options to control closeness in H().An effective control of closeness can be managed, in one way, via the metrics compatible with  and, in the other way, via the continuous functions from  to the positive real numbers.The idea of how to discriminate comes from the rationals.The clopen-open topology on H(Q) is the uniform topology induced by the Cech uniformity of Q, which in turn is the finest totally bounded uniformity compatible with Q.Consequently, being Q metrisable and separable, thus admitting compatible totally bounded metrics, the clopen-open topology on H(Q) can be reformulated as the supremum of all uniform topologies induced by totally bounded metrics compatible with Q.On the other hand, the fine or Whitney topology on H(Q) is a group topology [15].Hence, we demonstrate that it collapses on just the fine uniform topology [16], which in the case is the supremum of all uniform topologies deriving from metrics compatible with Q.These results pointout the suprema of uniform topologies deriving from metrics compatible with , running in a given class, as the right tool.The presentation in [19] of the fine uniform topology is a compelling motivation to generalise it in order to produce new admissible group topologies on H() and its subgroups.Given a class D() of metrics compatible with  and a group G() of self-homeomorphisms of , we refer to the uniform topology induced on G() by the supremum of the uniformities on  associated with the metrics in D() as the fine uniform topology on G() associated with, or generated by, D().Obviously, in this way the fine uniform topology is generated by the full homeomorphism group H() and by the class of all metrics compatible with .Blending in a group of self-homeomorphisms G() with a class D() of metrics compatible with  originates a new class of metrics compatible with , which reveals interesting and useful features.A class D() is invariant under the group G() if, whenever the distance between every two points of  is measured by a metric in D() applied to the pair of their images under a homeomorphic deformation of  belonging to G(), the new produced metric in this way belongs once again to D().We show that if D() is G()-invariant, then the fine uniform topology induced by D() on G() is a group topology.Justified by this result, we refer to the fine uniform topology on G() generated by the minimal G()-invariant enlargement of D() as the fine group topology on G() generated by D().A same group blended in with different classes of metrics gives rise to different fine group topologies.As for the rational case, for instance, the fine group topology generated on H(Q) by all totally bounded metrics compatible with Q and the fine group topology generated on H(Q) by all metrics compatible with Q are distinct from each other.Namely, the former one coincides with the clopen-open topology of H(Q) [7] and the latter one with the fine or Whitney topology on H(Q).And the clopen-open topology and the fine or Whitney topology on H(Q) do not agree, being the fine or Whitney topology strictly stronger than the clopen-open topology [7].Finally, we show that any admissible group topology on H(R × Q) is stronger than the fine group topology determined from the class of metrics on R × Q of the type  1 × 2 as  1 is the stereographic metric on R and  2 runs over all totally bounded metrics on Q [15].
The issues so far discussed lead us to show: a uniform topology on H() derived from a totally bounded uniformity on  is a group topology (hence an admissible group topology) if and only if it is derived from a totally bounded uniformity of  associated with a  2 -compactification of  with the lifting property [21].
On the other hand, if  is locally compact  2 , then the compact-open topology on H(), which is also the topology of uniform convergence on compacta derived from any uniformity on , is admissible and yields continuity of the product function.Unfortunately in general, the compact-open topology does not provide continuity of the inverse function.But, with the following additional property: ( * ) any point of  has a compact connected neighborhood, due to Dijkstra [22], the compact-open topology becomes a group topology and, as a consequence, the least admissible group topology of H().According to this issue the compact-open topology on H() is quoted as the most eligible one if  is a manifold of finite dimension or  is an infinite dimensional manifold modelled on the Hilbert cube [23].In looking for topologies of uniform convergence on members of a given family, containing all compact sets, which are admissible group topologies, we focus beyond local compactness.In order to do so, we follow as suggestive example that of bounded sets of an infinite dimensional normed vector space carrying as proximity the metric proximity associated with the norm.We emphasise first that local compactness of  is equivalent to the family of compact sets of  being a boundedness of  [24], which, jointly with any EF-proximity of , gives a local proximity space [25].As a consequence, we make this particular case fall within the more general one in which compact sets are substituted with bounded sets in a local proximity space, while the property ( * ) is replaced by the following one: ( * * ) for each nonempty bounded set  there exist a finite number of connected bounded sets  1 , . . .,   such that ≪  int( 1 ) ∪ ⋅ ⋅ ⋅ ∪ int(  ).So doing, we achieve the following issue: if (, B, ) is a local proximity space with the property ( * * ) and any homeomorphism of  preserves both boundedness and proximity, then the topology of uniform convergence on bounded sets derived from the unique totally bounded uniformity associated with  is an admissible group topology on H().
The uniform topologies so far considered are totally bounded, and the concept of totally bounded uniformity can be dually recast as EF-proximity and then as strong inclusion, [26].As a consequence, it is worthwhile to reformulate uniform topologies derived from totally bounded uniformities as proximal set-open topologies.Taking up the common proximity nature of set-open topologies as the compact-open topology, the bounded-open topology and the topology of convergence in proximity, Naimpally, jointly with the author, introduced as unifying tool the notion of proximal set-open topology, simply replacing the usual inclusion with a strong one [27].The proximal set-open topology relative to a network  and an EF-proximity , designed by the acronym PSOT , or, simply, PSOT  , when  is the set CL() of all non empty closed subsets of , is that having as subbasic open sets the ones of the following form: where  [21].
Again in local compactness, in the paper [28], unpublished as per my knowledge, Wicks gave necessary and sufficient conditions for the compact-open topology being a group topology by using nonstandard methods on one side and action on hyperspace on the other side, which is so inspiring [29].But, under local compactness is Dijkstra's property a necessary condition for the compact-open topology being a group topology?And is local compactness a necessary condition for the compact-open topology being a group topology that more makes the evaluation map jointly continuous?In both cases we give a negative answer by using as counterexample first a model of locally compact topologist's comb, a typical space that is not locally connected, and then a nonlocally compact one.Wicks proved that H() equipped with the compact-open topology  c.o being a topological group is equivalent to joint continuity of the evaluation map  : (, ) ∈ H() × CL → () ∈ CL with respect to  c.o and the Fell hypertopology   .Since for the compact-open topology three different formulations as set-open topology, as the topology of uniform convergence on compacta, and also as proximal set-open topology can be displayed, three possible generalizations in topology, proximity, and uniformity arise from those.After analyzing the compact case, we improve and contemporaneously generalize the compact case in the topological, uniform, and proximal frameworks by replacing the compact-open topology with a set-open topology based on a Urysohn family, with a topology of uniform convergence on a uniformly Urysohn family, with a proximal set-open topology relative to a proximity and a boundedness giving a local proximity space, respectively.Finally, we show that the topologicality of H() is equivalent to topologicality of the evaluation map  : (, ) ∈ H() ×  → () ∈ , as in the Wicks case, in each generalized case.We limit only to cite this final result since the paper containing it and others has to be published [29].

Background and Works
Firstly, we give some useful background and summarise a number of already stated basic facts.Definitions and terminology quoted below are drawn by [26,[30][31][32][33].
(i) Any admissible topology on H() induces a convergence that implies continuous convergence [34].
Topologies on H() compatible with the group operations are called group topologies.
(i) Let (, U) be a Weil uniform space.Then the topology of uniform convergence induced by U on H() provides continuity of the product at (, ) and of the inverse function at , where  is the identity function of .
(ii) Let  be a compact  2 space.Then the compact-open topology on H() is an admissible group topology on H().Furthermore, it is exactly the topology of continuous convergence [1,2].
(iii) Let (, ) be a compact metric space and d the supremum metric determined from  on H() by the usual formula Then the metric  * defined by the formula induces, as d does, the compact-open topology on H() and metrizes the two-sided uniformity so making H() into a Polish space [1].
(iv) Of course, every admissible group topology makes the evaluation function as a group action.
(v) There is always on H() a minimal convergence structure which provides continuity of the evaluation function and both the group operations.It is assigned by the formula The natural notation for it is as -convergence.The -convergence is not topological in general [9,10].
(vi) Of course, every admissible group topology on H() induces a convergence which implies the -convergence.
Let L  () stand for the set of all admissible group topologies on H() ordered by the usual inclusion.Since any topology finer than an admissible one is in its turn admissible and the join of subsets of group topologies is again a group topology, L  () is a complete upper semilattice.Obviously, the discrete topology is in L  () and is, indeed, the maximum.The existence in L  () of the minimum is equivalent to L  () being a complete lattice.The problem of the existence of a least element in L  () for noncompact space  goes back to Arens [2], who proved that: (i) if  is locally compact  2 , then the -topology, which is generated by the collection of all sets: where  is closed,  is open in  and  or − is compact, is the least admissible group topology.He also proved that, with the additional property of local connectedness for , the -topology agrees with the compact-open topology.Secondly, we differentiate the topologies on H() according to their derivation from the following: uniformities yielding uniform topologies, covers yielding the open-cover topology, the compatible metrics yielding the limitation topology, and the continuous functions to the positive reals yielding the fine or Whitney topology.

How Uniformities on 𝑋 Yield a Uniform Control on H(𝑋).
Let  stand for a Tychonoff space.Every Weil uniformity U compatible with  induces on H() the uniformity of the uniform convergence with respect to U, which admits as basic diagonal neighborhoods the sets as  runs over all diagonal neighborhoods in U.
with A being an open cover of  [16].

Closeness by Real Functions in the Metric Case:
The Fine or Whitney Topology.Let (, ) stand for a metric space.At any  ∈ H() the fine or Whitney topology on H(), that we will denote by   , admits as arbitrarily small neighborhoods the following sets, also called tubes: being a continuous function from  to the positive real numbers.
It is known that, having been given a topological characterisation, the fine topology   is independent of the metric  [20].

Closeness by Metrics:
The Limitation Topology.Let (, ) stand for a metric space again.At any  ∈ H() the limitation topology on H() admits as arbitrarily small open neighborhoods sets as the following: as  runs over all metrics compatible with  [1,19].
In [19] it has been proven that the limitation topology on H() is an admissible group topology.

Comparison.
In the metric setting three of the examined methods collapse in just one because of the two following circumstances.The former one is why the open-cover topology and the limitation topology agree: any open cover in a metric space  can be refined by the cover of balls of radius 1, {  (, 1) :  ∈ }, relative to a suitable metric  compatible with .The latter one is why the fine uniform topology and the limitation topology agree: the fine uniformity of a metric space  is the supremum of all metrisable uniformities compatible with .Accordingly, as for the metric setting, closeness in H() can be substantially controlled in two ways: via the metrics compatible with  or via the continuous functions from  to the positive real numbers.Usually, the fine or Whitney topology   is finer than the fine uniform topology   .Theorem 1.If   is a group topology, then   =   , [15].

Compact Extension Procedure
Implicitly due to Birkhoff, a natural way to get admissible group topologies works efficiently.Whenever  is Tychonoff, since any self-homeomorphism of  continuously extends to (), the Stone-Cech compactification of , then H() embeds as a subgroup in H().Analogously, whenever  is locally compact  2 , H() embeds as a subgroup in H( ∞ ), where  ∞ is the one-point compactification of .Thereby, the relativization to H() of the compact-open topology on H() and that on H( ∞ ) are both admissible group topologies.Accordingly, the previous significant examples strongly suggest investigating those uniform topologies on H() derived from totally bounded uniformities on  whose uniform completion is a  2 -compactification of  to which any self-homeomorphism of  continuously extends.We say that a  2 -compactification () of  has the lifting property if every self-homeomorphism of  continuously extends to ().Remember that whenever  is a Tychonoff, locally compact  2 , and rim-compact  2 space, any self-homeomorphism extends to a self-homeomorphism of its Stone-Cech compactification , its one-point compactification  ∞ , its Freudenthal compactification (), respectively.In other words ,  ∞ , and (), when they make sense, are all compactifications of  with the lifting property.
Theorem 2. Let () be a  2 -compactification of  with the lifting property.Then the relativization  () to H() of the compact-open topology on H(()) is an admissible group topology on H(), [7].
Starting with a totally bounded uniformity we construct a  2 -compactification with the lifting property as follows.
Let U be a collection of subsets of  × .For any  ∈ U and any ℎ ∈ H() put Furthermore, set Theorem 3. Let U be a uniformity on .Then the following hold: (a) The family S H is a subbase for a uniformity U H on , which is separated whenever U is so.

(b)
The uniformity U H is totally bounded whenever U is so.
(c) Any self-homeomorphism of  is a uniformly continuous function with respect to U H or equivalently U H has the lifting property.
(d) The uniformity U H is the least uniformity with the lifting property finer than U.
For every uniformity U the property (d) motivates us to refer to U H as the minimal H()-enlargement of U. Minimal H()-enlargements have interesting properties.Proposition 4. Let U be a totally bounded uniformity on .Then the uniform topology  U H on H() derived from U H is a group topology; hence it is an admissible group topology.
In the case U is totally bounded the previous result induces us to refer to the uniform topology  U H as the fine group topology associated with U.

Proposition 5. Let U be a totally bounded uniformity on 𝑋.
Then the uniform topology on H(),  U , derived from U is a group topology if and only if it agrees with the uniform topology  U H derived from U H .
The previous result can be summarised as follows.Theorem 6.A uniform topology on H() derived from a totally bounded uniformity on  is a group topology (hence an admissible group topology) if and only if it is derived from a totally bounded uniformity of  associated with a  2compactification of  with the lifting property, [21,29].

Completeness of L 𝐻 (𝑋) in Rim-Compactness
In the direction of extending Arens' result beyond the class of locally compact spaces, it comes as very natural idea to weaken local compactness into rim-compactness, since to a rim-compact  2 space  is attached the Freudenthal compactification () [3][4][5], to which any self-homeomorphism continuously extends [6].So, we focus our attention on rimcompact  2 spaces and in particular on their Freudenthal compactification.The Freudenthal compactification in rimcompactness plays a key role as the one-point compactification does in local compactness.A space  is rimcompact, peripherically compact, or semicompact if any point has arbitrarily small neighborhoods whose boundaries are compact.For example, removing from a compact metric space a totally disconnected   -set is a way to produce rimcompact  2 spaces [35].Of course, 0-dimensional spaces are rim-compact.We briefly summarize the characters of the Freudenthal compactification which we will refer to where  runs in the family of all closed sets whose boundaries are compact and  runs in the topology of .
Unfortunately, we have no hope for minimality of   without adding some more condition.In fact, there are rimcompact  2 spaces whose Freudenthal compactification does not determine a least admissible group topology, as for the space N of natural numbers, for instance.Since N is locally compact and locally connected, H(N) admits a least group topology which is just the compact-open topology [2] Then  is in H(N) and in If  > ,  is even, and  =  + ℎ, then ℎ has to be odd and () =  + 1 is odd.
For that, we focus our attention on the class of rimcompact  2 spaces whose Freudenthal compactification is locally connected at any ideal point.Naturally there exist rim-compact but not locally compact  2 spaces having their Freudenthal compactification locally connected at any ideal point.We can give as an example the subspace  obtained from  × , the unit square in the plane, by removing from it the points whose coordinates are both irrational.The space  is rim-compact  2 but not locally compact.Moreover, its Freudenthal compactification is just  × .
Trying to capture minimality in local connectedness we get a previous basic result.Let  be a Tychonoff space and () a  2 -compactification of .The space  is locally connected in () provided that any point in () −  admits arbitrarily small open neighborhoods  such that  ∩  is connected [36].Whenever a space  is connected, locally connected, locally compact, and second-countable  2 , then  is locally connected in () (Freudenthal's original construction).Naturally if  is locally connected in (), then () is locally connected at any ideal point.Theorem 9.If  is a locally compact  2 space, then () is locally connected at any ideal point if and only if  is locally connected in it.
A result about local compactness involving as particular case the real line and more generally connected non compact Lie groups is the following.Theorem 10.Let  be a rim-compact  2 and locally connected space.If () is an n-point compactification, then () is locally connected at any ideal point.

Theorem 11. Whenever 𝑋 is a rim-compact 𝑇 2 space and its Freudenthal compactification 𝐹(𝑋) is locally connected at any ideal point, then the group topology 𝜏 𝐹 induced on H(𝑋) from 𝐹(𝑋) is the least in the upper-semilattice L H (𝑋) of all admissible group topologies on H(𝑋).
A relationship with local compactness resides in the following.

Corollary 12. If 𝑋 is a locally connected space and its Freudenthal compactification has only a finite number of ideal points, then the group topology induced by the one-point compactification and the Freudenthal compactification agree.
In a more general context in which unfortunately the group topologies do not have a simple description and a convergence strategy, even though rather technical, has to be managed we have the following.

Theorem 13. If 𝑋 is rim-compact 𝑇 2 and admits a 𝑇 2 -compactification 𝛾(𝑋) with the lifting property, locally connected at any ideal point, in which 𝛾(𝑋) − 𝑋 zero-dimensionally embeds, then the group topology 𝜏 𝛾(𝑋) induced by 𝛾(𝑋) on H(𝑋) is the least of all admissible group topologies on H(𝑋).
By essentially using the previous basic result, then we construct in two steps a  2 -compactification of , (), in which () −  zero-dimensionally embeds and to which any self-homeomorphism of  continuously extends.At the first step,  comes densely embedded in the disjoint union of the Freudenthal compactifications of its components, (), which is a locally compact  2 space to which any selfhomeomorphism of  continuously extends.At the second step, in turn () becomes embedded in its one-point compactification (), and, as a matter of fact,  () is the least element of L  ().Theorem 14. Suppose  is a rim-compact  2 and locally connected space.Then: Whenever  is finite union of disjoint connected subspaces, as in particular if  is connected, the compactification () agrees with the Freudenthal compactification, but it is generally different as in the natural case.Under local compactness the previous construction works but is evidently redundant.
Example 15.Let   ,  ∈  + , be obtained from the rectangle [0, 1] × [  , 1/], where 1/( + 1) <   < 1/ after removing inside points whose coordinates are both rational.Put  = ∪{  :  ∈  + }.Add to  the segment  = {(, 0) : 0 ≤  ≤ 1}.Consider  = ∪ as subspace in the Euclidean plane.The space  is a rim-compact  2 space not locally compact, not connected, and not locally connected.Its Freudenthal compactification () agrees with the closure of the space ; then it is metrisable and locally connected at any ideal point.So H() admits a least admissible group topology which is induced by the supmetric deriving from the Euclidean metric on .

The Rational Case
The rational case apparently is singular.First, since any two nonempty open subspaces in Q are homeomorphic, H(Q) is a very big object.Anyway, H(Q), even though it admits no least admissible topology [2], still supports the clopen-open topology as the least admissible group topology.
Theorem 16.Let (Q) be an arbitrary  2 -compactification of Q but distinct from (Q).Then there always exists a self-homeomorphism of Q which does not continuously extend to (Q).
Remember that Q is strongly zero-dimensional, hence rim-compact.Its Stone-Cech compactification is zerodimensional and perfect, so is its Freudenthal compactification [37].The relativization to H(Q) of the compact-open topology on H(Q) is the closed-open topology.
Of course, the main issue in the rational case is the following one.[7].

Theorem 17. Any admissible group topology 𝜏 on H(Q) is finer than the closed-open topology
We now investigate whether the fine, strong, or Whitney topology on H(Q) [20] induces naturally an admissible group topology on H(Q) strictly finer than the closedopen topology.To make easier the relationship between the Whitney topology and the group operations, preliminarly we need to acquire the following two Lemmas.
Let (Q, R + ) denote the set of all real-valued positive continuous functions on Q. Lemma 18.Let  ∈ (Q, R + ).Then there exists a locally constant function  ∈ (Q, R + ) such that  < .
Let  ∈ (Q, R + ), and let  be the Euclidean metric on Q. Denote It is well known that { 1 () :  ∈ (Q, R + )} is a base for a uniformity U 1 on H(Q) which induces the fine, strong, or Whitney topology which is independent of the metric  since Q is paracompact [20].The fine or Whitney topology admits as typical basic neighborhoods for any where  ∈ (Q, R + ).

Theorem 20. The Whitney topology on H(Q) provides continuity of the usual product
It is easily verified that { 2 () :  ∈ (Q, R + )} is a base for a uniformity U 2 on H(Q), which induces a topology providing, in analogy with the previous result, continuity of the usual product.Jointly U 1 , U 2 generate a new uniformity U on H(Q) having as basic diagonal neighborhoods () =  1 () ∩  2 (),  ∈ (Q, R + ).
The uniformity U induces a topology on H(Q) whose typical basic neighborhoods for any  are where  ∈ (Q, R + ).We, justified from the following result, call it the fine group topology on H(Q).

Group Action on 0-Dimensional Spaces and Completeness
The full homeomorphism group H(Q) of the rational numbers space Q equipped with the Euclidean topology admits as least admissible group topology the closed-open topology induced by the Stone-Cech compactification of Q, which, in the case, agrees with the Freudenthal compactification of Q.In trying to extend a similar result to a larger class of zero-dimensional spaces we briefly review properties of some of their  2 -compactifications and in particular of their Freudenthal compactifications.A Tychonoff space  is zerodimensional if it admits a base of clopen sets.A clopen set in  is a subset of  that is at the same time closed and open.A zero-dimensional space is rim-compact.
In the rational case, the proof strategy is based on the property (⋆) any two non-empty clopen subspaces are homeomorphic.So we focus our attention on the class of spaces with this property and their products.This class includes all zero-dimensional spaces of diversity one (or divine) and all compact zero-dimensional spaces of diversity two (or semidivine), as introduced and investigated by Rajagopalan and others [12,13].An infinite Tychonoff space  is of diversity one if any two non-empty open subspaces are homeomorphic and is of diversity two if there exist two classes of homeomorphism for the open non-empty subspaces of .The rationals, the irrationals, and the Baire spaces are of diversity one by their topological characterizations.The Cantor discontinuum is of diversity two.In a compact space of diversity two any two non-empty clopen subspaces are homeomorphic.No space of diversity one can be compact or locally compact, connected or locally connected.Diversity one or two is not preserved under products.Every space of diversity one is rich of homeomorphisms that move any point, since it can be expressed as countable disjoint union of homeomorphic copies of itself.For further details see [12,13].Theorem 24.If  is a zero-dimensional, nonlocally compact space that satisfies the property (⋆), then its Freudenthal compactification () is the unique  2 -compactification of  with the lifting property and zero-dimensional growth.
Recall that a Tychonoff space  is strongly zero-dimensional if any two non-empty disjoint zero sets can be separated by the empty set.

Theorem 25.
If  is a strongly zero-dimensional, non-locally compact space that satisfies the property (⋆), then its Stone-Cech compactification () is the unique perfect  2 -compactification of  and also the unique  2 -compactification of  with the lifting property [8].
Supposing  is a zero-dimensional space, we call nice any base of clopen sets in  that is closed under complements and invariant under homeomorphisms of .Any base B of clopen sets embeds in the nice base {ℎ() :  ∈ B or  −  ∈ B, ℎ ∈ H()}, that is also the minimal nice base containing B. If B is a base of clopen sets, the minimal nice base containing B is referred to as the nice closure of B.
Recall that a Weil uniformity is non-Archimedean when it admits a base of diagonal neighborhoods that are equivalence relations in .For further details see [11].
(ii) The left, the right, and the two-sided uniformities associated with  B are all non-Archimedean.
(iii)  B is the topology of uniform convergence induced by a  2 -compactification of  with the lifting property [8].
Corollary 27.Let  be a zero-dimensional space and B a nice base of .Then the set-open topology  B determined from B is zero-dimensional.
Let {  :  ∈ } be a family of zero-dimensional spaces in each of which any two non-empty clopen subspaces are homeomorphic.Let  = ∏ ∈   be equipped with the product topology.We call standard nice base for  the nice closure of the standard clopen base generated by the subbasic clopen sets of the type   × ∏  ̸ =    , where   runs over all clopen sets in   and  in .We refer to the clopen-open topology generated by the standard base as the standard clopen-open topology.
Theorem 28.Let {  :  ∈ } be a family of zero-dimensional spaces in each of which any two non-empty clopen subspaces are homeomorphic.Let  = ∏ ∈   be equipped with the product topology.Then L  () is a complete lattice.The standard clopen-open topology is the minimum of L  () [8].
We conclude with the following.

Fine Group Topologies
Now, we carry on another efficient way to produce admissible group topologies in substitution of, or in parallel with, the compact extension procedure.Let  stand for a metrisable space.The presentation in [19] of the fine uniform topology is a compelling motivation to generalise it in order to produce admissible group topologies on H() and its subgroups.But first, we introduce a way to produce new metrics from old ones [15].We suitably combine a selfhomeomorphism ℎ of  with a metric  compatible with  and so generate a new metric  ℎ , which is once again compatible with .Namely, if the space  is subject to a homeomorphic deformation ℎ and we measure the distance between two points in  as the -distance of their ℎimages, we construct a new metric  ℎ defined more precisely by the following formula: compatible with  and further totally bounded when so is .
Let D() be a class of metrics compatible with  and G() a subgroup of H().We will refer to the uniform topology induced on G() by the supremum of the uniformities on  associated with the metrics in D() as the fine uniform topology on G() associated with (or generated by) D(), and we will denote it by  D,G .Of course, the fine uniform topology   is then generated by the full homeomorphism group H() and by the class of all metrics compatible with .
Whenever D() is closed under the scalar multiplication, it is easy to show that at any  ∈ G() the topology  D,G admits as subbasic open neighborhoods the sets of the kind  (, ) as  runs over D().Blending in a group of self-homeomorphisms G() with a class D() of metrics compatible with  gives rise to a new class of metrics compatible with , which reveals useful features.We say that a class D() is invariant under the group G() or G()-invariant if, whenever we submit the space  to any homeomorphic deformation ℎ in G() and we measure the distance between two points of  as the -distance of the pair of their ℎ-images, where  is a metric in D(), the new produced metric  ℎ , defined in (÷), belongs once again to D().
If D() is G()-invariant, then the fine uniform topology  D,G is a group topology on G().
Every class of metrics D() admits as G()-invariant enlargement the wider class { ℎ :  ∈ D(), ℎ ∈ G()}, which is also the minimal G()-invariant enlargement of D().The previous result enables us to define the fine uniform topology on G() generated by the minimal G()invariant enlargement of D() as the fine group topology on G() generated by D().

A Same Group Blended in with Different Classes of
Metrics Gives Rise to Different Fine Group Topologies.If  is metrisable and separable, thus admitting totally bounded compatible metrics, then the fine group topology generated on H() by all totally bounded metrics compatible with  is, in general, distinct from the fine uniform topology generated on H() by all metrics compatible with .The rational numbers provide the right counterexample that follows.The fine group topology generated on H(Q) by all totally bounded metrics compatible with Q and the fine group topology generated on H(Q) by all metrics compatible with Q are distinct from each other.Namely, the former one coincides with the clopen-open topology of H(Q) [7].The latter one has to coincide with the fine or Whitney topology on H(Q), since this one, in the case, is a group topology.And, as proven in [7], the clopen-open topology and the fine or Whitney topology on H(Q) do not agree, being the fine or Whitney topology strictly stronger than the clopen-open topology.

The Space R × Q
As rim-compactness is a weak and peripherical compactness property, one might think of any further relaxation as impossible.But, we show that rim-compactness for  is not a necessary condition for the existence of the least admissible group topology on H().More precisely, we show that the full group of self-homeomorphisms of the product space R × Q, where R and Q are the sets of the real and rational numbers, respectively, both carrying the Euclidean topology, admits a least admissible group topology even though notoriously R × Q is not rim-compact, [15].
Since, if  is closed and  is open in Q and  ⊆ , there exists a clopen set  such that  ⊆  ⊆ , then the sets like the following: Let us turn now our attention to R×Q.Since the boundary of any non-empty bounded open subset of R × Q is not compact, the product R × Q is not rim-compact when both R and Q carry the Euclidean metric.The study of a complex object as H(R × Q) is certainly simplified by splitting any self-homeomorphism  of R × Q into its two natural halves  1 ∘ ,  2 ∘ , where  1 ,  2 are the usual projections of R × Q over R and Q, respectively.The study of the two halves, separately, allows us to acquire their own features and their interplay.
Let us focus on the second half  2 ∘ .The following two facts are to be considered.The components of R × Q are the subsets of the type R × {}, as  runs over Q.Furthermore, every homeomorphism takes components to components.Consequently, for any given  in Q, the following occurs: This means that  2 ∘ is independent of the point  in R.This feature of  2 ∘  makes coherent its substitution with the map from Q to itself whatever is the point  in R. Accordingly, it seems natural to identify the self-homeomorphism  with the pair ( 1 ,  2 ), where Hence, if the inverse homeomorphism  −1 of  identifies with ( 1 ,  2 ), then This implies  2 =  −1 2 .Thus,  2 is in turn a homeomorphism of Q to itself whenever  is a homeomorphism of R × Q to itself.
The identification leads to a natural embedding of where (R × Q) is the set of all continuous functions from R × Q to the reals.
We now recall the notion of product metric on a product space.Let ( 1 ,  1 ), ( 2 ,  2 ) stand for two metric spaces.Then, their product  1 ×  2 can be metrised by the product metric  1 ×  2 , which is defined by If we suppose H(R × Q) embedded via the canonical identification, as described above, in (R × Q) × H(Q) and denote by  1 the stereographic metric on R, which measures the distance between two points in R as the geodesic distance of their images in the unit circle  1 of the Euclidean plane by the inverse of the stereographic projection, then the following holds true.

Locally Compact Extension Procedure
In looking for topologies of uniform convergence on members of a given family, containing all compact sets, which are admissible group topologies, we focus beyond local compactness.In order to do so, we follow as suggestive example that of bounded sets of an infinite dimensional normed vector space carrying as proximity the metric proximity associated with the norm.We emphasise first that local compactness of  is equivalent to the family of compact sets of  being a boundedness of  [24], which, jointly with any EF-proximity of , gives a local proximity space [25].As a consequence, we make this particular case fall within the more general one in which compact sets are substituted with bounded sets in a local proximity space, while the property ( * ) any point has a compact connected neighborhood is replaced by the following one: ( * * ) for each nonempty bounded set  there exist a finite number of connected bounded sets  1 , . . .,   such that  ≪  int( 1 ) ∪ ⋅ ⋅ ⋅ ∪ int(  ).9.1.Uniformity, Proximity, and  2 -Compactifications.Uniformities, proximities, and  2 -compactifications have an intensive reciprocal interaction.EF-proximity and totally bounded uniformity are dual concepts.Any uniformity U on  naturally determines an EF-proximity on  by setting for ,  ⊆ ,  ¡  U  if and only if there exists a diagonal neighborhood  ∈ U such that [] ∩  ̸ = 0.The class of all uniformities on  determining the same EF-proximity  on  contains a unique totally bounded uniformity, which is also the least element in the class.In the opposite, by the Smirnov compactification theorem [26], any EF-proximity  on  determines, up to homeomorphism, a  2 -compactification () of , whose unique compatible uniformity in turn induces on  a totally bounded uniformity U * , whose naturally associated proximity is just the starting .Both proximity and uniformity give rise to exhaustive procedures to generate all  2 -compactifications of a Tychonoff space.
Let (, ) be an EF-proximity space,   the natural underlying topology, U * the unique totally bounded uniformity compatible with , and () the uniform completion of (, U * ).Given that () is obviously the Smirnov compactification of (, ) up to homeomorphism, the following is easily acquired.It is remarkable that, for each positive integer , any metric uniformity compatible with the space R  , equipped with the Euclidean topology, for which any homeomorphism is uniformly continuous, or, which is equivalent, with the lifting property, is totally bounded [38].9.2.Strong Inclusion.The concept of EF-proximity can be recasted as strong inclusion, double containment, or nontangential inclusion.For any given EF-proximity  on a space  the relative dual strong inclusion is the binary relation over the power set Exp() of  defined as follows: Conversely, for any given binary relation over Exp(), ≪, which is a strong inclusion, the relative dual EF-proximity  is the binary relation over Exp() defined by The relations  and ≪  are interchangeable.Furthermore, later on we essentially use the following betweenness property.Let  be an EF-proximity.If ≪  , then there exists a   -closed set  such that ≪  int() ⊆ ≪  .

Proximal Set-Open Topologies on H(𝑋).
Let U be a uniformity compatible with  and let  stand for a family of nonempty subsets of .The topology of uniform convergence on members of  derived from U, which we denote by  , U , is that admitting as subbasic open sets at any  ∈ H() the following ones: (, , ) := {ℎ ∈ H () : ( () , ℎ ()) ∈ , ∀ ∈ } , where  runs through  and  varies in U.
Since the uniform topologies so far considered are relative to totally bounded uniformities, it is worthwhile to reformulate them as proximal set-open topologies.To unify the concepts of compact-open topology, bounded-open topology, and topology of proximity convergence [18], Naimpally, jointly with the author, introduced the unifying tool of proximal set-open topology relative to a network and a proximity [27].This recasting takes up the opportunity of reformulating topologies of uniform convergence on members of a network, when the range space carries a proximity.A collection  of subsets of a topological space  is said to be a network on  provided that for any point  in  and any open subset  of  containing  there is a member  in  such that  ∈  ⊆ .A network  is a closed network if any element in  is closed and is a hereditarily closed network if any closed subset of any element in  is again in .
Let (, ) be an EF-proximity space and  a network in , then the proximal set-open topology relative to  and , in short denoted by the acronym PSOT , or, simply, PSOT  when  is the network CL() of all non empty closed subsets of , is that admitting as subbasic open sets the following ones: [, ]  := { ∈ H () :  () ≪  } , (29) where  runs through  and  is open in .When  is the family of all compact subsets of , for any proximity we get the compact-open topology, which is the prototype within the class of set-open topologies.The proximal set-open topologies have remarkable properties [27].
Theorem 33.Let  be a closed, hereditarily closed network in  and  an EF-proximity on .Then  , is the topology of uniform convergence on members of  derived from the unique totally bounded uniformity compatible with .9.4.Boundedness plus Proximity.Blending proximity with boundedness gives local proximity.Local proximities play the same role in the construction of  2 local compactifications of a Tychonoff space  as that of EF-proximities in the construction of  2 -compactifications of .
Let  be a Tychonoff space.Any given  2 local compactification () of  takes up two features of .Whereas the former one is the separated EF-proximity on  induced by the one-point compactification of (), the latter one is the boundedness made by all subsets of  whose closures in () are compact.By joining proximity and boundedness in the unique concept of local proximity, Leader put this example in abstract [25].
A non empty collection B of subsets of a set  is called a boundedness in  if and only if The elements of B are called bounded sets.It is to be underlined that in [24] Hu proposed the notion of space with a boundedness as a natural generalisation of that of metric space.
We expressly remark that we look at a local proximity as localisation of an EF-proximity modulo of a free regular filter [25].A local proximity space (, B, ) consists of a set , together with an EF-proximity  on  and a boundedness B in  containing all singletons, which satisfies the following axiom: if  ∈ B,  ⊆ , and  ≪ , then there exists some  ∈ B such that  ≪  ≪ , where ≪ is the strong inclusion of .
It is remarkable that the boundedness in a local proximity space (, B, ) is also a uniformly Urysohn family w.r.t. the unique totally bounded uniformity naturally associated with  [30].In a local proximity space the closure of a bounded set is again bounded.Every compact subset of a local proximity space is bounded.Every local proximity space is also locally bounded.As a matter of fact, proximity spaces are just those ones where the underlying set  is bounded.Besides, the following holds true [25].
Theorem 34.For a Tychonoff space  there exists a bijection between the set of all, up to equivalence,  2 locally compact dense extensions of  and the set of all separated local proximities on  [27].If  is bounded, the  2 local compactification associated with (, B, ) is just the Smirnov compactification relative to , while, if  is unbounded, it can be obtained by removing from the Smirnov compactification relative to  the point determined in that by the free regular filter F = { \  :  ∈ B}. 9.5.Proximity and Homeomorphism Groups.Let (, ) be an EF-proximity space.It is easy to show the following.Proposition 35.Let G() be a subgroup of the full group H() of self-homeomorphisms of the underlying topological space .Assuming that G() is equipped with   , then the evaluation function  : (, ) ∈ G() ×  → () ∈  is continuous.
Furthermore, given that a proximity-isomorphism or isomorphism is a self-homeomorphism of  that preserves proximity in both ways, then the following holds.Proposition 36.If (, ) is an EF-proximity space, then   is a group topology on the full group of -isomorphisms of .
We summarise the previous two results as follows.
Theorem 37.If (, ) is an EF-proximity space, then the full group of -isomorphisms of , equipped with   , is a topological group which continuously acts on  by the evaluation function .Proposition 38.Whenever  is a  2 locally compact space, the PSOT associated with the Alexandroff proximity, known as the -topology, is the least admissible group topology on H().
Proposition 39.Whenever  is a  2 , rim-compact, and locally connected space, the PSOT associated with the Freudenthal proximity is the least admissible group topology on H().
Proposition 40.Whenever  is the rational numbers space Q, equipped with the Euclidean topology, the PSOT associated with the Cech proximity is the least admissible group topology on H(Q).Now, assume that a  2 local compactification has the lifting property if and only if any homeomorphism preserves both boundedness and proximity; that is, any homeomorphic image of a bounded set is bounded, and if ≪  , then ()≪  (), where  runs through H(),  is bounded, and  is open.
It is to be recalled that a local proximity space (, B, ) verifies the property ( * * ) if and only if for each non empty bounded set  there exist a finite number of connected bounded sets  1 , . . .,   such that ≪  int( 1 ) ∪ ⋅ ⋅ ⋅ ∪ int(  ).
Whenever (, B, ) is a local proximity space, then the subcollection of B of all closed bounded subsets of  is a closed, hereditarily closed network of .Accordingly, PSOT B, is the topology of uniform convergence on members of B derived from the unique totally bounded uniformity associated with .Unfortunately, PSOT B, is not in general an admissible group topology nor a group topology.
Nevertheless, what stated above is sufficient to draw the following final issue.

Theorem 22 .Theorem 23 .
If  is a zero-dimensional space, then the topology on H(),  () , induced by the Freudenthal compactification () is the clopen-open topology.If  has the property (⋆), then so does ().

Theorem 26 .
Let  be a zero-dimensional space, B a nice base of , and  B the set-open topology determined by B. Then the following holds:

Theorem 29 .Corollary 30 .
If  is a zero-dimensional space in which any two non-empty clopen subspaces are homeomorphic, then L  () is a complete lattice.The minimum is the clopen-open topology that is induced by the Freudenthal compactification.If  is a zero-dimensional metrisable space of diversity one, then L  () is a complete lattice.The minimum of L  () is the closed-open topology that is induced by the Stone-Cech compactification.

[
, ] := { ∈  (Q) :  () ⊆ }(21) as  runs over all clopen sets in Q, give arbitrarily small neighborhoods at the identity function of Q.This entails the coincidence of the closed-open topology with the clopenopen topology on H(Q).At the same time, the clopen-open topology on H(Q) is the uniform topology induced by the Cech uniformity of Q, which is the finest totally bounded uniformity compatible with Q.Consequently, the clopen-open topology on H(Q) can be reformulated as the supremum of all uniform topologies induced on H(Q) by totally bounded uniformities compatible with Q.Then, being Q metrisable and separable, the same is the supremum of all uniform topologies induced by totally bounded metrics compatible with Q.
(), that can be described as the set-open topology determined by all closed sets with compact boundaries contained in some component of .The latter, the rational one, is very singular indeed.First, since any two nonempty open subspaces in Q are homeomorphic, L  (Q) is a very big object.Next, Arens proved "given an admissible topology for the group of homeomorphisms H of the rational number system, one can construct another admissible topology for H which is not weaker (but now not stronger) than the first." And more, the minimal convergence structure on H(Q) which provides continuity of the evaluation function and both the group operations, denoted by -convergence and assigned by requiring () is the least element of L running in a given class, agrees with the uniform topology relative to the supremum uniformity in that class.Finally, if  is a metrisable separable space, which thus admits compatible totally bounded metrics, then the uniform topology on H() induced by the Cech uniformity of , which is also the finest totally bounded uniformity compatible with , is the supremum of all uniform topologies deriving from totally bounded metrics compatible with .
[16]uniformity of the uniform convergence w.r.t.U on H() generates in its turn the uniform topology or the topology of the uniform convergence w.r.t.U, that we will denote by   .Whenever the uniformity U is metrisable and  is a bounded metric compatible with it, then the uniform topology   is just the topology of the supremum metric d.The uniform topology induced on H() by the finest uniformity compatible with  is usually referred to as the fine uniform topology on H().Following[16], we will denote it by   .Moreover, the supremum of uniform topologies on H() relative to Weil uniformities on ,2.3.Closeness by Covers:The Open-Cover Topology.Let A be an open cover of  and ,  ∈ H().Then  is said to be A-close to  if for each  in  there exists some  ∈ A such that both (), () belong to .At any  ∈ H() the opencover topology admits as arbitrarily small neighborhoods the sets of the form: (14)Freudenthal compactification can be also described as the completion of the totally bounded uniformity determined by the covering uniformity generated from all binary coverings { − ,  − }, where  and  are open sets with compact boundaries.The Freudenthal compactification is the unique perfect  2compactification in which the growth zero-dimensionally embeds.A compactification () of a space  is called perfect if, for each point  ∈ () −  and each open neighborhood  of  in (), the set  ∩  is not a disjoint union of two open sets  and  such that  ∈ CL () () ∩ CL () ().We are now able to give a very simple description as setopen topologies for   , whenever  is normal, and for   .We recall that a set-open topology on H() admits as subbasic open sets those sets of the type[18][, ] = { ∈ H () :  () ⊆ } ,(14)where  runs in a fixed collection of closed sets ofand  is open in .When  runs over all closed sets in , then we get the closed-open topology.When  is  4 , the relativization   of the compact-open topology on H() is the closed-open topology.
. Any rim-compact  2 space  admits  2 -compactifications () whose growth () −  is zero-dimensionally embedded in () that is, every point in the growth () −  has arbitrarily small neighborhoods whose boundaries lie in .The Freudenthal compactification () is the maximal  2compactification of  whose growth () −  is zerodimensionally embedded in ().has the lifting property.Finally, the Freudenthal compactification is the Smirnov compactification associated to the Freudenthal proximity: two closed sets are far if and only if they can be separated by a compact set.If  is rim-compact  2 connected and locally connected, then its Freudenthal compactification is locally connected.[,] = { ∈ H () :  () ⊆ } , embeds in a  2 -compactification () which induces on H() the least admissible group topology  () , (ii)  () is the set-open topology determined by all closed sets with compact boundaries contained in some component of  [7].
as in (•).Of course, both  1 ,  2 are continuous.The identity map of R×Q identifies with the pair ( 1 ,  Q ), where  1 is again the usual projection of R × Q on R and  Q is the identity map of Q. Next, if  identifies with ( 1 ,  2 ) and  with ( 1 ,  2 ), then their composition  ∘  identifies with the pair (ℎ 1 , ℎ 2 ), where

Theorem 31 .
[15]y admissible group topology on H(R × Q) is stronger than the fine group topology generated on H(R × Q) by the class of all metrics on R × Q of the type  1 ×  2 , where  1 is the stereographic metric on R and  2 runs over all totally bounded metrics compatible with Q[15].
has the lifting property if and only if any self-homeomorphism of  continuously extends to it.According to the previous Lemma we naturally say that a proximity has the lifting property if it satisfies property (b) and that a uniformity has the lifting property if it satisfies property (c).
* .It is to be reminded that a  2 -compactification () of