THE STAR COMPACTIFICATION

The relationships between a convergence space and its star compactification is studied. Special attention is given to lifting properties of this compactification. In particular, it is shown that a natural extension of any continuous function to the respective compactification spaces is θ-continuous.


DECOMPOSITION SERIES.
The reader is asked to refer to [7] for convergence space notation and terminology not given here, as well as additional information about the star compactification.As in [7], space will always mean convergence space, and the abbreviation "u.f." is often used for "ultrafilter".The separation axioms T I (singletons are closed), T 2 (convergent filters have unique limits), and T 3 (regular plus T 2) will be used, but no separation axioms are assumed unless such is explicitly stated.
Given a space X, let F(X) (resp.U(X)) be the set of all filters (resp., ultrafilters) on X.Let X' be the set of all non-convergent members of (X), and X* X U X'.If A c X, define A' { 6 X': A E,}, and A* A U A'.If E F(X), and F' # for all F , then let be the filter generated by {F': F } let * be the filter generated by {F* :F 3 }.If 3' exists, then 3" 3 3'; otherwise, 3" 3 We omit the easy proofs of the first two propositions.PROPOSITION 2.1.The following equalities hold for any subsets A, B of X: A' U B' (AUB)' A' B' (AB)'; A* U B* (A UB)* A* B* =(AB)*.
Let X be a space, F(X*), and X' Define ^to be the filter on X generated by {A !X A' E }.  (X)and 3' exists, then .')^3 (c) If (X*) and X' 6 , then 6 6 (X).

X*
A convergence structure is defined on as follows: For x X, -+ x in X* iff there is + x in X such that >-*; for X' + iff >_ * i* Let denote the identity embedding of X into X*-, it is proved in [II] that <* (X*,i*) is a compactification of X which is T 2 whenever X is T 2. It is immediate from the construction that, for any non- compact space X, X*-X is a T 2 pretopological space; thus X* is pretopological whenever X is pretopological.The universal property of K* established in [II] will be obtained in Section 3 as a corollary of a much more general result.
A subset A of space X is bounded if each ultrafilter containing A is convergent.X is said to be locally bounded if each convergent filter contains a bounded set.X is essentiall7 bounded if 3 ( X' implies that the filters and ( X' : % 3 } contain disjoint sets.The next proposition is proved in [7].
We shall next consider the relationship between the closure operators of X and X*.Let cl X be the closure operator on a space X.For an ordinal number , we deflne:

<
The smallest ordinal u such that clA clu+IA for all A c X is called the X X length of the.decomposition series of X and denoted by ED(X).The relationship between ED(X) and ED(X*) can be obtained with the help of several lemmas.
For the remainder of this section, we shall assume that X is an arbitrary space; (X*,i*) will always' denote the star compactification of X.Let be the smallest infinite ordinal number.
PROOF.It suffices to prove this result for n--2.Note that CI2xA U (cl XA)' c__ cl2x , A is obvious.If x {(cl2x ,A) 0 X then there is >_ Thus (X*) such that > 8" and X Q implies QIx 2 A 0 X' then it is easy to see that cl XA 3 and x CI2xA.If { clx, so : (c].XA)'.B We next describe cl, B for B c__ X'.For this purpose, it is necessary to introduce some additional terminology.If B c X' let 0 3: { B}; note B that B F(X). Define B { 8 X': 8 >_ B} and B {x X 3 { (X) Bv.
PROOF.For n--I, the statement becomes Clx, B B U B'.If 8{ (clx* B) 0 X', then there is { (X*) such that / 8 in X* and B Thus Q >-8', and so B O G' # for all G 8  This implies 8 >-B' and so

B
If x (clx, B) X, then there is an u.f.containing B and a filter + x such that >_ ' By Proposition 2.2, > 3 and so 6 + x in X.Also, 6 6  (X) and >-(6^)'.Letting 8   we have B N G' # for all G 8. Thus 8 >-B' and x { B.
Conversely, if x B", then there is { (X) such that / x and >_ For each F 6 , choose 8 F B such that F { 8 F Let be the filter of Then >_ * impliesx in X*, and so sections of the net (SF) F x clx, B. A similar argument shows that B~c__ clx, B. This establishes the result for n I. 2 If n--2, then clx, B clx, (B U Bv) (B~) U (B~) U (cl xBv) follows with the help of Lemma 2.4.By the remarks preceding Lemma 2.5, (B~) This establishes the result for n--2.The generalization to arbitrary n is now a trivial induction argument.
COROLLARY 2.6.If A_c X, then clx, A clx, A* PROOF.Clx, A* clx, A U clx, A'.By Lemma 2.5, clx, A' It is easy to check that (A') A' _c clx, A, and (A') LEMMA 2.7 (a) If A c X and u is a non-limit ordinal, then PROOF.Using transfinite induction along with Lemma 2.4, the results follow easily from both limit and non-limit ordinals, except for the case u 7 +I, where y is a limit ordinal.In this case we have c'Y'+l B"e where B and B c__ (c17 XA)'; we omit the details.
The symbol AX represents the topological modification of X, i.e., the finest topological space on X coarser than X.
PROOF.Let X I AX*Ix Then AX >_ X 1 is clear.Let A c__ X be %X-closed.
Then clx, A clxA U A' AU A' A* clxA* by Corollary 2.6.Thus A* is closed in AX* and A A*0 X, which implies A is X 1 -closed.
We define a to be a regular space with the property that cl X -for all { X'; this concept (but not the terminology) was introduced by Gazik [4].Discrete spaces and compact regular spaces are the most obvious examples of G-spaces.Another class of G-spaces are the ultraspaces, which are the topological spaces having exactly one convergent ultrafilter.The next theorem is proved in [4] in the case where X is T2; removal of the T 2 assumption causes no problems in the proof.THEOREM 2.11.X* is regular iff X is a G-space.
A regular space is symmetric if + x whenever y andx.Examples of symmetric spaces include T 3 spaces, regular topological spaces, and regular convergence groups.It is shown in [14] that a compact symmetric space has the same ultrafilter convergence as a compact regular topological space.PROPOSITION 2.12.If X is a symmetric G-space, then X* is symmetric.

PROOF. Let
and + 8 in X* By the construction of X* we can con- clude that both e and 8 are in X.Since + , there is / e in X such that >_ *.Since X is symmetric, / 8 in X, and so / 8 in X*.THEOREM 2.13.(a) If X is a G-space, then D(X*) _< 2 and D(X) _< PROOF.(a) If X is a G-space, then X* is a compact regular space by Theorem 2.11, and it follows by Theorem 2.4(a), [14], that D(X*) 2. The second inequality follows from the first and Theorem 2.9.
(c) If X is a topological G-space, then X is symmetric, and so D(X*) _< I by (b).If D(X*) _< I, then X is a compact T 2 topological space, and hence X* is regular.By Theorem 2.11, X is a G-space.%D(X*) 0 iff X is a finite discrete space.If X is not a finite discrete space, we can replace "D(X*) _< l"by "D(X*) I" in (b) and (c) of Theorem 2.13.
In this section, we summarize some results on the R-serles of a space (originally studied in [13] and [14]), obtain a few results on the R-serles of X*, and lay the groundwork for many of the results of Section 4.
Starting with a space X, an ordinal family of spaces {r X} is defined on the same underlying set as follows: r0(X X x in rlX iff there exist n N and + x in X such that >_ cl n X + x in rX iff there exist n N,x in X and < such that >_ cln_x The family r X} is called the R-series of X.If is the least ordinal such that rX r+ 1 X, then ryX is denoted Xr, and is called the length of the R-serles of X and denoted by R(X).Note that X is regular iff R(X) 0. It is shown in [13] that X is the finest regular space coarser than X, and X is called r r the regular modification of X.
Of course X will not in general be T 2 even when X is T 2. A T 3 space r associated with X (we use T 3 to mean regular plus T I) is constructed as follows.
First, define an equivalence relation among the elements of X by x by iff / y in X Let X be the set of equivalence classes with the quotient convergence r s structure derived from X r If f X -Y, let X + Y be the (unique) function which makes the s s where the maps from X to X r and Y to Y are the respective identity maps, and r X' Y are the respective quotient maps.
Let oX denote the symmetric modification of X, i.e., the finest symmetric space coarser than X.The next proposition follows immediately from results of [13] and [14].
PROPOSITION 3. I.If f X -Y is continuous, then each map in the following commutative diagram (in which all non-labeled vertical maps are f and all non- labeled horizontal maps are identities) is continuous. X For any space X, rlX is a subspace of rlX*.
PROOF.Let X I be the restriction of rlX* to X Since i* X --+ X* is continuous, it follows from Proposition 3.1 that i* :r I Xr I X* is continuous, and thus r I X >_ X I. On the other hand, let -+ x in X I. Then there is Q --+ x , in X and n N such that cl, Since --+ x in X* there is --+ x in X such that >_ * and so we have > ClX,n , clx,n cl (cl-l '), by Lemma 2.4 and Corollary 2.6.Since F(X), >_ cl , and so --+ x in rlX.PROPOSITION 3.3.If X is a locally compact T 3 space, then r 2 X is a subspace of r 2 X*.
PROOF.It is sufficient to show that for any A c__ X, clix A (clix, A)N X for all n 6 N, and'this will be proved by induction.For n I, the equality follows by Proposition 3.2.If the equality holds for n and x n+l A) 0 X, (Clr I X* then there is --+ x and k { N such that (clxk, F) (clIX, A) # for all F Since X is locally compact and T3, we may assume without loss of From this observation, along with the induction hypothesis, we may conclude that x In+l A c+/-rl X X* To establish this It is not true in general that rX is a subspace of r fact, we need to make use of some theorems from [9].
A space X is defined to be completely regu.larif it is a subspace of a symmetric compact space.It is shown in [9] that X is completely regular iff it is a symmetric space with the same ultrafilter convergence as a completely regular topological space.Let mX denote the finest completely regular space coarser than X A space X is defined to be 0-reular if --+ x implies claX --+ x.It is proved in [9] that X is m-regular iff X is a subspace of a compact regular space.
The m-regular spaces include the completely regular spaces and also the c-embedded spaces of Binz [I].PROPOSITION 3.4.(a) If X is a regular space which is not m-regular, then X r is not a subspace of X* R(X*) r and >_ 2.
(b) If X is a locally compact T 3 space which is not m-regular, then R(X*) > 3.
PROOF.The first part of (a) follows from the aforementioned character- ization of m-regular spaces as subspaces of compact regular spaces.The two statements concerning R(X*) follow by Proposition 3.2 and 3.3, respectively.
For any space X, let C(X) be the set of all continuous real-valued functions on X.A T 3 topological space X for which C(X) consists only of constant functions is an example of a regular space which is not m-regular; for this space, R(X)=0 and R(X*) >_ 2.An example of a regular space X for which R(X*) >_ 3 is obtained with the help of the following lemma.
LEMMA 3.5.A locally compact T 3 space X is m-regular iff C(X) separates points in X.
PROOF.If X 0X, then wX is T 2 and so C(X) separates points in X.

Conversely, if
--+ x in X, then contains a compact set A. Since separates points in X, wX is T 2 and so the subspaces XIA and XIA have the same ultrafilter convergence.It follows that cl X Clx --+ x in X, and therefore X is w-regular.EXAMPLE 3.6.Let X be the set [0, I]   If s is a sequence on the set X, let denote the filter generated by s in the usual way.Let be the filter generated by the sequence ().Define a convergence on X as follows: (I) If x {0, I}, then -+ x iff there is a sequence s converging to x in the usual topology such that >_ (2) --+ 0 iff there is a sequence s con- verging to 0 in the usual topology, but not a subsequence of (), such that >- where s is a sequence converging to 1 in s s the usual topology, or else >_ I.One may easily verify that the space X is locally compact and T3, but C(X) will not separate the points 0 and i.Thus, by Lepta 3.5, X is not w-regular, and it follows from Proposition 3.4 that R(X*) >_ 3, whereas R(X) 0; this result contrasts with the conclusion of Theorem 2.9.One can also show (we omit the details) that 6--+ I in r 3 X*.Since r 3 X =X, it follows that r 3 X is not a subspace of r 3 X*.This shows that the con- clusion of Proposition 3.3 cannot be improved without imposing additional conditions.
Gazik showed in [4] that a T 3 pretopologlcal G-space is a completely regular topological space.Another result along these lines is PROPOSITION 3.7.(a) A symmetric G-space is completely regular.
(b) Every G-space' is w-regular.
PROOF.These statements follow immediately from Theorem 2.11, Proposition 2.12, and the characterization of w-regular spaces, obtained in [9].
THEOREM 3.8.Let X be a space.
X* (a) X is regular iff X is a subspace of r I (b) X is e-regular iff X is a subspace of X* r (c) X is completely regular iff X is a subspace of X*.
For topological spaces X and Y, function f X --+ Y is defined to be @-continuous if, for every x E X and every neighborhood V of f(x), there is a neighborhood of f(x) such that f(c %< c_ Cly V.
PROPOSITION 3.9.Let f X --+ Y, where X and Y are topological spaces.
Then f X --+ Y is @-continuous iff f r I X --+ rlY is continuous.
PROOF.Let f X --+ Y be @-continuous, and let --+ x in rlX.Then there is --+ x in X such that >-cl x >-cl x b(x), where (x) is the neighborhood filter at x.By @-continuity, f(cl X b (x) >-cly b(f(x)), and so f() >_ Cly b(f(x)).The latter filter rlY-converges to f(x), and so f :rlX --+ rlY is continuous.For the converse argument, one easily see that f rlX --+ rlY implies f(cl x (x)) > Cly b(f(x)) for all x E X, which is equivalent to 8-continuity of f X --+ Y.
The characterization of 8-continuity given in Proposition 3.9 is not suitable for a purely topological investigation, since rlX may fail to be topological even when X is topological.Perhaps this suggests that convergence spaces are the natural realm for the study of 8-continuity.But in any event, we shall define a function f X --+ Y between arbitrary convergence spaces to be @-continuous if f rlX --+ rlY is continuous.
More generally, if P is any function property, then f X --Y is defined to have property @-P if f rlX --+ rlY has property P. Thus, one can speak of 8-open maps, @-quotient maps, etc.Some of these "@-properties" will be studied COROLLARY 4.3.If f X --+ Y is continuous, then each map in the following commutative diagram (in which all non-labeled vertical maps are f, and all non- labeled horizontal maps are identities) is continuous.
X* --+ --+ X*r --+ oX* T< X* s r I X* The next resu.tclosely resembles, but is more general than, the extension property of the star compactification obtained in [II].first statement is establis'hed.The second follows from an earlier remark.
In the next section we shall see that continuity of f :X-Y does not X* Y* guarantee the continuity of f, then by Proposition 3.2 <~(X~, i*) is a compactification of X.
Our study of the compactification <~will be limited to the following proposition.
PROPOSITION 4.5.Let X and Y be regular spaces. (a PROOF.Statement (a) follows immediately from Theorem 4.2.
(b) If X is a T 2 G-space, then X* is regular by Proposition 2.10, and so X X* is T 2. If X is not a G-space, then there is X' such that > cl X where X (X) and # ]If --+ x, then --+ in X~.If X', then the on X* generated by converges in X* to both and filter 1 It is shown in [12] that every completely regular T 2 space has a Stone-ech compactificatlon.Th_-is compactification is regular and T2, has the universal property relative to the class of completely regular spaces, and agrees with the COROLLARY 4.7.If X is a completely regular T 2 space, then (X%, i,-) is the Stone-Cech copactification of X.
If X is a Tychonoff topological space, then Corollary 4.7 gives a new method for constructing 8X.Indeed, 8X is in this case the pretopological modification 5. CONTINUITY OF NATURAL EXTENSIONS.
We next consider conditions under which a natural extension f, of a continuous function f is continuous.For this purpose, we use some additional notation and terminology.
Let f X --+ Y be a continuous function, and let f, X* --+ Y* be a natural Ff,() 6 F(Y) to be the filter generated by {Ff,(F) F }; F(X) is said to be f,-closed if Ff,() f(3).
PROPOSITION 5.1.Let f be a continuous map. (I) f, is continuous at x f-l(y) iff -+ x in X implies that Ff, --+ f(x) in Y.
COROLLARY 5.2.Let f X --+ Y be continuous, and Y a regular space.Then -i (I) f, is continuous at all points of f, (Y).
(3) If X is essentially bounded, then f, is continuous.
PROOF.Statements (I) and (2) follow immediately from Proposition 5.1 and the fact that Y is regular.The assumption that X is essentially bounded (see Section 2 for this definition) guarantees that each f,-I (y,) is f,-closed.PROOF.If f is weakly proper and A c__ X, then A'f .Thus each filter F(X) is f,-closed.The conditions (I) and (3) of Proposition 5.1 for conti- nuit# of f, are thus satisfied, while condition (2) is satisfied vacuously.
COROLLARY 5.4.If X is a closed subspace of Y and f :X --+ Y is the identity embedding, then f, X --+ Y is also an embedding.
PROOF.Since X is closed in Y, f is weakly proper; thus f, is continuous by Proposition 5.3.f is clearly one-to-one, and by Lemma 4.1, f,(5*) f()* for all F(X).From this equality, it follows easily that f, is an embedding.
PROPOSITION 5.5.The following statements about a regular space Y are equivalent.
(a) Y is a G-space.
(b) Every natural extension of every continuous function into Y is continuous.(c) (a).If Y is not a G-space, then there is Y' such that cly # Since Y is T 1 and Z is discrete, it follows that F f # for some natural extension f, Thus is not f,-closed, and by Proposition 5.1 f, is not It is clear that a proper map is weakly proper, and that a weakly proper map onto a T 2 space is proper.
In all of the propositions that follow, f denotes a function from a space * y* X into a space Y, and f, X --+ denotes a natural extension of f.PROPOSITION 6.1.If f is a proper map, then f, is also a proper map.
PROOF.Suppose that is an u.f. on X such that f,() c in Y* Then there exists an u.f. on Y such that f() >_ N*, where * a In Y* Suppose that G---t in X* then there exists a f.lter .on X such that --+ in X* and >_ *.Hence f() >_ f(*) f()* since f is a proper map, and thus f(.) and N* are not disjoint filters on This implies that f() and are not disjoint filters on Y; consequently, f() E.
If Y, then it follows that f,() .If E Y, then since f is a proper map, .x in X for some x E f-l(c), and thus ---+ x in X*.It follows that f, is a proper map.PROPOSITION 6.2.If f is a convergence quotient map, then f, is a conver- gence quotient map iff f, is continuous.
PROOF.The relation f(A)* _ f,(A*) is satisfied for each subset A of X, and hence f,( <_ f( )* for each filter on X. Suppose that * --+ in Y* and Y; let be any u.f. on X such that f() = Then f,() and f,(*) _< *.If E Y, then there exists E F(X) and x E f-l() such that --+ x in X and f() since f is a convergence quotient map.Since f,( <_ *, it follows that f, is a convergence quotient map precisely when f, is a continuous map.COROLLARY 6.3.f, is a perfect map whenever f is a perfect map.Let 6 (Y*) such that --+ in Y*, and let 8 f, 1 ().Then there is (Y) such that >_ * and * --+ in Y*.If Y, then 8 X, and hence there exists exactly one u.f. 3 on X such that 3--+ 8 in X*.Thus f() , and since f is a proper map, f,( 3*) * -< 6 Let 8 (X*) contain both 3" and f, 1 () then f, (8) and 8 --+ 8 in X*.
If e Y then, since f is a proper map, 8 6 X, and since f is an open map, there is ?(X)such that --+ 8 in X and f (3)   Then, as in the argument of the preceding paragraph, there exists an u.f. 8 --+ 8 in X* such that f, (8) this establishes that f, is an open map.
We omit the straightforward proof the next proposition.PROPOSITION 6.5.If f is perfect 8-proper map, then f, is a 8-perfect map.Propositions 6.4 and 6.5 yield the following corollary.COROLLARY 6.6.If f is an open perfect map, then f, is a 8-perfect map.PROPOSITION 6.7.If f is a convergent quotient map which is closure preserving, then f, is a 8-convergence quotient map.
, PROOF.Suppose that --+ in rlY Then there is F(Y) such that --+ e and >_ cln, .If Y, then it may be assumed that e--Y'.

COROLLARY 4 . 4 .
If f X --+ Y is continuous and Y is compact and regular, X* then f, --+ Y is continuous.If Y is also T2, then the extension f, is unique.

PROPOSITION 5 . 3 .
If f X --+ Y is continuous and weakly proper, then f, is continuous.
(c) If Z and Y have the same set, Z is discrete, and f Z Y is the identity, then f, is continuous.PROOF.(a) (b).If f X --+ Y is continuous and Y is a G-space, then Y X* * is regular by Theorem 2.11 and so f, --+ Y is continuous by Corollary 4.4.(b) (c).Obvious.

PROPOSITION 6 . 4 .
If f is an open, proper map, then f, is open, proper, 0-open, and 0-proper.PROOF.It follows by Proposition 6.1 that f, is proper.If f, is also open, then it follows from Theorem 4.2,[14], that f, is also o-open and o-proper.Thus it remains only to show that f, is open.