QUANTIFICATION OF TOPOLOGICAL CONCEPTS USING IDEALS

We introduce certain ideals of real-valued functions as a natural generalization of filters. We show that these ideals establish a canonical framework for the quantification of topological concepts, such as closedness, adherence, and compactness, in the setting of approach spaces. 2000 Mathematics Subject Classification. 54A05, 54D30.


Introduction.
In [3], Kuratowski introduced what he called the measure of noncompactness for complete metric spaces.The purely topological concept of compactness was quantified in the setting of metric spaces in order to measure the discrepancy a metric space may have from being compact.Since then several variants, such as Hausdorff's ball measure of noncompactness have been introduced.For an extensive account on applications of these measures in the setting of Banach spaces, we refer to Banaś and Goebel [1].All what these measures have in common is that they involve a (pseudo-)metric space.
The introduction of approach spaces (see [5]), established a more general setting for the quantification of topological concepts.Approach spaces, which are a unification of topological spaces and metric spaces, express both qualitative and quantitative information.If these kinds of information are combined in a relevant (though canonical) way, then the numerical information can be used to express to what extent some qualitative aspect is or is not fulfilled.In [4], it is shown that compactness of topological spaces and total boundedness of metric spaces are special instances of a unifying concept for approach spaces, yielding a measure of compactness for approach spaces.Moreover, this measure is a generalization of the Hausdorff measure for metric spaces, mentioned above.Also, a measure of connectedness is defined, generalizing connectedness and Cantor-connectedness.For a recent and full account on approach spaces, we refer to [6].
Consider for instance, the following example in [6].Let X be a separable metrizable space and suppose the set ᏹ(X) of probability measures on X is equipped with the so-called weak approach structure (which is a canonical generalization of the weak topology on ᏹ(X)).Further, let Ᏼ be a weakly compact subset of ᏹ(X) and let be an arbitrary subset of ᏹ(X).Then the collection of contaminated probability measures need no longer be compact, although from a statistical point of view, Ᏼ and Ᏼ are indiscernible for ε sufficiently small.The fact that Ᏼ is almost compact can be expressed formally by saying that the measure of compactness is µ c (Ᏼ ) ≤ 2ε (see [6,Example 6.1.16]),that is, Ᏼ is weakly compact "up to 2ε."Using the measure of compactness we regain a lot of information compared to the classical situation of topological spaces.Nevertheless, we can do better.It is intuitively quite clear that, in general, the probability measure (1−ε)P +εQ contributes more to the noncompactness of Ᏼ than (1−(ε/2))P +ε/2Q does.In fact, every element in Ᏼ causes µ c (Ᏼ ) to deviate a certain amount from zero.So we can consider functions from Ᏼ to [0, ∞], mapping every element in Ᏼ to a number that equals (or is smaller than) this deviation.In the sequel, we will call such a function compact.
To that end, we need a numerification of filters, called approach ideals, which will be the subject of the first section, where we will also introduce prime approach ideals (generalizing ultrafilters).
2. Approach ideals.In [5,8], it is shown that if one wants to formulate canonical numerifications of topological or uniform concepts, it is useful to consider functions in [0, ∞] X instead of subsets of X, pointwise order instead of inclusion, ideals of functions instead of filters of sets and so forth.Consequently, we will introduce the following quantification of filters.Definition 2.1.Let X be a set.An approach ideal (a-ideal, for short) on X is an ideal containing ∞, satisfies only the condition that for all φ 1 ,φ 2 ∈ G, there exists φ 3 ∈ G : φ 1 ∨ φ 2 ≤ φ 3 , that is, if G is an ideal basis, then G is called an approach ideal basis.The approach ideal generated by G is denoted by G .
is an a-ideal on X.
Example 2.3.Let F and G be a-ideals on X such that Then are a-ideals on X.
is an a-ideal on Y .If f is onto and G is an a-ideal on Y , then is an a-ideal on X.
In the setting of real-valued functions, the concept of ideals is in some situations too weak.Different ideals can be "almost equal" in the sense that the members of one ideal can be "uniformly approximated" by the members of another ideal and vice versa.Therefore, we will look at ideals which are maximal in this respect.Definition 2.5.Let F be an a-ideal on X.Then F is called saturated if and only if Lemma 2.6.If F is an a-ideal on X, then is an a-ideal on X.
The a-ideal F is called the saturation of F. Clearly, F is saturated if and only if F = F. Different useful topological and the like structures can be defined in the setting of a-ideals (cf.[7]).We give one important example, which we will pursue in the sequel.Definition 2.7 (see [5]).An approach system on X is a family of saturated a-ideals (Ꮽ(x)) x∈X such that for every x ∈ X (A1) for all φ ∈ Ꮽ(x) : φ(x) = 0, (A2) for all φ ∈ Ꮽ(x), for all ε > 0, for all N < ∞, for all z ∈ X, there exists φ z ∈ Ꮽ(z) : Then the pair (X, (Ꮽ(x)) x∈X ) is called an approach space.
In the sequel we will want to build filters with sections of functions φ ∈ [0, ∞] X , that is, sets of the form {φ < ε} := {x ∈ X | φ(x) < ε} for some ε ∈ R + .If F is an aideal, then we will call the smallest number ε such that all sections {φ < ε} for φ ∈ F are nonempty, the height of F. Definition 2.8.Let F be an a-ideal on X.Then is called the height of F.
In the sequel, we will often abbreviate inf x∈X φ(x) by inf φ.If for each φ ∈ F, inf φ = 0, that is, if h(F) = 0, then F is said to be of zero height.For instance, the a-ideals Ꮽ(x) in an approach system are of zero height.If h(F) < ∞, then F is said to be of bounded height.
With an a-ideal on X we can associate a sheaf of (ordinary) filters on X in more than one canonical way.Proposition 2.9.Let F be an a-ideal on X of bounded height h.Then for every ε such that h < ε ≤ ∞ and for every ε such that h ≤ ε < ∞ are filters on X.
Conversely, with a classical filter on X we can associate in a natural way a-ideals of different heights.
Proposition 2.10.Let Ᏺ be a filter on X and let ε < ∞.Then are a-ideals on X.
For every A ⊂ X we define It is also possible to write down an explicit basis for Ᏺ ε , as we did for Ᏺ ε in Proposition 2.11.The result is however quite involved, and can be inferred from [8].
Example 2.12.Let (X, (Ꮽ(x)) x∈X ) be an approach space.Then the collection (Ꮽ(x) ε ) x∈X,ε∈R + (defined as in Proposition 2.9) is a collection of filters on X satisfying the following conditions for every x ∈ X: (B1) for all ε ∈ R + , for all V ∈ Ꮽ(x) ε : x ∈ V , (B2) for all ε, ε ∈ R + , for all V ∈ Ꮽ(x) ε+ε , there exists V ε x ∈ Ꮽ(x) ε , there exists  (Ᏺ ε x ) x∈X,ε∈R + is a collection of filters satisfying the above conditions, then (2.15) defines an approach system such that for every ε ∈ R + we have Ꮽ(x) ε = Ᏺ ε x .This means that an approach space can be described by a sheaf of pre-neighbourhood filters at every point x ∈ X satisfying the quantified open kernel condition (B2).
In the sequel, we will denote the pre-closure operator associated with the pretopology (Ꮽ(x) ε ) x∈X by cl ε .Sometimes we will consider an approach space (X, (cl ε ) ε ) in terms of these pre-closure operators instead of the equivalent structure (X, (Ꮽ(x)) x∈X ).
Example 2.13 (see [5]).Let (ᏺ(x)) x∈X be a family of neighbourhood filters on X, turning X into a topological space.Then (ᏺ(x) 0 ) x∈X is an approach structure on X.This construction yields an embedding functor from Top into Ap (the category of approach spaces and contractions).Moreover, this embedding is coreflective, the coreflection of any (Ꮽ(x)) x∈X being (Ꮽ(x) 0 ) x∈X .Proposition 2.14.Let F and G be a-ideals on X, let Ᏺ be a filter on X and let ε < ∞.Then we have the following: Proof.Immediate.
Proposition 2.15.Let F be an a-ideal on X and let f : X → Y be a function.Then Proof.We see that (2.16) 3. Prime approach ideals.Following Gierz et al. [2] and Lowen et al. [7], we define an a-ideal to be prime if it is a prime ideal in [0, ∞] X .Definition 3.1.An a-ideal F on X is said to be prime if for each φ, ψ ∈ [0, ∞] X we have Prime a-ideals are the numerification of ultrafilters, which is illustrated by the following two propositions.Proposition 3.2.Let Ᏺ be a filter on X and let ε < ∞.Then the following are equivalent: (1) Ᏺ is an ultrafilter, (2) Thus Ᏺ is an ultrafilter.The equivalence of ( 1) and ( 3) is shown analogously.

Proof. To show the only if part, suppose
In order to show the if part, first notice that for all and, again using Proposition 3.3, this implies that G is prime.
We will write P(F) := P | P is a prime a-ideal and F ⊂ P . (3. 2) The collection P(F) is closed under refinement (Proposition 3.4) but it does not have maximal elements of which are prime.The collection P(F) has minimal elements though.These will be investigated later.Proposition 3.5.Let F be an a-ideal on X and let f : If F is prime, then by Proposition 3.3, F ∞ is an ultrafilter, and thus f (F) ∞ = f (F ∞ ) is an ultrafilter too.Again by using Proposition 3.3, this means that f (F) is prime.
As we will see in the sequel, P(F) is too big a set for our purposes.Therefore, we will extract a subset of P(F) which still contains all the necessary information.
We will consider the set of minimal prime ideals containing F, that is, The fact that M(F) is nonempty is a consequence of Zorn's lemma and the following result.
Lemma 3.6.Every totally ordered subcollection of P(F) has a lower bound.
Proof.Let ᏼ be a totally ordered subset of P(F), and put P 0 := ∩ P∈ᏼ P. Obviously, P 0 is an a-ideal and F ⊂ P 0 .To see that P 0 is prime, suppose φ ∧ ψ ∈ P 0 .If for all P ∈ ᏼ we have φ ∈ P, then φ ∈ P 0 and we are done.If not, then there exists some The collection M(F) still contains all the relevant information, in the sense of the following proposition.Proposition 3.7.Let F be an a-ideal on X.Then F = P∈M(F) P. (3.4) Proof.We have The first identity is a well-known fact; the second is a consequence of Lemma 3.6.
In order to show a useful characterization of minimal prime a-ideals (Theorem 3.9), we need one lemma first.Lemma 3.8.Let Ᏺ be a filter and let F be an a-ideal on X.If F ∨ Ᏺ 0 exists, then Proof.The first part of the lemma follows from the observation that for each F ∈ Ᏺ, and for each φ ∈ F, we have {φ ∨ θ F < ∞} = {φ < ∞} ∩ F .
In order to prove the second part, suppose Ᏺ is an ultrafilter.Then Then by the first part of the lemma, The following theorem establishes a characterization of minimal a-ideals which will turn out to be of great use in the sequel.If Ᏺ is a filter, then we write U(Ᏺ) := {ᐁ | ᐁ is an ultrafilter and Ᏺ ⊂ ᐁ}.
(3.8) Theorem 3.9.Let F be an a-ideal on X.Then Proof.See the proof of Proposition 1.5 in [7].
The next proposition is an illustration of the fact (which we mentioned before) that by considering minimal prime a-ideals instead of prime a-ideals no relevant information gets lost.Proposition 3.10.If F is an a-ideal on X, then there exists some P ∈ M(F) such that h(F) = h(P).
Proof.First, suppose that F is of bounded height h.Let ᐁ be an ultrafilter containing F h and put P := F ∨ ᐁ 0 .Since ᐁ ⊃ F h ⊃ F ∞ , we obtain from Theorem 3.9 that P ∈ M(F).Obviously, h(P) ≥ h.To show the converse inequality, let ε > 0 and φ ∈ F.
From Proposition 3.10 we obtain that h(F) = min P∈M(F) h(P).If we replace the minimum by a supremum, we obtain a new characteristic number for F. The following proposition gives a workable description of prime height.
Proposition 3.12.Let F be an a-ideal of bounded prime height.Then Proof.For every µ > 0 we have by Theorem 3.9 that which yields the desired result.
Proposition 3.13.Let F be an a-ideal of bounded prime height, let Ᏺ be a filter and let ε < ∞.Then we have Proof.Straightforward verification.
Example 3.14.Although equality can occur in Proposition 3.13(c) (as is illustrated in part (d) and (e) of the same proposition), the inequality is strict in general.Suppose F is a nontrivial a-ideal on X and Since F = Ᏺ 0 , we have m( F) = 0.By Proposition 3.12 however, m( F) = ∞.
In order to show an analogue for Proposition 2.15 for prime height, we need to show a couple of lemmas first.Lemma 3.15.Let F be an a-ideal on X and for each P ∈ M(F), let φ P ∈ P. Then there exists a finite set M 0 ⊂ M(F) such that inf P∈M 0 φ P ∈ F.
Proof.By Theorem 3.9, it is possible to find ψ P ∈ F, ᐁ P ∈ U(F ∞ ) and U P ∈ ᐁ P such that φ P ≤ ψ P ∨θ U P .Then there exists a finite set For suppose it is not, then F ∞ ∪{X \U P | P ∈ M(F)} has the finite intersection property, and therefore it is contained in some ultrafilter ᐁ ⊃ F ∞ , ᐁ = ᐁ P say.But then X \U P ∈ ᐁ P , which is impossible.
Lemma 3.16.Let F be an a-ideal on X, and let f : X → Y be a function.Then Proof.Suppose G ∈ M(f (F)) and for every P ∈ M(F), f (P) ⊂ G. Choose for every P ∈ M(F) some φ P such that (φ P ) f ∈ G.By Lemma 3.15, there is a finite set M 0 such that inf P∈M 0 φ P ∈ F, and thus inf P∈M 0 (φ P ) f ∈ f (F) ⊂ G. From the fact that G is prime, it follows that there is some P ∈ M(F) such that (φ P ) f ∈ G, which is a contradiction.Therefore the existence of some P ∈ M(F) such that f (P) ⊂ G is guaranteed.Moreover, f (P) being prime (by Proposition 3.5) and G being minimal prime, we have f (P) = G, which concludes the proof.Proposition 3.17.Let F be an a-ideal on X and let f : X → Y be a function.Then m(f (F)) ≤ m(F).
Proof.Using Lemma 3.16 and Proposition 2.15, respectively, we find that (3.17) There is no counterpart of Proposition 3.10 for prime height.The following example establishes an a-ideal F such that m(F) = ∞, while every P ∈ M(F) is of finite height.
Example 3.18.For every n ∈ N, consider and for every finite J ⊂ N, let ξ J := sup j∈J ξ j .Consider the ideal First we show that m(F) = ∞.To that end, notice that for every n ∈ N we have that ṅ ⊃ {R + } = F ∞ , and therefore ( ṅ) 0 ∨ F ∈ M(F) by Theorem 3.9.A short computation reveals that h(( Second, we show that for every prime ideal P ∈ M(F), we have m(P) < ∞.Let ᐁ ∈ U(F ∞ ) be such that P = ᐁ 0 ∨ F and let Ᏼ denote the filter generated by the sets {k ∈ N | k ≥ n} for all n ∈ N.
• If ᐁ ⊃ Ᏼ, then every U ∈ ᐁ is an infinite subset of N, and consequently, Then for every U ∈ ᐁ such that U ⊂ U 0 , and for every ξ J ∈ F, we see that inf x∈U ξ J (x) ≤ n.

Adherence and limit operator.
In an approach space (X, (Ꮽ(x)) x∈X ) for every filter Ᏺ, Lowen [6]  In fact, the adherence operator α and the limit operator λ both determine the approach structure (Ꮽ(x)) x∈X (see [6, Propositions 1.8.1 and 1.8.2]).The value λᏲ(x) (or αᏲ(x)) is interpreted as the distance that the point x is away from being a limit point (cluster point) of Ᏺ.These notions can be generalized in the setting of a-ideals.We list some basic properties of these operators for future reference and prove some characterizations of the adherence and the limit operator.By abuse of notation, we define h(F + G) to be ∞ if F + G does not exist.Proposition 4.2.Let (X, (Ꮽ(x)) x∈X ) be an approach space and let F be an a-ideal on X.Then αF(x) = h(Ꮽ(x) + F) for every x ∈ X. Proof.These assertions follow directly from the definitions.
Proposition 4.4.Let (X, (Ꮽ(x)) x∈X ) be an approach space and let F be an a-ideal on X.Then λF(x) = sup Proof.The first assertion follows from the definition and Proposition 4.3(e).In order to show the second assertion, notice that from Lemma 3.15 it follows that for any σ ∈ P∈M(F) P, there is a finite set M σ ⊂ M(F) such that inf P∈Mσ σ (P) ∈ F. Consequently, we obtain by applying complete distributivity that inf  The other inequality follows from Proposition 4.3(a).Now it is easy to verify that Definition 4.1 establishes an extension of the adherence and limit of ordinary filters, in the following sense.Proposition 4.5.Let (X, (Ꮽ(x)) x∈X ) be an approach space and let Ᏺ be a filter on X.Then αᏲ 0 = αᏲ and λᏲ 0 = λᏲ.
Proof.Using the characterization of Ᏺ 0 in Proposition 2.11, we see that for every we have, by applying the first part of the proposition, that for every x ∈ X λᏲ 0 (x) = sup (4.9) 5. Closure and level-adherence.Yet another characterization of approach spaces can be formulated in terms of the so-called hull operator h : (see [6]).The hull operator is a natural quantification of closure in ordinary topology.A function φ is called regular if h(φ) = φ.Regular functions are a generalization of closed sets.Notice that αF = sup φ∈F h(φ).Another numerification of closure and closedness, can be obtained by considering a slight modification of the hull operator, which we will call the closure, defined by φ(x) := sup η∈Ꮽ(x) inf y∈X (φ ∨ η)(y) (for all x ∈ X).
(5.2) This is essentially not innovating, since we abuse the word "function" in this context as an abbreviation for "sheaf of sets" in the spirit of Proposition 2.9.Nevertheless, the modified concept will prove to be useful in the sequel.If we define the leveladherence of an a-ideal F by α l F = sup φ∈F φ, then-mutatis mutandis-all the results in the previous section remain true for α l instead of α.
The fact that the closure operator too is an extension of closure in topological spaces, which will be a consequence of the following observation.
Consequently, we obtain the following result, which holds as well for the hull operator.
Corollary 5.2.Let X be a topological approach space.Then θA = θ cl(A) for every A ⊂ X.
Proof.From the definition it is clear that θA only attains the values 0 and ∞.Moreover, by applying Proposition 5.1, we obtain (5.5) The closure operator behaves like a topological pre-closure operator (whence the terminology) as is illustrated by the following proposition.
Proposition 5.3.Let X be an approach space let φ, ψ, Proof.These assertions are easy consequences of the definition.
In general, it is however not true that φ = φ.Nonetheless, we define the following.
Definition 5.4.Let X be an approach space and let φ ∈ [0, ∞] X .Then φ is said to be closed if φ = φ.This is an extension of closedness in topological spaces, in the following sense.

Proposition 5.5. A set A in a topological space is closed if and only if θ A is closed in the associated topological approach space.
Proof.If A is closed, then by Corollary 5.2 we have that θA = θ cl(A) = θ A , whence θ A is closed.Conversely, if θ A is closed, then again by Corollary 5.2, we see that θ A = θA = θ cl(A) , whence A = cl(A).
From Proposition 5.3, we obtain the following results, which are to be expected.Proposition 5.6.Let X be an approach space and let φ, ψ, (φ i ) i∈I be functions in Closed functions turn out to be exactly those functions that have closed sections at every level, whence the terminology.Proposition 5.7.Let X be an approach space, and let φ ∈ [0, ∞] X .Then φ is closed if and only if for every ε ∈ R + , {φ ≤ ε} is closed with respect to cl ε .
Corollary 5.9 shows the relationship between the closure and the hull operator.Given φ ∈ [0,M] X for some M < ∞, a family (φ ε ) ε>0 of functions that attain only finitely many values, is called a development of φ if for every ε > 0 we have that Lemma 5.8.The closure operator is completely determined by the closure of functions of the form θ A .In particular, for any M < ∞, any φ ∈ [0,M] X , and any development ( Proof.We see, by Proposition 5.3(b) and Proposition 5.

6.
Compactness.The aim of this section is to generalize compactness in topological spaces.Let X be an approach space and let A ⊂ X.Let the collection of all (ordinary) filters on X be denoted by F(X) and the collection of all filters on X containing A by F(A).If Ᏺ ∈ F(X) and for each F ∈ Ᏺ, F ∩ A ≠ ∅, then the restriction of Ᏺ to A will be denoted by The measure of compactness of A can be characterized as The number µ c (A) expresses to what extent the set A differs from being compact.
If F is an a-ideal on X and φ ∈ [0, ∞] X , and if for all ψ ∈ F, φ ∨ ψ ≠ ∞, then we define the restriction of F to φ by By abuse of notation, we put h(F | φ) := ∞ in case F | φ does not exist.A set A is called compact (with respect to a topology) if every filter containing A has an adherence point.We can generalize this notion in the following canonical way.Definition 6.1.Let (X, δ) be an approach space and let φ ∈ [0, ∞] X .Then φ is said to be compact if and only if for every a-ideal F on X we have that inf(φ∨αF) ≤ h(F | φ); and φ is said to be strongly compact if and only if for every a-ideal F on X, there is some Clearly, if φ is strongly compact, then φ is compact.The following proposition pinpoints the precise difference between compactness and strong compactness.Proposition 6.2.Let (X, δ) be an approach space and let φ ∈ [0, ∞] X .Then the following are equivalent: (1) φ is strongly compact, (2) φ is compact and for every regular function ψ, the function φ ∨ ψ attains its minimum.
Proof.To show that (1)⇒( 2), first observe that if φ is strongly compact, then it must be compact.(6.4)To show that (2)⇒(1), it suffices to remark that αF = sup ξ∈F h(ξ) is a regular function by [6,Definition 1.7.1].Therefore, and by compactness of φ, there is some x ∈ X such that φ(x) ∨ αF(x) = inf(φ ∨ αF) ≤ h(F | φ), which shows that φ is strongly compact.Therefore, we will not consider strong compactness in the sequel.Most propositions remain true for strong compactness.
On the analogy of equivalent characterizations in the classical case (e.g., A is compact if every ultrafilter containing A converges), here too we can restrict ourselves to particular classes of a-ideals.Theorem 6.3.Let X be an approach space and let φ ∈ [0, ∞] X .The following are equivalent: (1) φ is compact.
In order to generalize the measure of compactness mentioned above, we are to consider the compactness notion associated with the level-adherence operator, which was introduced in the previous section.Definition 6.4.Let (X, δ) be an approach space and let φ ∈ [0, ∞] X .Then φ is said to be level-compact if and only if for every a-ideal F on X we have that inf(φ The analogue of Theorem 6.3 remains true for level-compactness.Moreover, we have the following relationship between Definition 6.1 and the measure of compactness.Proposition 6.5.Let X be an approach space, and let A ⊂ X.Then for every ε > 0, We can generalize different well-known results on compact sets.For instance, the intersection of a finite number of compact sets is again compact.Proposition 6.6.Let X be an approach space and let φ, ψ ∈ [0, ∞] X .If φ and ψ are (level-)compact, then φ ∧ ψ is (level-)compact.
An analogous argument proves the statement for level-compactness.
Corollary 6.7.Let X be an approach space and let A, B ⊂ X.Then µ c (A ∪ B) ≤ µ c (A) ∨ µ c (B).Consequently, in a topological space, the union of two compact sets is compact, and, in a metric space, the union of two totally bounded sets is totally bounded.
Proof.If µ c (A) ∨ µ c (B) ≤ ε, then µ c (A) ≤ ε and µ c (B) ≤ ε and then ε + θ A and ε + θ B are level-compact by Proposition 6.5.Calling on Proposition 6.6 we see that Suppose A and B are compact in some topological space.Then we obtain from [4] that µ c (A) = µ c (B) = 0 in the associated topological approach space, and thus µ c (A ∪ B) = 0 by the first part of the proposition.This yields the compactness of A ∪ B.
An analogous argument holds for total boundedness in metric spaces.
Proposition 6.8.Let X be an approach space and let φ, ψ ∈ whence φ ∨ ψ is compact.Corollary 6.9.Let X be an approach space and let A, B ⊂ X.If B is closed with respect to the topological reflection of X, then µ c (A ∩ B) ≤ µ c (A).Consequently, in a topological space, the intersection of a compact set and a closed set is compact.
Proposition 6.10.Let X and Y be approach spaces, let f : X → Y be a contraction and let Hence, φ f is compact.
Corollary 6.11.Let X and Y be approach spaces, let f : X → Y be a contraction, and let A ⊂ X.Then µ c (f (A)) ≤ µ c (A).Consequently, in a topological space, the continuous image of a compact set is compact.
Proof.The proof goes along the same lines as the proof of Corollary 6.9.
Finally, we want to show a Tychonoff-like theorem.To that end, we need two lemmas.If for each i The canonical projections will be denoted by π j : i∈I X i → X j : x = (x i ) i x j .If F is an a-ideal on i∈I X i , then we write F i := π i (F) for every i ∈ I.If different approach spaces X 1 ,X 2 ,... are involved, we will denote all their adherence operators by α, all the hull operators by h and so on, in order to avoid involved notation.Lemma 6.12.For every i ∈ I, let X i be an approach space, and let F be a prime a-ideal on i∈I X i .Then αF = i∈I αF i .
Proof.Suppose φ ∈ F. If for every i ∈ I, µ i is a regular function on X, then for every finite J ⊂ I we have that µ := inf i∈J µ i • π i is a regular function on i∈I X i .Let denote the collection of all regular functions µ ≤ φ that can be constructed in this manner.If µ ∈ , then µ ∈ F and by primality of F, there is some j ∈ J such that µ j •π j ∈ F, and so µ j ∈ F j .Consequently, µ ≤ µ j •π j = h(µ j )•π j ≤ αF j •π j ≤ i∈I αF i .Hence, αF = sup µ∈ µ ≤ i∈I αF i .
Proof.Let φ = i∈I φ i , X = i∈I X i and let F i := π i (F).Suppose ξ ∈ F. Since φ i ≤ φ π i , we find that φ i ∨ ξ π i ≤ φ π i ∨ ξ π i ≤ (φ ∨ ξ) π i .Therefore, h(F j | φ j ) = sup ξ∈F inf y∈X i φ i ∨ ξ π i (y) ≤ sup ξ∈F inf y∈X i (φ ∨ ξ) π i (y) = sup ξ∈F inf x∈X (φ ∨ ξ)(x) = h(F | φ).By arbitrariness of i ∈ I, this proves the claim.Theorem 6.14.For every i ∈ I, let X i be an approach space and φ i ∈ [0, ∞] X i .If φ i is compact for all i ∈ I, then i∈I φ i is compact.Conversely, if i∈I φ i is compact and inf φ i = 0 for all i ∈ I, then φ i is compact for all i ∈ I.
Proof.Write φ := i∈I φ i and X = i∈I X i .Let F be a prime ideal on X and let ε > 0. If for every i ∈ I, φ i is compact, then inf(φ i ∨ αF i ) ≤ h(F i | φ i ) by definition.Choose some x = (x i ) i ∈ X such that for every i ∈ I, φ i (x i ) ∨ αF i (x i ) < h(F i | φ i ) + ε.Then, calling on Lemmas 6.12 and 6.13, respectively, we see that inf(φ ∨ αF) ≤ φ(x) ∨ αF(x) = sup i∈I φ i x i ∨ sup (6.10) which by arbitrariness of ε proves that φ is compact.
To show the second part of the assertion, we will show that under the condition that inf φ j = 0 we have that φ j = φ π j .It is easy to see that φ j ≤ φ π j .Conversely, let ε > 0 and let y j ∈ X j be arbitrary.For every i ∈ I \ {j}, choose some y i ∈ X i such that φ i (y i ) ≤ ε and write y = (y i ) i .Then φ π j (y j ) ≤ φ(y) = sup i∈I φ i (y i ) = sup i∈I\{j} φ i (y i )∨φ j (y j ) ≤ ε ∨φ j (y j ).By arbitrariness of ε, this yields φ π j (y j ) ≤ φ j .
So, if φ is compact then every φ j = φ π j is compact by Proposition 6.10.
For the converse implication, the extra condition that all φ i "have zero height" is necessary.For instance, if φ = ∞ and ψ is not compact, then φ × ψ = ∞ is nonetheless compact.Corollary 6.15.A product of topological spaces is compact if and only if each factor space is compact.Question 6.16.Is there a Tychonoff-like theorem for level-compactness?

Definition 3 .
11. Let F be an a-ideal on X.Then m(F) := sup P∈M(F) h(P) (3.12) is called the prime height of F.

Definition 4 . 1 .
Let (X, (Ꮽ(x)) x∈X ) be an approach space and let F be an a-ideal on X.Then we define the adherence of F by αF(x) the limit of F by λF(x) := sup P∈M(F) αP(x), ∀x ∈ X.(4.4)