compactoid and compact filters - Project Euclid

PACIFIC JOURNAL OF MATHEMATICS Vol. 117, No. 1,1985

COMPACTOID AND COMPACT FILTERS SZYMON DOLECKI, GABRIELE H. GRECO AND ALOJZY LECHICKI We study compactoid and compact filters which generalize the concepts of convergent filters and compact sets. In particular, we investigate their properties in subregular and regular spaces, their localizations, and their countable variants. Several classical results follow (e.g., theorems of Tychonoff, Kuratowski, Choquet). More recent results on preservation of compactness (e.g., Smithson) and local compactness (e.g. Lambrinos) are extended and refined.

0. Introduction. Several topological properties of relations may be expressed in terms of some filters on the image space. Therefore, certain investigations of relations may be reduced to the study of filters, thus simplifying the arguments. There arise compactoid, compact, semiconvergent, locally compact, cocompact, subregular, regular and other filters. Compactoid and compact filters generalize both convergent filters and compact sets. Frequently they may be used where convergence or compactness of sets is too strong an assumption (see, for instance, the classical Theorem 7.10 of Kuratowski [15]). Compactoid filters find numerous applications in optimization, generalized differentiation, differential equations, fixed point theory (see e.g. [20] and [2] where compactoid filters are hidden under the guise of measure of noncompactness) and elsewhere, primarily in the context of existence results. To our knowledge, the concept of compactoid filter appeared for the first time in [24] by Topsoe, formulated with the aid of nets in topological spaces.1 Unfamiliar with this, two of us reintroduced compactoid filters in [9]. The same idea occurred independently to Penot; in [20] he presents an extensive collection of applications. Going back to the twenties one finds compact sequences of Urysohn [25]. They constitute a sequential counterpart of compactoid filters. Compactoid and compact filters are also of considerable theoretical interest. They provide a broader comprehension of the notion of compactness and enable one to establish new and subtler results (for instance, on preservation of compactness). notion of compactoid filter appears also in M. P. Kac, Characterization of some classes of pseudotopological linear spaces, in Convergence Structures and Applications to Analysis, Akademie-Verlag, Berlin, 1980, 115-135. 69

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SZYMON DOLECKI, GABRIELE H. GRECO AND ALOJZY LECHICKI

We found it natural to carry out our study of compactoid filters in pseudotopological spaces. This enables us to better see their nature and to distinguish topological properties from pretopological and pseudotopological ones. Later we discovered an intimate and essential relationship between pseudotopologies and compactness [8]. A filter is compactoid if every ultrafilter of it converges. Compactoid and compact sets are special cases of compactoid and compact filters. We establish the relationship between compactoid and overcoverable filters. It is demonstrated that if J^is compactoid (compact) in a pretopology TΓ, then its neighborhood filter Jf(ϊF) is comopactoid (compact) in τr2. A filter is subregular if its closure filter is compactoid. The adherence of every subregular filter is compactoid. In subregular spaces every compactoid filter is subregular. In regular spaces the closure of every convergent filter converges. This paper discusses regular spaces primarily in the context of compactness and compactoidness. An extension of the Tychonoff theorem (given already in [20] in topological spaces) says that a filter in a product pseudotopological space is compactoid if and only if every projection of it is compactoid. We refine and extend results of Smithson [23] on subcontinuous relations, and relations mapping compact sets into compact sets. It is shown that a relation is subcontinuous if and only if it maps every compactoid filter into a compactoid filter. A filter J^is pseudocompactoid if the lower limit along J^of every lower semicontinuous function is greater than - oo. We show that every countably compactoid filter is pseudocompactoid. In Dieudonne complete topological spaces, J^is finer than the neighborhood filter of a closed set A if and only if the trace of IF on the complement of A is compactoid and the ^boundary of A is a subset of A. An alternative result in metrizable spaces, but without the assumption that A be closed, generalizes Choquet's [5, Thm 3]. 1. Filters and Grills. This section has a preliminary character: its aim is to recall a few facts that we shall use later on. The basic reference is [4], from which we slightly differ in terminology and presentation. For a nonvoid family of subsets of X, s/c

2X, we define its conjugate

s/* by (1.1)

ίe#,

iff

c

B £s/.

s#** = j^and for a collection of families,

(ικ)* = n-*?, (n •*;•)* = ικ*

COMPACTOID AND COMPACT FILTERS

71

The grill9 s/\ of si is the family of all subsets of X intersecting every element of J ^ . A family s/ is called based if 0 £ j^and B e j / , B c A implies that i G j / . A subfamily dί of a based family J^is a base ofs/(s/is generated by ^ ) if for every A e j^there is B e ^ such that 5 c Λ. If j^is based, then J ^ # = s/*. A based family is called countably based if it has a countable base. We write ^ (iii). Let ^ " b e a compactoid filter and °U an ultrafilter finer than β. By Corollary 2.2, Jίiβl) meets SF. An ultrafilter ^finer than Jfiβί) V ^"is convergent because #"is compactoid. On the other hand, ^ meets JΓ(βl) and, by Corollary 2.2, ^ meets ^ , thus f 3 f". By (ii), o^is compactoid, hence lλm (i). Every convergent filter is compactoid. As a consequence of Theorems 4.2 and 4.4, we have COROLLARY 4.5. The pretopological adherence of every compactoid filter in a subregular space is compactoid.

4.6. The closure of each compactoid subset of a subregular space is compactoid. COROLLARY

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81

We complement now Proposition 3.12 by COROLLARY 4.7. Each compactoid subset of a subregular topological space is relatively compact.

Proof. Let A be a compactoid set; then J^L{A) is compactoid and, by subregularity, Cl A is (closed and) compactoid, hence compact. We conclude that in subregular topological spaces, compactoid and relatively compact sets coincide. In subregular spaces we have a reinforcement of Proposition 3.18 that generalizes [20, Thm 15] PROPOSITION 4.8. A compactoid filter in a subregular pretopological space semiconverges to a compactoid set. If a filter in a topological space semiconverges to a compactoid set, then it is compactoid.

There are examples that the second part of the above proposition need not be true in pretopological spaces. A pseudotopological space is called regular if for every filter J*", Lim J^= Lim J*\ Accordingly a pretopological space is regular if and only if its every neighborhood filter is regular. A subspace of a regular space is regular. Our definition does not require that the space be Hausdorff (or 7^). It is immediate that PROPOSITION

4.9. Every regular space is subregular.

An example of a subregular space which is neither regular nor compact is given by the topological infinite sum of copies of the space of natural numbers with the cofinite topology (open sets are the complements of finite sets). THEOREM

4.10. For a pretopological space X the following are equiva-

lent: (i) X is regular. (ii) For every A c X and x £ Cl A, JV(A) andJf(x) are disjoint. (iii) Cl JT(A) = Cl A for each A c X. (iv) Cl JΓ{s/) = Cl s/for eachs/o 2X. (v) For each A a X and each filter &'semiconvergent to A, Adh J^= ClJ^c C\A.

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SZYMON DOLECKI, GABRIELE H. GRECO AND ALOJZY LECHICKI

Proof, (i) => (ϋ) Let x ί C M . By (2.10), Ac e J\Γ{x) and, in view of regularity, there is M e Jf(x) such that Cl M c Ac. In other words, A c Int Mc or, equivalently, Mc e Jf{A). (ii) => (iii). We always have Cl A c Cl JT{A). Assume (ii) and let x f(4) such that β c e Jf{x) and, by (2.10), x € C l β ) , thus x £ C\JΓ(A\ (iii) => (iv) In view of (2.11) and (2.7), Cl JT(J*) = Π ^ ^ C l JT{A\ which, by (iii), is equal to Cl j * . (iv) => (v) If J ^ D ^Γ(yί) then Cl J ^ c Cl ^T(^) = C\{A) by (iv). (v) => (iii) We have JT(A) ^ JT(A)\ thus Cl ^ = Cl ^ ( ^ ) c Cl JT(A) c Cl ^ . (iii) => (ii) Let j c ί Q i l , thus, by (iii), x ί Cl JΓ{A\ that is, ^Γ(x) and JΓ{A) are disjoint, (ii) => (i) Let Q e ^Γ(x), i.e., Λ: € C l β c . By (ii) there ί s M e ^Γ(x) such that Mc e ^Γ(β c ), hence β c c Int Λfc or, equivalently, Cl M c Q, proving that Q JP The implication (i) => (v) entails [9, Thm 5]. As in (v) of Theorem 4.10, in the case of Hausdorff topological spaces, Adh J ^ c A whenever J^is semiconvergent to A and A is a compact set ([20; Proposition 13, a]). However, unlike (v), this need not be true for pretopological spaces. 4.11. Let X= R, A = {-1/n: n e N) U {0} and let { ^ J V ^ ( { ( Π + 1, +oo): Λ > 0}) and define EXAMPLE

Λ

jrr(o)cΛ ~°°

%m

6.6. Letf: I - ^ R U { + oo}. If{x:f(x)

< inf(/)} Φ 0,then

each filter is countably compactoid for τ(epi / ) . // {x: f(x) < inf(/)} = 0 ,

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SZYMON DOLECKI, GABRIELE H. GRECO AND ALOJZY LECHICKI

then a filter & is countably compactoid for τ(epi/), if and only if li^f > inf(/) We infer that THEOREM

6.7. Every countably compactoid filter is pseudocompactoid.

A space X is pseudocompact if the filter {X} is pseudocompactoid, that is, whenever for every lower semicontinuous function /: I - > R U {oo}, inΐx(ΞXf(x) > — oo. In the case of topological pseudocompact spaces this infimum is attained [11, p. 263]. This is not the case of pseudocompactoid filters where neither supremum nor infimum in (6.2) need be attained. However, PROPOSITION 6.8. If & is a countably compactoid filter and f is a lower semicontinuous function, then there is JC0 e Adh Φsuch that f(x0) = li^/

Proof. Let { rn } π e N be a sequence strictly decreasing to li^/. The filter generated by {{x:f(x) < rn}}n^N meets 3?and, Adh{{x:f(x) < rn}}nξΞN meets Adh^. Thus f\ i l w {x: f(x) Xi^f). If the infimum were not attained, Fo would be a subset oί {x: f(x) > li^f}. Let {rn}n r J } n e N is an overcover of Fo and, by countable compactness, there is n e N such that {x: f(x) > rn) belongs to 3F. This is a contradiction. 7. Totally bounded and semiconvergent filters. Let (X, U) be a uniform space. A filter ^"in X is said to be totally bounded if for every U (ϋ). Every J^(x) contains a compact set, thus is a compactoid filter. Thus X is subregular. (iii) => (i). Every neighborhood filter JV*(X) contains a compactoid set Ax. By Corollary 4.7 the closure of Ax is compact and closed. PROPOSITION 8.5. The following properties of a topological space X are equivalent: (i) Every neighborhood filter of X has a base composed of closed compact sets. (ii) X is hereditary locally compact and regular. (iii) X is hereditary locally compactoid and regular.

Proof. A consequence of Theorem 8.1 and Propositions 8.4 and 4.9. Subregularity in Proposition 8.4 may be replaced by the condition (8.1) the closure of every compactoid set is compact. Schnare [22] shows that the following similar condition, (8.2) the closure of every compact set is compact, amounts to (i) of Proposition 8.4 in locally compact spaces. However, in general, in locally compactoid space, (8.2) does not imply (i) (of Proposition 8.4), since in [16, Ex. 1] there is a Hausdorff locally compactoid space which is not locally compact. It is important to note that in regular spaces all properties of Propositions 8.4 and 8.5 are equivalent. Indeed, one has 8.6. If a regular space is locally compactoid, then it is hereditary locally compactoid. PROPOSITION

To this we add information on regular spaces. PROPOSITION 8.7. If a Hausdorff space is either locally compact or hereditary locally compactoid, then it is regular.

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SZYMON DOLECKI, GABRIELE H. GRECO AND ALOJZY LECHICKI

[16, Ex. 2] shows that in Proposition 8,7 hereditary local compactoidness must not be replaced by local compactoidness. 9. Closure of composed relations. As an example of applications consider topological spaces X, 7, Z and relations T: X ^ Y and Δ: 7 4 Z . Their composition is the relation ΔΓ: X ^t Z defined by ΔΓJC=

U

Δy>

y R (Δ = epi/). The composition (epi/) ° Γ is graph-closed if and only if the marginal function fT (fT(x) = mfγ R be a lower semicontinuous function and Γ: X zX Y a graph-closed subcontinuous relation. Then the marginal function fT is lower semicontinuous and f attains its infimum on every Tx.

Note that we know that Γ is upper semicontinuous and closed-valued, but for nonsubregular spaces we may not have compact values of Γ. A function / is well-conditioned if it is lower semicontinuous and its level relation (epi/)" 1 is subcontinuous. This definition is equivalent to [20, Def. 16] and is less stringent than that of well-posedness.

COMPACTOID AND COMPACT FILTERS

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COROLLARY 9.4. If f is well-conditioned and Γ is graph-closed, then / Γ is lower semicontinuous and f attains its infimum on every Tx.

The above is not surprising, since a well-conditioned function attains its infimum on every closed set. For other conditions on lower semicontinuity of marginal function see [10]. Another special case is that of linear (graph-) closed operations Γ, Δ in topological vector spaces. Then the subcontinuity of Δ" 1 implies that Δ""1 is an operator. 9.5. Let Γ, Δ be graph-closed linear operators. ΔΓ is graph-closed if either Γ is continuous or Δ has the continuous inverse. COROLLARY

REFERENCES

[I] [2] [3] [4] [5] [6] [7] [8] [9] [10] [II] [12] [13] [14] [15] [16] [17] [18]

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SZYMON DOLECKI, GABRIELE H. GRECO AND ALOJZY LECHICKI

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38050 Povo (TRENTO) ITALY AND PEDAGOGICAL UNIVERSITY SZCZECIN, POLAND