Continuous Selections and finite C-spaces

Topology and its Applications, 1997

For an infinite cardinal a, we say that a subset B of a space X is Ca-compact in X if for every continuous function f : X -~ II~ ~, f [B] is a compact subset of II~ ~. This concept slightly generalizes the notion of a-pseudocompacmess introduced by J.F. Kennison: a space X is a-pseudocompact if X is Ca-compact in itself. If a = w, then we say C-compact instead of C~-compact and ~v-pseudocompactness agrees with pseudocompactness. We generalize Tamano's theorem on the pseudocompactness of a product of two spaces as follows: let A C_ X and B C_ Y be such that A is z-embedded in X. Then the following three conditions are equivalent: (1) A x B is C~compact in X x Y; (2) A and B are Ca-compact in X and Y, respectively, and the projection map ~r : X x Y ~ X is a za-map with respect to A x B and A; and (3) A and B are Co-compact in X and Y, respectively, and the projection map ~':X x Y -+ X is a strongly z~-map with respect to A x B and A (the z~-maps and the strongly z~-maps are natural generalizations of the z-maps and the strongly z-maps, respectively). The degree of C,~-compactuess of a C-compact subset B of a space X is defined by: p(B, X) = cx~ if B is compact, and if B is not compact, then p(B, X) = sup{a: B is Ca-compact in X}. We estimate the degree of pseudocompacmess of locally compact pseudocompact spaces, topological products and E-products. We also establish the relation between the pseudocompact degree and some other cardinal function's. In the context of uniform spaces, we show that if A is a bounded subset of a uniform space ~X,~/), then A is Ca-compact in X, where (X,~) is the completion of (X,/,/) iff f(A) is a compact subset of R ~ from every uniformly continuous function from X into ~'~; we characterize the C~-compact subsets of topological groups; and we also prove that if {Gi: i E I} is a set of topological groups and A~ is a Ca-compact subset of G,~ for all i E I, then I-L~ A~ is a Co-compact subset of 1-[iel Gi. © 1997 Elsevier Science B.V.

Set-Valued Analysis, 2003

A characterization of n-dimensional spaces via continuous selections avoiding Z n -sets is given, and a selection theorem for strongly countable-dimensional spaces is established. We apply these results to prove a generalized Ostrand's theorem, and to obtain a new alternative proof of the Hurewicz formula. It is also shown that our selection theorem yields an easy proof of a Michael's result.

The following selection theorem is established:\\ Let $X$ be a compactum possessing a binary normal subbase $\mathcal S$ for its closed subsets. Then every set-valued $\mathcal S$-continuous map $\Phi\colon Z\to X$ with closed $\mathcal S$-convex values, where $Z$ is an arbitrary space, has a continuous single-valued selection. More generally, if $A\subset Z$ is closed and any map from $A$ to $X$ is continuously extendable to a map from $Z$ to $X$, then every selection for $\Phi|A$ can be extended to a selection for $\Phi$. This theorem implies that if $X$ is a $\kappa$-metrizable (resp., $\kappa$-metrizable and connected) compactum with a normal binary closed subbase $\mathcal S$, then every open $\mathcal S$-convex surjection $f\colon X\to Y$ is a zero-soft (resp., soft) map. Our results provide some generalizations and specifications of Ivanov's results (see \cite{i1}, \cite{i2}, \cite{i3}) concerning superextensions of $\kappa$-metrizable compacta.

Proceedings of the American Mathematical Society, 1969

AIP Proceedings

The aim of this paper is to study fuzzy extensions of some covering properties defined by A. V. Arhangel'skii and studied by other authors.

Journal of the Korean Mathematical Society, 2005

G. Gruenhage gave a characterization of paracompactness of locally compact spaces in terms of game theory ([6]). Starting from that result we give another such characterization using a selective version of that game, and study a selection principle in the class of locally compact spaces and its relationships with game theory and a Ramseyan partition relation. We also consider a selective version of paracompactness.

2005

We prove that if X is a σ -compact Polish space, then the space C k (X) of all continuous realvalued functions on X with the compact-open topology is a µ-space, and hence is M 1 , i.e., it has a σ -closure-preserving base. We also construct an explicit σ -closure-preserving base for C k (X).  2004 Elsevier B.V. All rights reserved.

Journal of Mathematics

A space X is said to be set selectively star-ccc if for each nonempty subset B of X , for each collection U of open sets in X such that B ¯ ⊂ ∪ U , and for each sequence A n : n ∈ ℕ of maximal cellular open families in X , there is a sequence A n : n ∈ ℕ such that, for each n ∈ ℕ , A n ∈ A n and B ⊂ St ∪ n ∈ ℕ A n , U . In this paper, we introduce set selectively star-ccc spaces and investigate the relationship between set selectively star-ccc and other related spaces. We also study the topological properties of set selectively star-ccc spaces. Some open problems are posed.

European Journal of Pure and Applied Mathematics, 2021

A C-paracompact is a topological space X associated with a paracompact space Y and a bijective function f : X −→ Y satisfying that f A: A −→ f(A) is a homeomorphism for each compact subspace A ⊆ X. Furthermore, X is called C2-paracompact if Y is T2 paracompact. In this article, we discuss the above concepts and answer the problem of Arhangel’ski ̆i. Moreover, we prove that the sigma product Σ(0) can not be condensed onto a T2 paracompact space.

Economic Theory, 2021

We root this tribute to Nicholas Yannelis in Chapter II of his 1983 Rochester Ph.D. dissertation, and in his 1983 paper with Prabhakar: this work strengthens the lower semicontinuity assumption of Michael's continuous selection theorem to open lower sections, and leads to correspondences defined on a paracompact space with values on a Hausdorff linear topological space. We move beyond the literature to provide a necessary and sufficient condition for upper semi-continuous local and global selections of correspondences, and apply our result to four domains of Yannelis' contributions: Berge's maximum theorem, the Gale-Nikaido-Debreu lemma, the Sonnenschein-Shafer non-transitive setting, and the Anderson-Khan-Rashid approximate existence theorem. The last also resonates with Chapter VI of Yannelis' dissertation, and allows a more general framing of the pioneering application of the paracompactness condition to his current and ongoing work in mathematical economics. Some of the results reported here were first presented by Khan at the Summer Workshop in Economic Theory (SWET) on October 25, 2018 under the title "Nicholas Yannelis and Equilibrium Theory: Salient Contributions to Economics and Mathematics." He thanks Bernard Cornet, Ed Prescott and Anne Villamil for discussion and encouragement at his talk; he should also like to acknowledge Greg Duffie and his JHU colleagues for a departmental discussion on proper names in connection with the renaming of the departmental Ely Lectures. This final submission has benefited substantially from stimulating conversation and correspondence with