Paracompact Research Articles

Semiregular Property on Generalized Compact and Paracompact Spaces

Misalkan $(X, \tau)$ ruang topologi dan $(X, \tau^\ast)$ semiregularannya. Maka sifat topologi ${\cal P}$ adalah semiregularan asalkan $\tau$ mempunyai sifat ${\cal P}$ jika dan hanya jika $\tau^\ast$ mempunyai sifat yang serupa. Dalam kertas ini, kami menyelidiki sifat semiregularan di atas beberapa ruang padat dan para--padat teritlak, iaitu ruang para-padat, hampir padat, padat lemah dan para--padat lemah. Katakunci: Semiregularan; sifat semiregularan; ruang padat dan ruang para-padat. We let $(X, \tau)$ be a topological space and $(X, \tau^{\ast})$ its semiregularization. Then a topological property ${\cal P}$ is semiregular provided that $\tau$ has property ${\cal P}$ if and only if $\tau^{\ast}$ has the same property. In this work, we study semiregular property on some generalizations of compact and paracompact spaces; namely, paracompact, nearly paracompact, weakly compact and weakly paracompact spaces. Keywords: Semiregularity; semiregular property; compact and paracompact spaces.

Completeness properties of the generalized compact-open topology on partial functions with closed domains

The primary goal of the paper is to investigate the Baire property and weak α -favorability for the generalized compact-open topology τ C on the space P of continuous partial functions f :A→Y with a closed domain A⊂X . Various sufficient and necessary conditions are given. It is shown, e.g., that ( P ,τ C ) is weakly α -favorable (and hence a Baire space), if X is a locally compact paracompact space and Y is a regular space having a completely metrizable dense subspace. As corollaries we get sufficient conditions for Baireness and weak α -favorability of the graph topology of Brandi and Ceppitelli introduced for applications in differential equations, as well as of the Fell hyperspace topology. The relationship between τ C , the compact-open and Fell topologies, respectively is studied; moreover, a topological game is introduced and studied in order to facilitate the exposition of the above results.

Monotone insertion of continuous functions

We consider the problem of inserting continuous functions between pairs of semicontinuous functions in a monotone fashion. We answer a question of Pan and in the process provide a new characterization of stratifiability. We also provide new proofs of monotone insertion results by Nyikos and Pan, and Kubiak. We then investigate insertion theorems for hedgehog-valued functions providing monotone versions of two theorems due to Blair and Swardson. From this we provide new characterizations involving hedgehogs of monotonically normal spaces, stratifiable spaces, normal, countably paracompact spaces, and perfectly normal spaces. The proofs are mostly geometric in nature.

On selection theorems with decomposable values

The main result of the paper asserts that for every separable measurable space $(T,\mathfrak F,\mu)$, where $\mathfrak F$ is the $\sigma$-algebra of measurable subsets of $T$ and $\mu$ is a nonatomic probability measure on $\mathfrak F$, every Banach space $E$ and every paracompact space $X$, each dispersible closed-valued mapping $F: x \rightsquigarrow L_1(T,E)$ of $X$ into the Banach space $L_1(T,E)$ of all Bochner integrable functions $u: T\to E$, admits a continuous selection. Our work generalizes some results of Gon\v carov and Tol'stonogov.

Free equivariant extensors

We prove for a finite group G and a compact metric G -space Y that the conditions (1) Y∈LC n−1∩C n−1 , and (2) Y∈G - AE(X) , for every normal n -dimensional space X endowed with a free numerable action of the group G , are equivalent. As a corollary we obtain: (A) For the space X endowed with a free action of the finite group G the conditions (1) the space X is normal, dim X⩽n and K(X;G)⩽n+1 ; (2) the space X is normal, dim X⩽n and K(X;G)<∞ ; (3) G∗⋯∗G∈G - AE(X) , are equivalent. (B) For a paracompact space X with a free action of the finite group G the inequality K(X;G)⩽ dim X+1 holds.

On finitistic spaces

Finitistic spaces form a natural class containing compact and finite-dimensional spaces. Introduced and investigated by fixed-point theorists, finitistic spaces found an application in cohomological dimension theory. In the paper, two characterizations of paracompact, finitistic spaces are proved. These characterizations allow to create a mechanism of generalizing results of finite dimension theory. As an application we obtain results on compact group actions on paracompact spaces which were previously known for compact Lie group actions.

The separated cellularity of a topological space and finite separation spaces

A separated cell A of a topological space X is a family of pairwise disjoint open sets with the property that, for any partition A= A 1∪ A 2 of A , ∪ A 1 and ∪ A 2 are completely separated. The separated cellularity sc( X) of X is the supremum of all cardinals of separated cells. A space X has finite separation if it has no infinite separated cells. With X compact, every separated cell of X has size < κ if and only if no regular closed subset has a continuous surjection onto βκ. It is shown that for normal, first-countable spaces finite separation is equivalent to sequential compactness. Compact finite separation spaces are examined, and compared to other classes of compact spaces which occur in the literature. It is shown that all dyadic spaces have finite separation. Likewise, every compact scattered space, every compact countably tight space, and every compact hereditarily paracompact space has finite separation.

Easier proofs of coloring theorems

We present a simpler proof of the known theorem that a fixed-point free homeomorphism on an n-dimensional paracompact space can be colored with n+3 colors.

Partition-complete paracompact [formula omitted]-spaces are preserved by closed maps

The theorem in the title is proved, together with some related results.

Local paraconvexity and local selection theorems

Theorems on the local extendability of selections for non-convex-valued maps of paracompact spaces into Banach spaces, i.e., infinite-dimensional analogs of the finite-dimensional Michael selection theorem are proved. We were able to obtain these results under an appropriate metric control of the local degree of nonconvexity on the valuesF(x), which naturally leads us to introduce the notion of equi-locally paraconvex families of sets. It is shown that all convex subsets of the integral curves of the differential equationy′=f(x,y) with a continuous right-hand sidef and the isometric images of such subsets form an equi-locally paraconvex family of subsets of a Euclidean space.

Approximations of Upper Semicontinuous Maps on Paracompact Spaces

Approximations of Upper Semicontinuous Maps on Paracompact Spaces

Normal spaces and fixed points of Čech-Stone extensions

We present two examples of nice normal spaces X having the property that for some fixed-point free homeomorphism on X its Čech-Stone extension has a fixed point. One of the spaces presented here is locally countable, locally compact, separable, normal, countably paracompact and weakly zero-dimensional. The other one is hereditarily normal and strongly zero-dimensional. Our construction of this example however requires the Continuum Hypothesis. Since for paracompact spaces with finite covering dimension every fixed-point free homeomorphism has a fixed-point free Čech-Stone extension, these results are “best possible”.

The Normality in Products with a Countably Compact Factor

AbstractIt is known that the product ω1 × X of ω1 with an M1-spacemay be nonnormal. In this paper we prove that the product κ × X of an uncountable regular cardinal κ with a paracompact semi-stratifiable space is normal iff it is countably paracompact. We also give a sufficient condition under which the product of a normal space with a paracompact space is normal, from which many theorems involving such a product with a countably compact factor can be derived.

A selection theorem for C-spaces

We prove a selection theorem which characterizes C-spaces among paracompact spaces and give two applications: 1. (1) if f: X → Y is a map between compact spaces with k-dimensional fibres and Y has property C, then f can be approximated by maps with fibres contained in k-dimensional polyhedra; 2. (2) if f: X → Y is an open map with infinite fibres between compact metric spaces, then the map P( f): P( X) → P( Y) between the spaces of probability measures is a trivial Q-bundle, provided that Y has property C.

Nonshrinking open covers and K. Morita's Third Conjecture

For every regular cardinal κ we construct a hereditarily normal, countably paracompact space X κ which has an increasing open cover W = { W μ : μ ϵ ω 1} such that W has no refinement by ⩽ κ closed sets. By work of Chiba, Przymusiński and Rudin, the existence of such a space proves a conjecture of Morita that σ-locally compact metrizable spaces are precisely the spaces whose product with every normal, countably paracompact space is normal. The spaces X κ are the first known examples of hereditarily normal ω 1-Dowker spaces.

Invariance of strong paracompactness under closed-and-open maps

Invariance of strong paracompactness under closed-and-open maps

A Selection Theorem for Strongly Regular Multivalued Mappings

We prove the following generalization of a theorem of Ferry concerning selections of strongly regular multivalued maps onto the class of paracompact spaces: Let Φ : X →(Z, ρ) be a map of a paracompact space X into a metric space (Z, ρ) whose values Φ(x) are complete subspaces of Z and absolute extensors (AE), for every x ∈ X. Suppose that Φ can be represented as Φ = Γ ○ ϕ, where ϕ : X → Y is a continuous singlevalued map of X onto some finite-dimensional paracompact space Y and Γ : Y →(Z, ρ) is a strongly regular map. Then for every closed subset A ⊂ X and every selection r : A → Z of the map Φ |A : A → Z, there exists an extension \(\hat r\): X → Z of r such that \(\hat r\) is a selection of the map Φ. We also prove a local version of this theorem.

There is a paracompact Q-set space in ZFC

There is a paracompact Q-set space in ZFC

Versions of Monotone Paracompactness

ABSTRACT:We consider the notion of monotone paracompactness introduced by Gartside and Moody. We show, in particular, that not all “possible” definitions for monotone paracompactness coincide, and we provide a new characterization of proto‐metrizability.

On expandable discrete collections

Expandability-type properties, which are more general than both normality and countable paracompactness, are considered. We give a common generalization of theorems related to closed mappings of normal or countably paracompact spaces onto a q-space. We also consider these properties in product spaces and prove theorems which are parallel to Katĕtov's theorem on hereditary normality in product spaces. An application to a space of closed subsets is given too.