Paracompact Hausdorff Space Research Articles

Comparison of persistent singular and Čech homology for locally connected filtrations

We show that the interleaving distance between the persistent singular homology and the persistent Čech homology of a homologically locally connected filtration consisting of paracompact Hausdorff spaces is 0.

Fixed point free actions of spheres and equivariant maps

The concept of index and co-index of a paracompact Hausdorff space X equipped with free involutions relative to the antipodal action on spheres were introduced by Conner and Floyd [2]. In this paper, we extend the notion of index and co-index for free G-spaces X, where X is a finitistic space and G=S1 (with complex multiplication) and G=S3 (with quaternionic multiplication). We prove that the index of X is less than or equal to the mod 2 cohomology index of X. We also compute the ring cohomology of the orbit space X/G, where G=S1 or G=S3 and X is a finitistic space whose cohomology ring is the same as the product of spheres Sn×Sm,1≤n≤m. Using these cohomological calculations, we obtain some Borsuk-Ulam type results.

Borsuk–Ulam theorem for filtered spaces

Abstract Let X and Y be pathwise connected and paracompact Hausdorff spaces equipped with free involutions T : X → X {T:X\to X} and S : Y → Y {S:Y\to Y} , respectively. Suppose that there exists a sequence ( X i , T i ) ⁢ ⟶ h i ⁢ ( X i + 1 , T i + 1 ) for ⁢ 1 ≤ i ≤ k , (X_{i},T_{i})\overset{h_{i}}{\longrightarrow}(X_{i+1},T_{i+1})\quad\text{for }% 1\leq i\leq k, where, for each i, X i {X_{i}} is a pathwise connected and paracompact Hausdorff space equipped with a free involution T i {T_{i}} , such that X k + 1 = X {X_{k+1}=X} , and h i : X i → X i + 1 {h_{i}:X_{i}\to X_{i+1}} is an equivariant map, for all 1 ≤ i ≤ k {1\leq i\leq k} . To achieve Borsuk–Ulam-type theorems, in several results that appear in the literature, the involved spaces X in the statements are assumed to be cohomological n-acyclic spaces. In this paper, by considering a more wide class of topological spaces X (which are not necessarily cohomological n-acyclic spaces), we prove that there is no equivariant map f : ( X , T ) → ( Y , S ) {f:(X,T)\to(Y,S)} and we present some interesting examples to illustrate our results.

Čech systems that induce approximate inverse systems

In previous work we showed that if a paracompact Hausdorff space contains a nontrivial component, then none of the Čech systems of the nerves of its open covers can be an approximate (inverse) system. This extended a theorem of the first author who, in answering a question of S. Mardešić, proved this result in the restricted case that the given Hausdorff space was arc-like (and hence is nontrivial, compact and connected). We will demonstrate that the same is true for a nonempty discrete space: none of the Čech systems of the nerves of its open covers can be an approximate (inverse) system. In our main theorem, we are going to show in contradistinction that the completely opposite phenomenon occurs in the case that the space in question is strongly 0-dimensional (meaning that it has covering dimension 0) and has a limit point by proving that at least one of its Čech systems can support the structure of an approximate system. We will also show that such a space can be written as the inverse limit, in the classical sense, of an inverse system of discrete spaces.

An ideal version of the star-C-Hurewicz covering property

A space X is said to have the star-C-I-Hurewicz (SCIH) property if for each sequence (Un : n ? N) of open covers of X there is a sequence (Kn : n ? N) of countably compact subsets of X such that for each x ? X, {n ? N : x ? St(Kn,Un)} ? I, where I is a proper admissible ideal of N. We investigate the relationships among the SCIH and related properties. We study the topological properties of the SCIH property. This paper generalizes several results of [21, 24] to the larger class of spaces having the SCIH property. The star-C-I-Hurewicz game is introduced. It is shown that, in a paracompact Hausdorff space X, if TWO has a winning strategy in the SCIH game on X, then TWO has a winning strategy in the I-Hurewicz game on X.

Ideal analogues of some variants of the Hurewicz property

In this paper, we continue the study on the ideal analogues of several variations of the Hurewicz property introduced by Das et al. [4, 6, 7] for example, the I-Hurewicz (IH), the star I-Hurewicz (SIH), the weakly I-Hurewicz (WIH) and the weakly star-I-Hurewicz (WSIH). It is shown that several implications in the relationship diagram of their concepts are reversible under certain conditions, for instance; (1) If a paracompact Hausdorff space has the WSIHproperty, then it has theWIHproperty. (2) If the complement of dense set has the IHproperty, then theWIHproperty implies the IHproperty and (3) If the complement of dense set has the SIH property, then the WSIH property implies the SIH property. In addition, we introduce the ideal analogues of some new variations of the Hurewicz property called the mildly I-Hurewicz and the star K-I-Hurewicz properties and explore their relationships with other variants of the I-Hurewicz property. We also study the preservation properties under certain mappings.

Proximately chain refinable functions

We define proximately chain refinable functions as a generalization of refinable maps and investigate some of their properties for Hausdorff paracompact spaces. We prove that the proximate fixed point property is preserved by proximate near homeomorphisms in paracompact Hausdorff spaces. This generalizes a previous result of E. Grace.

A note on base-paracompact and monotone base-covering properties

In the first part of this note we show that if X is a paracompact Hausdorff space and there is a locally compact closed subspace Y of X such that for every x ? X \ Y there exists an open neighborhood Ox of x in X such that O?x is base-paracompact, then the space X is base-paracompact. In the second part of this note we introduced notions of monotonically base-paracompact (base-metacompact, base-Lindel?f) and discuss some of their properties.

Insertion of poset-valued maps with the way-below and -above relations

In this paper, with the way-below relation ≪ and the way-above relation ≪d, we give a poset-valued insertion theorem by a pair of semi-continuous maps. Moreover, we show a poset-valued insertion theorem by a continuous map as follows: Let X be a paracompact Hausdorff space, P a bi-bounded complete, bicontinuous, pathwise connected, topological poset. For each upper semi-continuous map f:X→P with a lower bound and each lower semi-continuous map g:X→P, if 〈f,g〉 has interpolated points pointwise, there exists a continuous map h:X→P such that f≪dh≪g. A generalized Dowker–Katětov's insertion theorem, by using the way-below and -above relations on bicontinuous posets P, is also given.

A note on productively paracompact spaces

Let X be a paracompact Hausdorff space which is paracompact at infinity (i.e., βX∖X is paracompact). We show that X×Y is paracompact for every paracompact space Y if, and only if, Player I has a winning strategy in Telgársky's game G(DC,X).

Thick Spanier groups and the first shape group

We develop a new route through which to explore $\ker \Psi_{X}$, the kernel of the $\pi_{1}$-shape group homomorphism determined by a general space $X$, and establish, for each locally path connected, paracompact Hausdorff space $% X$, $\ker\Psi_{X}$ is precisely the Spanier group of $X$.

A Fixed Point Theorem and Equivariant Points for Set-valued Mappings

We give a proof of a coincidence theorem for a Vietoris mapping and a compact mapping and prove the Lefschetz fixed point theorem for the class of admissible mappings which contains upper semi-continuous acyclic mappings. When a source space is a paracompact Hausdorff space with a free involution and a target space is a closed topological manifold with an involution, the existence of equivariant points is proved for the class of admissible mappings under some conditions. When a source space is a Poincaré space with a finite covering dimension, the covering dimension of the set of equivariant points is determined.

Geometric cycles, index theory and twisted K-homology

We study twisted \mathrm{Spin}^c -manifolds over a paracompact Hausdorff space X with a twisting α : X → K(ℤ, 3) . We introduce the topological index and the analytical index on the bordism group of α -twisted \mathrm{Spin}^c -manifolds over (X, α) , taking values in topological twisted K-homology and analytical twisted K-homology respectively. The main result of this article is to establish the equality between the topological index and the analytical index for closed smooth manifolds. We also define a notion of geometric twisted K-homology, whose cycles are geometric cycles of (X, α) analogous to Baum–Douglas's geometric cycles. As an application of our twisted index theorem, we discuss the twisted longitudinal index theorem for a foliated manifold (X, F) with a twisting α : X → K(ℤ, 3) , which generalizes the Connes–Skandalis index theorem for foliations and the Atiyah–Singer families index theorem to twisted cases.

Generalized universal covering spaces and the shape group

If a paracompact Hausdorff space $X$ admits a (classical) universal covering space, then the natural homomorphism $\varphi:\pi_1(X)\rightarrow \check{\pi}_1(X)$ from the fundamental group to its first shape homotopy group is an isomorphism. We present a

Strong bonding homology and cohomology

In this paper we define new strong bonding homology and strong bonding cohomology groups of an arbitrary Tychonoff (i.e., completely regular) topological space X with coefficients in an Abelian group G, denoted by H ¯ m b d ( X ; G ) and H ¯ b d m ( X ; G ) , respectively, m ∈ Z . These groups connect strong homology groups with compact supports H ¯ m c ( X ; G ) and strong homology groups H ¯ m ( X ; G ) of X and the Alexander–Spanier cohomology groups H m ( X ; G ) and strong cohomology groups H ¯ m ( X ; G ) of X. We show that for paracompact Hausdorff spaces X, which are homologically locally connected ( hlc G ∞ ) with respect to strong homology with compact supports, H ¯ m b d ( X ; G ) are trivial. As a consequence, H ¯ m c ( X ; G ) ≈ H ¯ m ( X ; G ) , for m ∈ Z . Similarly, for strongly cohomologically locally connected ( sclc G ∞ ) spaces X with respect to strong cohomology, H ¯ b d m ( X ; G ) are trivial and thus, H m ( X ; G ) ≈ H ¯ m ( X ; G ) , for m ∈ Z . It is also shown that the paracompactness condition is indispensable. The subject of the paper is interesting because of close connections of local and global properties of X. It is proved that any resolution p ̲ : X → X ̲ = ( X λ , p λ λ ′ , Λ ) of X consisting of paracompact Hausdorff spaces X λ , which are hlc G ∞ and at the same time are homologically locally connected ( HLC G ∞ ) with respect to singular homology, i.e., X λ ∈ hlc G ∞ ∩ HLC G ∞ , λ ∈ Λ , determines the strong homology groups H ¯ m ( X ; G ) , m ∈ Z , even if it is not an ANR - resolution of X. Another consequence of this theory is the following nontrivial result: If ( X , A ) is a closed pair of hereditarily paracompact Hausdorff spaces such that X , A ∈ hlc G m (respectively, X , A ∈ sclc G m and X , A ∈ clc G m ), then the quotient space X / A ∈ hlc G m (respectively, X / A ∈ sclc G m and X / A ∈ clc m ), where hlc G m , shlc m and clc G m are the homological local m-connectedness with respect to strong homology with compact supports, the strongly cohomological local m-connectedness with respect to strong cohomology and the cohomological local m-connectedness with respect to Čech cohomology, respectively. Two examples of separable metric spaces X are exhibited such that H ¯ m c ( X ; G ) ≉ H ¯ m ( X ; G ) , which solves a problem from [S. Mardešić, Strong Shape and Homology, Springer-Verlag, Berlin, Heidelberg, New York, 2000, p. 457].

Continuity of the cone functor

This paper looks at the continuity of a class of functors that includes as special cases the cone functor Γ and the suspension functor Σ. The purpose of the paper is to highlight a sufficient topological property satisfied by paracompact Hausdorff spaces, which guarantees the continuity. Since paracompact Hausdorff spaces constitute a large class of topological spaces studied in mathematics, we regard this as a strong result. The impetus for the present paper came from a certain confusion encountered in the book General Topology and Homotopy Theory by James [General Topology and Homotopy Theory, Springer-Verlag, 1984]. We give a counterexample showing that the cone functor is not continuous in the category of regular spaces, as stated in the book. Although the results in this paper concern functors, the emphasis of this paper is more on classical point set topology.

Intersection properties for coverings of G-spaces

The Ljusternik–Schnirelmann–Borsuk theorem for antipodal maps on the sphere can be stated as an intersection property of coverings of the sphere. We generalized this theorem to free finite-group actions on paracompact Hausdorff spaces and discuss how this result might be improved. We illustrate our results by a free action of the Klein four-group on the torus.

Dieudonné completeness of a space of closed subsets and extensions of an open-closed map

We prove: if X is a paracompact Hausdorff space, then the space 2 X of all nonempty closed subsets of X with the Vietoris topology is Dieudonné complete. If f: X → Y is an open-closed continuous onto map and 2 μ(X) is Dieudonné complete, then the extension μ( f): μ( X) → μ( Y) is open-closed, where μ( X) is the Dieudonné completion of X.

Variations on a theorem of lusternik and schnirelmann

A fixed-point free map f: X → X is said to be colorable with k colors if there exists a closed cover β of X consisting of k elements such that C∩ f( C) = φ for every C in β. It is shown that each fixed-point free involution of a paracompact Hausdorff space X with dim X ≤ n can be colored with n + 2 colors. Each fixed-point free homeomorphism of a metrizable space X with dim X ≤ n is colorable with n + 3 colors. Every fixed-point free continuous selfmap of a compact metrizable space X with dim X ≤ n can be colored with n + 3 colors

Paracompactness on L-fuzzy topological spaces

Abstract In this paper, we introduce a new kind of paracompactness (called ∗-paracompactness) on L-fuzzy topological spaces and obtain some of its properties: such as that it is a good extension, hereditary with respect to closed subsets, and that the Cartesian product of an N-compaact space and a ∗-paracompact space is a ∗-paracompact space, as well as that a *-paracompact Hausdorff space is both a regular and normal space.