Paracompact Research Articles

Characterizations of Paracompactness and Subparacompactness Using Star Reducibility

Characterizations of Paracompactness and Subparacompactness Using Star Reducibility

Another dowker product

The continuum hypothesis implies that there are (normal) countably paracompact spaces X and Y such that the product X × Y is normal and not countably paracompact.

Property Q

Some properties of property Q are stated, some new results are proved and implications to totally metacompact and totally paracompact are obtained.

Ergodic properties of markov processes driven by a set of vector fields

In this paper we describe the decomposition of a connected paracompact manifold M into uniquely ergodic components of a Markov process driven by a set of vector fields. Let Xr, O≤r ≤p be smooth and...

A characterization for paracompactness

A characterization for paracompactness

Locally compact, countably paracompact spaces in the constructible universe

It is proven that under V = L, every locally compact, countably paracompact space is collectionwise normal with respect to compact sets. From this it can be shown that under V = L, submetacompact implies paracompact in the class of locally compact, countably paracompact spaces.

Abstract Riemannian stratifications

RIEMANNIAN STRATIFICATIONS GIACOMO MONTI BRAGADIN Vol. 133, No. 1 March 1988 PACIFIC JOURNAL OF MATHEMATICS Vol. 133, No. 1, 1988 ABSTRACT RIEMANNIAN STRATIFICATIONSRIEMANNIAN STRATIFICATIONS GIACOMO MONTI BRAGADIN Let A be an abstract stratification. Assume that every stratum X is a riemannian manifold. We first give some conditions, under which it is possible to endow A with a suitable metric extending a well known technique of riemannian manifolds. Then stronger conditions are introduced in order that the metric space A becomes a G-space. An abstract stratification with all the above conditions is called an abstract riemannian stratification. Whitney stratifications are, in a natural way, riemannian. Introduction. An abstract stratification is the disjoint union of a locally finite set of smooth manifolds, the strata, glued together according to certain rules (see [9], [13], [15], [16]). The basic tools of differential calculus, smooth functions, vector fields and differential forms, can be also defined for an abstract stratification by means of collections of tools. Classical results can be extended to abstract stratifications (e.g. de Rham's theorems, see [15], [8]) in this way. On the other hand a controlled approach seems to be useless when we take into account a riemannian structure: control conditions on the metric of the strata are too strict (bundle-like metrics, see [11], are not controlled in general). The purpose of this paper is to obtain weaker conditions in order that riemannian metrics of the strata glue together in an appropriate manner. In §1 sufficient conditions are given for extending the fundamental theorem of the riemannian geometry; in §2 a definition of compatibility among the riemannian structures of the strata is given and connections with the structure of G-space are studied. It is shown that the axioms of finite compactness, convexity, local prolongability are satisfied in mild hypotheses, whereas local uniqueness and existence of geodesic coordinates must be required (the former is an open question even in manifolds with boundary, see [1]). 0. Notation. 1. By a manifold we shall always mean a smooth manifold; topologically it is a connected, paracompact space. For any

On the Bitopological Extension of the Bing Metrization Theorem

AbstractBased on a Junnila's paracompactness characterization we give a definition of pairwise paracompact space which permits us to prove that a bitopological space is quasi-metrizable if, and only if, it is a pairwise developable and pairwise paracompact space. An easy consequence of this result is the biquasi-metric form of the Morita metrization theorem. We also give some results on open mappings and strong quasi-metrics.

Smooth extensions of Lipschitzian real functions

In this short note we point out that any Lipschitzian real function f f defined in a subset K K of a Banach space E E , with span ¯ (K) ≠ E \overline {{\text {span}}} {\text {(K)}} \ne {\text {E}} , can be extended to a surjective, open and Lipschitzian real function g g on E E in such a way that, for every r ∈ R r \in {\mathbf {R}} , the set g − 1 ( r ) {g^{ - 1}}(r) is arcwise connected. In fact, a more refined result is proved.

On the cardinality of a topology

Let o ( X ) o\left ( X \right ) denote the cardinality of topology of a space X X . I. Juhasz proves that o ( X ) ω = o ( X ) o{\left ( X \right )^\omega } = o\left ( X \right ) for regular hereditarily paracompact spaces. We prove it for more general classes of spaces.

Total Paracompactness of Real Go-Spaces

Total Paracompactness of Real Go-Spaces

On the product of a perfect paracompact space and a countable product of scattered paracompact spaces

On the product of a perfect paracompact space and a countable product of scattered paracompact spaces

On the Smale theorem in weak shape theory

We modify various lemmas from Dydak's paper on the Vietoris-Smale theorem to obtain more general results. We consider closed subsets X and Y of paracompact spaces M, N respectively, and a map F: M→ N such that F∥ X: X→ Y is closed and surjective and X= F -1( Y) to obtain the following result. (Theorem). If F -1(y)ϵAC n M( K) for each y ϵ Y and N is LC n+1 , then the morphism in-pro- π k[F]:π kU K M(X,x)→π kU K N(Y,y) induced by the morphism [F]:U K M(X,x)→U K N(Y,y) is an isomorphism of in-pro-Grp for k⩽ n and an epimorphism for k= n+1.

A new bitopological paracompactness

AbstractIn this paper we define generalization of paracompactness for bitopological spaces. (X, τ1, τ2) is Δ-pairwise paracompact if and only if every τi open cover admits a τ1 ∨ τ2 open refinement which is τ1 ∨ τ2 locally finite. Every quasimetric space (X, τp, τq) is Δ-pairwise paracompact. An analogue of Michael's characterization of regular paracompact spaces is proved for Δ-pairwise paracompact spaces.

REDUCTION TO GENERAL POSITION OF A MAPPING OF A ONE-DIMENSIONAL POLYHEDRON, DEPENDING CONTINUOUSLY ON A PARAMETER

This paper is devoted to a proof of the fact that by refining the triangulation of a one-dimensional polyhedron, one can approximate a given mapping of that polyhedron into Rk by a piecewise linear mapping having no more than a zero-dimensional violation of general position; and that all this can be carried out continuously with respect to a parameter running through a strongly paracompact space. Spaces of triangulations of one-dimensional simplexes are also investigated, and the structure of spaces of semilinear mappings of a one-dimensional polyhedron into Euclidean space is considered. Figures: 6. Bibliography: 6 titles.

Normality of product spaces and Morita's conjectures

It is well-known that Z is a perfectly normal space (normal P-space) if and only if X× Z is perfectly normal (normal) for every metric space X. Conversely, denote by Q (resp. N ) the class of all spaces X whose products X× Z with all perfectly normal spaces (all normal P-spaces) Z are normal. It is natural to ask whether Q and N necessarily coincide with the class M of metrizable spaces. Clearly, M⊂ N⊂ Q . We prove that first countable members of Q are metrizable and that under V= L the classes M and N coincide, thus giving a consistency proof of Morita's conjecture. On the other hand, even though Q contains non-metrizable members, it is quite close to M : the class Q is countably productive and hereditary, and all members X of Q are stratifiable and satisfy c( X)= l( X)= w( X). In particular, locally Lindelöf or locally Souslin or locally p-spaces in Q are metrizable. The above results immediately lead to the consistency proof of another Morita's conjecture, stating that X is a metrizable σ-locally compact space if and only if X× Y is normal for every normal countably paracompact space Y. No additional set-theoretic assumptions are necessary if X is first countable. In our investigation, an important role is played by the famous Bing examples of normal, non-collectionwise normal spaces. Answering Dennis Burke's question, we prove that products of two Bing-type examples are always non-normal.

Paracompactness in the class of closed images of GO-spaces

Paracompactness in the class of closed images of GO-spaces

A local strong 𝑈𝑉^{∞}-property of the homeomorphism groups of 𝑅^{∞}- (𝑄^{∞}-) manifolds

We will show in this note that if M M is an R ∞ - (or Q ∞ - ) {R^\infty }{\text { - (or }}{Q^\infty }{\text { - )}} manifold having the homotopy type of a finite complex, then the homeomorphism group of M M , endowed with the compact-open topology, has the local strong U V ∞ U{V^\infty } -property with respect to the classes of pseudo CW complexes C W ( C ) \mathcal {C}\mathcal {W}(\mathcal {C}) and C W ( M ) \mathcal {C}\mathcal {W}(\mathcal {M}) .

Separation in Countably Paracompact Spaces

We study the question "Are discrete families of points separated in countably paracompact spaces?" in the class of first countable spaces and the class of separable spaces.

Strong homology of inverse systems. III

In previous parts I and II of this paper [4] strong homology groups of inverse systems were introduced and studied. In this part III of the paper we define strong homology groups of inverse systems of pairs and show that they have suitable exactness and excision properties. As a consequence of these results the Steenrod-Sitnikov homology [1] for pairs ( X,A), where X is a paracompact space and A is a closed subset of X, is exact and satisfies the excision axiom.