On Quasi-Order Spaces, Normality, and Paracompactness

Abstract

IntroductionThe aim of this paper is a clarification and extension of the work of V. I. Ponomarev ([8]) on projective spectra. Ponomarev characterizes paracompact Hausdorff spaces as the spectral limits of inverse systems of simplicial complexes. In this paper the spectral limit of an inverse system of simplicial complexes is replaced by the true limit in the categorical sense of an inverse system of certain quasi-order spaces called special quasi-order spaces. In this way a characterization of the wider class of paracompact regular spaces is obtained (Theorem 3.2). In addition there is a characterization of normal regular spaces and the strongly paracompact regular spaces are identified amongst the paracompact spaces. The main technique in the proof of the theorems is the association of two 'dual' quasi-orders with each covering of a space. These are studied in § 2. Special quasi-order spaces have a very simple structure particularly from the point of view of dimension theory and the author will exploit this in subsequent work on the dimension of paracompact spaces. The definition of a special quasi-order space is given in § 2. Some properties of normal, paracompact and strongly paracompact spaces are described in § 1; these are easily obtained from well-known results ([7], [8], [9]). In the final paragraph the relation of this work to Ponomarev's theory of spectra is examined.The author wishes to thank the referee for helpful comments on an earlier version of this paper. Canonical coverings, normality and paracompactnessA canonical covering of a topological space X is a locally finite closed covering {F A } AeA of X such that for each X e A, F x = G x where {G x } is a collection of mutually disjoint open sets. Finite canonical coverings were introduced by P. S. Alexandrav and V. I. Ponomarev ([3]). Canonical coverings have been considered subsequently by Ponomarev ([8]) and others. For lack of a suitable reference we derive in the following three lemmas those properties of canonical coverings we shall need later.

Closed Covers, Dimension, and Quasi-Order Spaces

“…A quasi-order space X will be called a special quasi-order space if, for each x e X, every subset of x consisting of inequivalent elements of X is finite. This is a strengthening of the definition made in [11]. The main results of that paper would hold with this definition.…”

“…Also, X is said to be the regular limit of the inverse system X if for each open set U of X and x e U, for some a E Q, there exists a closed covering ^ of X a such that St(a;,7r a~1 (J^)) c: U. When each X a is a special quasi-order space, these definitions coincide with those made in [11]. In that paper it was proved that a topological space is paracompact and regular if and only if it is the regular uniform limit of an inverse system of special quasi-order spaces.…”

“…It follows from Lemma 1.3 of [11] that any directed set of canonical coverings can be given the structure of a strongly directed family of coverings. The terms regular, uniform, et cetera, are applied to families of coverings in the obvious way.…”

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In [11], Pears characterizes paracompact spaces as inverse limits of quasi-order spaces. Here the techniques of [11] are applied to the study of dimension theory. In §1, the basic properties of inverse systems of quasi-order spaces are developed.

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Topological spaces and Abian structures

“…open. Spaces with this property have been considered by Pears [5] who calls them quasi-ordered spaces.…”