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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Covering dimension in finite-dimensional metric spaces


Author: Japheth Hall
Journal: Proc. Amer. Math. Soc. 41 (1973), 274-277
MSC: Primary 54F45
MathSciNet review: 0322828
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Abstract: Let $ P:{2^V} \to {2^V}$ be a structure in a topological space $ V$ such that $ P(\emptyset ) = \emptyset ,P(\{ x\} ) = \{ x\} $ if $ x \in V$, and $ P(Z)$ is closed if $ Z \subseteq V$. If $ G$ is a covering of $ V$, let $ {G_x} = \{ X \in G:x \in X\} $. If $ X$ is a set and $ Y$ is a set, let $ \vert X\vert$ denote the cardinal number of $ X$ and $ X - Y = \{ x \in X:x \notin Y\} $. Let $ n$ be an integer such that $ n \geqq - 1.{\dim _P}V$ is defined as follows: $ {\dim _P}V = - 1$ if $ V = \emptyset $. If $ V \ne \emptyset $, then $ {\dim _P}V = n$ if (1) for each finite open covering $ G$ of $ V$, there is an open refinement $ H$ of $ G$ such that $ \vert{H_x}\vert \leqq n + 1$ if $ x \in V$; and (2) there is a finite open covering $ G$ of $ V$ such that if $ H$ is an open refinement of $ G$, then $ \vert{H_x}\vert \geqq n + 1$ for some $ x \in V$. We say that $ P$ has property ($ ( \ast )$) if for each nonempty open $ Y \subseteq V$ and each $ X \subseteq V$ such that $ P(X) \ne V$ and $ x \notin P(X - \{ x\} )$ whenever $ x \in X$ and each $ x \in [V - P(X)],[Y - P(X)] \cap P(X \cup \{ x\} ) \ne \emptyset $. THEOREM. If $ V$ is a metric space, $ P$ has property ($ ( \ast )$), $ B \subseteq V,B$ is finite,$ P(B) = V$ and $ x \notin P(B - \{ x\} )$ if $ x \in B$, then $ {\dim _P}V = \vert B\vert - 1$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1973-0322828-2
PII: S 0002-9939(1973)0322828-2
Keywords: Covering dimension, covering dimension relative to structures in topological spaces, finite-dimensional metric spaces
Article copyright: © Copyright 1973 American Mathematical Society