Covering dimension in finite-dimensional metric spaces

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 $|X|$ 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 $|{H_x}| \leqq n + 1$ 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 $|{H_x}| \geqq n + 1$ 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 ${\dim _P}V = |B| - 1$.