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An approach to covering dimensions. (English) Zbl 0834.54019
Several kinds of dimensions of topological spaces are introduced by means of a device used in some questions of entropy theory. There are eleven dimensions being introduced: \(\Gamma\)-dim, dim, \(\gamma\)-dim, \(\Gamma\)- Dim, Dim, \(\gamma\)-Dim, \(\Gamma\)-dim*, dim*, \(\gamma\)-dim*, \(p\)-dim, \(p\)- Dim. Many of these dimensions coincide on normal spaces and on quasi- discrete ones. The properties of these dimensions are examined. Some of the results are as follows.
For a hereditarily normal space, all the dimensions coincide. For every space \(S\), \(\varphi (S)\leq \varphi (X)+ \varphi (Y)+1\), if \(S= X\cup Y\) \((\varphi= \gamma\)-dim, \(\gamma\)-Dim). For every finite poset \(S\), \(p \text{-Dim } S=\gamma \text{-Dim } S\). For some dimensions monotonicity (\(\Gamma\)-Dim, Dim \(\gamma\)-Dim), addition formula (\(\gamma\)-dim, \(\gamma\)-Dim) and sum formula (Dim, \(\gamma\)-Dim) are proved. The partition dimension \(p\)-Dim coincides with the Čech-Lebesgue dimension on normal spaces and with the height dimension on posets. If \(S\) is a poset then, for any \(n\in \mathbb{N}\), \(\varphi (S)\leq n\) iff \(\varphi (X)\leq n\) for every relatively bounded (from below) finite \(T\subset S\) (\(\varphi= \Gamma\)-dim, dim, \(\gamma\)-dim). If \(S\) is a nonvoid normal space, then the Čech-Lebesgue dimension of \(S\) is equal to the least \(n\in \mathbb{N}\) such that every rigged finite open cover of \(S\) is strongly refined by some continuous \(f: S\to T\), where \(T\) is a finite poset with height dimension not exceeding \(n\). A number of open questions is stated.
54F45 Dimension theory in general topology
06A06 Partial orders, general
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