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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.
##### MSC:
 54F45 Dimension theory in general topology 06A06 Partial orders, general
##### Keywords:
Čech-Lebesgue dimension
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