Some aspects of the theory of dimension of completely regular spaces. (English) Zbl 1055.54015
Let \(\mathcal S\) denote the class of separable metrizable spaces. For any \(X\) from \(\mathcal S\), the three dimension functions, dim, Ind and ind have a common value, say \(d(X)\), which lies in \(N'=\{\dots,-1,0,1,2,\dots\}\cup\{\infty\}\) [see W. Hurewicz and H. Wallman, “Dimension theory” (1941; Zbl 0060.39808)]. It is also true that d satisfies certain properties (e.g., the logarithmic and additive properties). The authors list eight such properties and then three questions one might ask about such a dimension function enjoying some, or all, of them. In this paper, it is the third question, denoted (\(\gamma\)), that they consider, so let us state it here.
Question: Do there exist \(N'\)-valued topologically invariant functions defined on classes of spaces wider than the class \(\mathcal S\) and having the above-mentioned eight properties simultaneously? In particular, does a function \(d\) of that type exist on the class of all Tychonoff spaces? If not, then what can one say about the existence of \(N'\)-valued topologically invariant functions defined on the given wider class of spaces and having simultaneously the properties from any preassigned subsystem of these eight?
The aim of the paper is to consider this Question for the class \(\mathcal T\) of all completely regular spaces. The authors list eight different classes of functions which correspond respectively to the eight properties that such a dimension function \(d\) might have. The problem then is to select some (possibly proper) subset of these eight classes and then to ask if there is a function \(d\) common to these classes, i.e., to ask if this is “realizable”. Such a function \(d\) is (awkwardly) called a “realizator”. Of course for the class \(\mathcal S\), everything works. But the question of whether this is true with the expanded universe \(\mathcal T\) of completely regular spaces can have different answers.
The third section of the paper is dedicated to the nonrealizable cases. In at least one case, Martin’s axiom is employed. In the fourth section some positive results are given along with specific functions \(d\) which are realizators. Section 5 consists of a table of known and unknown results, and Section 6 contains some questions.
Question: Do there exist \(N'\)-valued topologically invariant functions defined on classes of spaces wider than the class \(\mathcal S\) and having the above-mentioned eight properties simultaneously? In particular, does a function \(d\) of that type exist on the class of all Tychonoff spaces? If not, then what can one say about the existence of \(N'\)-valued topologically invariant functions defined on the given wider class of spaces and having simultaneously the properties from any preassigned subsystem of these eight?
The aim of the paper is to consider this Question for the class \(\mathcal T\) of all completely regular spaces. The authors list eight different classes of functions which correspond respectively to the eight properties that such a dimension function \(d\) might have. The problem then is to select some (possibly proper) subset of these eight classes and then to ask if there is a function \(d\) common to these classes, i.e., to ask if this is “realizable”. Such a function \(d\) is (awkwardly) called a “realizator”. Of course for the class \(\mathcal S\), everything works. But the question of whether this is true with the expanded universe \(\mathcal T\) of completely regular spaces can have different answers.
The third section of the paper is dedicated to the nonrealizable cases. In at least one case, Martin’s axiom is employed. In the fourth section some positive results are given along with specific functions \(d\) which are realizators. Section 5 consists of a table of known and unknown results, and Section 6 contains some questions.
Reviewer: Leonhard R. Rubin (Norman)
MSC:
| 54F45 | Dimension theory in general topology |
Citations:
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