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Homogeneous subspaces of products of extremally disconnected spaces. (English) Zbl 07286530
In this exceptional paper the author establishes several results that extend or relate to results of Arhangel’skii, Frolík, Kunen, and others concerning homogeneity, products, extremally disconnected spaces (and the more general notion of a \(\beta\omega\)-space), and subspaces thereof. Maximal homogeneity, countable compactness, and pseudocompactness are also studied.
A. V. Arkhangel’skij [C. R. Acad. Sci., Paris, Sér. A 265, 822–825 (1967; Zbl 0168.43702)] proved that any compact subset of an extremally disconnected topological group is finite. In addition, Z. Frolík proved in [Commentat. Math. Univ. Carol. 8, 757–763 (1967; Zbl 0163.44503)] that homogeneous extremally disconnected compact spaces are finite. In the present paper the author obtains a simultaneous generalization of these two theorems by showing that all compact subsets of homogeneous extremally disconnected spaces are finite. In fact this result can be strengthened.
The main results include:
1. Any compact subspace of an homogeneous \(\beta\omega\)-space is finite (this is more general than what is mentioned in the abstract),
2. Under CH, if \(Y\) is a \(\beta\omega\)-space and \(X\subseteq Y^\omega\) is homogeneous, then every compact subspace of \(X\) is metrizable,
3. Under CH, if \(Y\) is a \(\beta\omega\)-space , and \(X\) is a homogeneous subspace of \(Y^n\) for some \(n\in\omega\), then any compact subspace of \(X\) is finite,
4. Let \(Y\) be an \(F\)-space, and let \(X\subseteq Y^3\) be a homogeneous space. Then any compact subspace of \(X\) is finite, and
5. If \(Y\) is an \(F\)-space, \(n\in\omega\), and \(X\subseteq Y^n\) is a homogeneous compact space, then \(X\) is finite.
The Keisler-Rudin order is used heavily in these results, as well as extensive notation introduced by the author involving sequences, sequences of open sets, and \(p\)-limits.
It is asked in Question 1 whether CH can be removed from 2 or 3 above, perhaps under the stronger assumptions that \(Y\) is an \(F\)-space or an extremally disconnected space. Question 2 is as follows: Let \(Y\) be an \(F\)-space, and let \(X\subseteq Y^4\) be a homogeneous space. Is it true that any compact subspace of \(X\) is finite?
54B10 Product spaces in general topology
54G05 Extremally disconnected spaces, \(F\)-spaces, etc.
54B05 Subspaces in general topology
54E99 Topological spaces with richer structures
54H99 Connections of general topology with other structures, applications
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