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Observations on spaces with property \((DC(\omega_1))\). (English) Zbl 1398.54009

Following S. Ikenaga, a topological space \(X\) is said to have property \((DC(\omega_{1}))\) if it has a dense subspace \(D\) such that every uncountable subset of \(D\) has a limit point in \(X\). Clearly, all separable spaces as well as all spaces with countable extent possess this property. The authors prove that the cardinality of a Hausdorff space \(X\) with property \((DC(\omega_{1}))\) is bounded by \(\mathfrak{c}\) (i.e., \(|X| \leq \mathfrak{c}\)) provided that \(X\) satisfies one of the following additional conditions: (1) \(X\) is normal and has a rank-2-diagonal in the sense of A. V. Arhangel’skii and R. Z. Buzyakova [Commentat. Math. Univ. Carol. 47, No. 4, 585–597 (2006; Zbl 1150.54335)]; (2) \(X\) is a perfect space with a rank-2-diagonal; (3) \(X\) is a regular perfect space with countable tightness or (4) \(X\) has a rank-3-diagonal in the sense of [loc. cit.]. In a recent paper, they have identified a further condition: (5) \(X\) is a first countable space with a \(G_{\delta}\)-diagonal [the authors, Commentat. Math. Univ. Carol. 58, No. 1, 131–135 (2017; Zbl 1463.54070)]. If \(X\) is merely a regular perfect space with property \((DC(\omega_{1}))\), then \(|X| \leq 2^{\mathfrak{c}}\).

MSC:

54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
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Full Text: Euclid