\(D\)-spaces, \(aD\)-spaces and finite unions. (English) Zbl 1162.54012
Summary: We prove that if a space \(X\) is the union of a finite family of strong \(\Sigma \)-spaces, then \(X\) is a \(D\)-space. This gives a positive answer to a question posed by Arhangel’skii in [A. V. Arhangel’skii, Proc. Am. Math. Soc. 132, 2163–2170 (2004; Zbl 1045.54009)]. We also obtain results on \(aD\)-spaces and finite unions. These results improve the corresponding results in [A. V. Arhangel’skii and R. Z. Buzyakova, Commentat. Math. Univ. Carolin. 43, No. 4, 653–663 (2002; Zbl 1090.54017)] and [Liang-Xue Peng, Topology Appl. 154, No. 2, 469–475 (2007; Zbl 1110.54014)].
References:
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