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compact spaces; $G_\delta $-sets; resolvability
It is well-known that compacta (i.e. compact Hausdorff spaces) are maximally resolvable, that is every compactum $X$ contains $\Delta(X)$ many pairwise disjoint dense subsets, where $\Delta(X)$ denotes the minimum size of a non-empty open set in $X$. The aim of this note is to prove the following analogous result: Every compactum $X$ contains $\Delta_\delta(X)$ many pairwise disjoint $G_\delta$-dense subsets, where $\Delta_\delta(X)$ denotes the minimum size of a non-empty $G_\delta$ set in $X$.
[1] Čech E., Pospíšil B.: Sur les espaces compacts. Publ. Fac. Sci. Univ. Masaryk 258 (1938), 1–14. Zbl 0019.08903
[2] Comfort W.W., Garcia-Ferreira S.: Resolvability: A selective survey and some new results. Topology Appl. 74 (1996), 149–167. DOI 10.1016/S0166-8641(96)00052-1 | MR 1425934 | Zbl 0866.54004
[3] El'kin A.G.: Resolvable spaces which are not maximally resolvable. Vestnik Moskov. Univ. Ser. I Mat. Meh. 24 (1969), no. 4, 66–70. MR 0256331 | Zbl 0243.54018
[4] Juhász I.: Cardinal functions in topology – 10 years later. Mathematical Centre Tracts, 123, Mathematisch Centrum, Amsterdam, 1980.
[5] Juhász I.: On the minimum character of points in compact spaces. in: Proc. Top. Conf. (Pécs, 1989), 365–371, Colloq. Math. Soc. János Bolyai, 55, North-Holland, Amsterdam, 1993. MR 1244377 | Zbl 0798.54005
[6] Juhász I., Szentmiklóssy Z.: Convergent free sequences in compact spaces. Proc. Amer. Math. Soc. 116 (1992), 1153–1160. DOI 10.2307/2159502 | MR 1137223 | Zbl 0767.54002
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