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Two results concerning cardinal functions on compact spaces

Authors: I. Juhász and Z. Szentmiklóssy
Journal: Proc. Amer. Math. Soc. 90 (1984), 608-610
MSC: Primary 54A25; Secondary 54D30
MathSciNet review: 733414
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Abstract: We show that for $X$ compact ${T_2}:\left ( i \right )d\left ( X \right ) \leqslant s\left ( X \right ) \cdot \hat F\left ( X \right )$; (ii) if the pair $\left ( {\kappa ,\hat F\left ( X \right )} \right )$ is a caliber of $X$ then $\pi \left ( X \right ) < \kappa$. These strengthen results of Šapirovskii from [3 and 5], respectively. Moreover, (i) settles a problem raised in [2] implying that there are no compact ${T_2}$ $\kappa$-examples for any singular cardinal $\kappa$.

References [Enhancements On Off] (What's this?)

    I. Juhász, Cardinal functions in topology—ten years later, Math. Centre Tracts, Vol. 123, Math. Centrum, Amsterdam, 1980.
  • I. Juhász and W. Weiss, The determination of $\pi $-weight by subspaces of singular cardinality, C. R. Math. Rep. Acad. Sci. Canada 3 (1981), no. 5, 257–260. MR 630941
  • B. Šapirovskii, Canonical sets and character. Density and weight in compact spaces, Soviet Math. Dokl. 15 (1974), 1282-1287. ---, Special types of embeddings in Tychonoff cubes. Subspaces of $\Sigma$-products and cardinal invariants, Topology, Vol. II (Proc. Fourth Colloq., Budapest, 1978), Colloq. Math. Soc. János Bolyai, Vol. 23, North-Holland, Amsterdam, 1980, pp. 1055-1086.
  • B. È. Shapirovskiĭ, Cardinal invariants in compacta, Seminar on General Topology, Moskov. Gos. Univ., Moscow, 1981, pp. 162–187 (Russian). MR 656957

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Article copyright: © Copyright 1984 American Mathematical Society