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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?)

  • [1] I. Juhász, Cardinal functions in topology--ten years later, Math. Centre Tracts, Vol. 123, Math. Centrum, Amsterdam, 1980.
  • [2] I. Juhász and W. Weiss, The determination of 𝜋-weight by subspaces of singular cardinality, C. R. Math. Rep. Acad. Sci. Canada 3 (1981), no. 5, 257–260. MR 630941
  • [3] B. Šapirovskii, Canonical sets and character. Density and weight in compact spaces, Soviet Math. Dokl. 15 (1974), 1282-1287.
  • [4] -, 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.
  • [5] 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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