Two results concerning cardinal functions on compact spaces
Authors:
I. Juhász and Z. Szentmiklóssy
Journal:
Proc. Amer. Math. Soc. 90 (1984), 608610
MSC:
Primary 54A25; Secondary 54D30
DOI:
https://doi.org/10.1090/S00029939198407334141
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 $\kappa$.

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), 12821287. , 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, NorthHolland, Amsterdam, 1980, pp. 10551086.
 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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© Copyright 1984
American Mathematical Society