Rad HAZU, Matematičke znanosti, Vol. 30 (2026), 169-176. \( \)

THE MAXIMUM CARDINALITY OF ESSENTIAL FAMILIES IN REGULAR OR NORMAL SPACES

Leonard R. Rubin

Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019, USA
e-mail:lrubin@ou.edu

Abstract.   Let \(X\) be a regular or normal space (\(\mathrm{T}_1\) not required) with infinite weight and \(\mathcal{C}\) be an essential family in \(X\). We will show that \({\operatorname{card}}\, \mathcal{C}\leq{\operatorname{wt}}\, X\). This implies that every essential family in a separable metrizable space is countable.

2020 Mathematics Subject Classification.   54F45

Key words and phrases.   Cardinality, essential family, normal space, regular space, weight

Full text (PDF) (free access)

https://doi.org/10.21857/yl4okfkgv9

References:

  1. M. Lynam and L. Rubin, Characterizing strong infinite-dimension, weak infinite-dimension, and dimension in inverse systems, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 29 (2025), 299–318.
    MathSciNet    CrossRef

  2. J. van Mill, Infinite-Dimensional Topology, North-Holland, Amsterdam, 1989.
    MathSciNet

  3. L. Rubin, R. Schori, and J. Walsh, New dimension-theory techniques for constructing infinite-dimensional examples, General Topology Appl. 10 (1979), 93–103.
    MathSciNet    CrossRef

  4. L. Rubin, Hereditarily strongly infinite-dimensional spaces, Michigan Math. J. 27 (1980), 65–73.
    MathSciNet    CrossRef

  5. L. Rubin, Non-compact hereditarily strongly infinite-dimensional spaces, Proc. Amer. Math. Soc. 79 (1980), 153–154.
    MathSciNet    CrossRef

  6. L. Rubin, Totally disconnected spaces and infinite cohomological dimension, Topology Proc. 7 (1982), 157-166.
    MathSciNet

  7. L. Rubin, More compacta of infinite cohomological dimension, Contemp. Math. 44 (1985), 221–226.
    MathSciNet    CrossRef

  8. K. Sakai, Geometric Aspects of General Topology, Springer Monographs in Mathematics, Springer, Tokyo, 2013.
    MathSciNet    CrossRef

  9. J. Walsh, An infinite dimensional compactum containing no \(n\)-dimensional \((n\geq1)\) subsets, Topology 18 (1979), 91–95.
    MathSciNet    CrossRef
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