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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The density character of unions


Authors: W. W. Comfort and Teklehaimanot Retta
Journal: Proc. Amer. Math. Soc. 65 (1977), 155-158
MSC: Primary 54A25
MathSciNet review: 0445441
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Abstract: We consider only completely regular, Hausdorff spaces. Responding to a question of R. Levy and R. H. McDowell [Proc. Amer. Math. Soc. 49 (1975), 426-430] we show that for $ \omega \leqslant \gamma \leqslant {2^{{2^\omega }}}$ there is a separable space equal to the (appropriately topologized) disjoint union of $ \gamma $ copies of the ``Stone-Čech remainder'' $ \beta N\backslash N$. More generally, denoting density character by d and weight by w, we prove this

Theorem. The following statements about infinite cardinal numbers $ \gamma $ and $ \alpha $ are equivalent: (a) $ {2^\alpha } \leqslant {2^\gamma }$ and $ \gamma \leqslant {2^{{2^\alpha }}}$; (b) For every family $ \{ {X_\xi }:\xi < \gamma \} $ of spaces, with $ w({X_\xi }) \leqslant {2^\alpha }$ for all $ \xi < \gamma $, the set-theoretic disjoint union $ X = { \cup _{\xi < \gamma }}{X_\xi }$ admits a topology such that $ d(X) \leqslant \alpha $ and each $ {X_\xi }$ is a topological subspace of X.

The following observation (a special case of Theorem 3.1) suggests that it may be difficult to achieve a stronger result: If $ \alpha \geqslant \omega $ and $ {X_0}$ and $ {X_1}$ denote copies of the discrete space of cardinality $ {\alpha ^ + }$, then the disjoint union $ X = {X_0} \cup {X_1}$ admits a topology (making each $ {X_i}$ a topological subspace) such that $ d(X) \leqslant \alpha $.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0445441-9
PII: S 0002-9939(1977)0445441-9
Keywords: Density character, weight, Tychonoff space
Article copyright: © Copyright 1977 American Mathematical Society