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Journal Article

# The Density Character of Unions

W. W. Comfort and Teklehaimanot Retta
Proceedings of the American Mathematical Society
Vol. 65, No. 1 (Jul., 1977), pp. 155-158
DOI: 10.2307/2042013
Stable URL: http://www.jstor.org/stable/2042013
Page Count: 4
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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 $X = \bigcup_{\xi < \gamma}X_\xi$ 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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