The density character of unions

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 $d(X) \leqslant \alpha$.