Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

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
  • Read Online (Free)
  • Download ($30.00)
  • Subscribe ($19.50)
  • Cite this Item
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
The Density Character of Unions
Preview not available

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 $d(X) \leqslant \alpha$.

Page Thumbnails

  • Thumbnail: Page 
155
    155
  • Thumbnail: Page 
156
    156
  • Thumbnail: Page 
157
    157
  • Thumbnail: Page 
158
    158