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Zbl 0646.54005
Bella, A.; Cammaroto, F.
On the cardinality of Urysohn spaces. (English)
[J] Can. Math. Bull. 31, No.2, 153-158 (1988). ISSN 0008-4395; ISSN 1496-4287/e

Some cardinal inequalities for Urysohn spaces are established. In particular the following two theorems are proved: (i) if $A\subset X$ then $$ \vert [A]\sb{\theta}\vert \le \vert A\vert\sp{\chi (X)}, $$ where $[A]\sb{\theta}$ denotes the $\theta$-closed hull of A, i.e. the smallest $\theta$-closed subset of X containing A; $$ (ii)\quad \vert X\vert \le 2\sp{\chi (X)aL(X,X)}, $$ where aL(X,X) is the smallest cardinal number m such that for every open cover ${\cal U}$ of X there is a subfamily ${\cal U}\sb 0\subset {\cal U}$ for which $X=\cup\sb{U\in {\cal U}\sb 0}\bar U$ and $\vert {\cal U}\sb 0\vert \le m$.

MSC 2000:: *54A25 Cardinality properties of topological spaces
54D50 k-spaces

Cited in: Zbl 1027.54006 Zbl 0747.54003

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