On d-separability of powers and $C_p(X)$

István Juhász and Zoltán Szentmiklóssy

A space is called $d$-separable if it has a dense subset representable as the union of countably many discrete subsets. We answer several problems raised by V. V. Tkachuk by showing that

1. $X^{d(X)}$ is $d$-separable for every $T_1$ space $X$;

2. if $X$ is compact Hausdorff then $X^\omega$ is $d$-separable;

3. there is a 0-dimensional $T_2$ space $X$ such that $X^{\omega_2}$ is $d$-separable but $X^{\omega_1}$ (and hence $X^\omega$) is not;

4. there is a 0-dimensional $T_2$ space $X$ such that $C_p(X)$ is not $d$-separable.

The proof of (2) uses the following new result: If $X$ is compact Hausdorff then its square $X^2$ has a discrete subspace of cardinality $d(X).$

Key words and phrases: compact space, discrete subspace, $d$-separable space, power of a space, $C_p(X)$

2000 Mathematics Subject Classification: 54A25, 54B10

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