Diagonals and discrete subsets of squares

Burke, Dennis ; Tkachuk, Vladimir V.

Commentationes Mathematicae Universitatis Carolinae, Tome 54 (2013) no. 1, pp. 69-82.

Résumé

In 2008 Juhász and Szentmiklóssy established that for every compact space $X$ there exists a discrete $D\subset X\times X$ with $|D|=d(X)$. We generalize this result in two directions: the first one is to prove that the same holds for any Lindelöf $\Sigma$-space $X$ and hence $X^\omega $ is $d$-separable. We give an example of a countably compact space $X$ such that $X^\omega $ is not $d$-separable. On the other hand, we show that for any Lindelöf $p$-space $X$ there exists a discrete subset $D\subset X\times X$ such that $\Delta = \{(x,x): x\in X\}\subset \overline{D}$; in particular, the diagonal $\Delta $ is a retract of $\overline{D}$ and the projection of $D$ on the first coordinate is dense in $X$. As a consequence, some properties that are not discretely reflexive in $X$ become discretely reflexive in $X\times X$. In particular, if $X$ is compact and $\overline{D}$ is Corson (Eberlein) compact for any discrete $D\subset X\times X$ then $X$ itself is Corson (Eberlein). Besides, a Lindelöf $p$-space $X$ is zero-dimensional if and only if $\overline{D}$ is zero-dimensional for any discrete $D\subset X\times X$. Under CH, we give an example of a crowded countable space $X$ such that every discrete subset of $X\times X$ is closed. In particular, the diagonal of $X$ cannot be contained in the closure of a discrete subspace of $X\times X$.

Détail

Classification : 54C10, 54C25, 54D25, 54H11
Keywords: diagonal; discrete subspaces; $d$-separable space; discrete reflexivity; Lindelöf $p$-space; Lindelöf $\Sigma $-space; finite powers; Corson compact spaces; Eberlein compact spaces; countably compact spaces