Dennis Burke, Vladimir V. Tkachuk
Diagonals and discrete subsets of squares

Comment.Math.Univ.Carolin. 54,1 (2013) 69-82.

Abstract:In 2008 Juh\'asz and Szentmikl\'ossy 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 Lindel\"of $\Sigma$-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 Lindel\"of $p$-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 Lindel\"of $p$-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 $X\times X$.

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