Diagonals and discrete subsets of squares

Commentationes mathematicae Universitatis Carolinae

Dennis Burke¹ 

关键词: 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;   

DOI  :  

学科分类：物理化学和理论化学

来源: Univerzita Karlova v Praze * Matematicko-Fyzikalni Fakulta / Charles University in Prague, Faculty of Mathematics and Physics

【 摘 要 】

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 $X\times X$.

【 授权许可】

CC BY