Diagonals and discrete subsets of squares

Burke, Dennis, and Tkachuk, Vladimir Vladimirovich. "Diagonals and discrete subsets of squares." Commentationes Mathematicae Universitatis Carolinae 54.1 (2013): 69-82. <http://eudml.org/doc/252473>.

@article{Burke2013,
abstract = {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 $\Delta = \lbrace (x,x): x\in X\rbrace \subset \overline\{D\}$; 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 $X\times X$.},
author = {Burke, Dennis, Tkachuk, Vladimir Vladimirovich},
journal = {Commentationes Mathematicae Universitatis Carolinae},
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; discrete subspace; -separable space; discrete reflexivity; Lindelöf -space; Lindelöf -space; Corson compact space; Eberlein compact space; countably compact space},
language = {eng},
number = {1},
pages = {69-82},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Diagonals and discrete subsets of squares},
url = {http://eudml.org/doc/252473},
volume = {54},
year = {2013},
}

TY - JOUR
AU - Burke, Dennis
AU - Tkachuk, Vladimir Vladimirovich
TI - Diagonals and discrete subsets of squares
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2013
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 54
IS - 1
SP - 69
EP - 82
AB - 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 $\Delta = \lbrace (x,x): x\in X\rbrace \subset \overline{D}$; 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 $X\times X$.
LA - eng
KW - 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; discrete subspace; -separable space; discrete reflexivity; Lindelöf -space; Lindelöf -space; Corson compact space; Eberlein compact space; countably compact space
UR - http://eudml.org/doc/252473
ER -