Closed discrete subsets of separable spaces and relative versions of normality, countable paracompactness and property $\left(a\right)$

Silva, Samuel Gomes da. "Closed discrete subsets of separable spaces and relative versions of normality, countable paracompactness and property $(a)$." Commentationes Mathematicae Universitatis Carolinae 52.3 (2011): 435-444. <http://eudml.org/doc/246662>.

@article{Silva2011,
abstract = {In this paper we show that a separable space cannot include closed discrete subsets which have the cardinality of the continuum and satisfy relative versions of any of the following topological properties: normality, countable paracompactness and property $(a)$. It follows that it is consistent that closed discrete subsets of a separable space $X$ which are also relatively normal (relatively countably paracompact, relatively $(a)$) in $X$ are necessarily countable. There are, however, consistent examples of separable spaces with uncountable closed discrete subsets under the described relative topological requirements, and therefore the existence of such uncountable sets is undecidable within ZFC. We also investigate what are the outcomes of considering the set-theoretical hypothesis “$2^\{\omega \} < 2^\{\omega _1\}$” within our discussion and conclude by giving some notes and posing some questions.},
author = {Silva, Samuel Gomes da},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {relative normality; relative countable paracompactness; relative property $(a)$; closed discrete subsets; separable spaces; relative normality; relative countable paracompactness; relative property ; closed discrete subsets; separable spaces},
language = {eng},
number = {3},
pages = {435-444},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Closed discrete subsets of separable spaces and relative versions of normality, countable paracompactness and property $(a)$},
url = {http://eudml.org/doc/246662},
volume = {52},
year = {2011},
}

TY - JOUR
AU - Silva, Samuel Gomes da
TI - Closed discrete subsets of separable spaces and relative versions of normality, countable paracompactness and property $(a)$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2011
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 52
IS - 3
SP - 435
EP - 444
AB - In this paper we show that a separable space cannot include closed discrete subsets which have the cardinality of the continuum and satisfy relative versions of any of the following topological properties: normality, countable paracompactness and property $(a)$. It follows that it is consistent that closed discrete subsets of a separable space $X$ which are also relatively normal (relatively countably paracompact, relatively $(a)$) in $X$ are necessarily countable. There are, however, consistent examples of separable spaces with uncountable closed discrete subsets under the described relative topological requirements, and therefore the existence of such uncountable sets is undecidable within ZFC. We also investigate what are the outcomes of considering the set-theoretical hypothesis “$2^{\omega } < 2^{\omega _1}$” within our discussion and conclude by giving some notes and posing some questions.
LA - eng
KW - relative normality; relative countable paracompactness; relative property $(a)$; closed discrete subsets; separable spaces; relative normality; relative countable paracompactness; relative property ; closed discrete subsets; separable spaces
UR - http://eudml.org/doc/246662
ER -