On dense subspaces satisfying stronger separation axioms

Ofelia Teresa Alas; Mihail G. Tkachenko; Vladimir Vladimirovich Tkachuk; Richard Gordon Wilson; Ivan V. Yashchenko

Czechoslovak Mathematical Journal (2001)

  • Volume: 51, Issue: 1, page 15-28
  • ISSN: 0011-4642

Abstract

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We prove that it is independent of ZFC whether every Hausdorff countable space of weight less than c has a dense regular subspace. Examples are given of countable Hausdorff spaces of weight c which do not have dense Urysohn subspaces. We also construct an example of a countable Urysohn space, which has no dense completely Hausdorff subspace. On the other hand, we establish that every Hausdorff space of π -weight less than 𝔭 has a dense completely Hausdorff (and hence Urysohn) subspace. We show that there exists a Tychonoff space without dense normal subspaces and give other examples of spaces without “good” dense subsets.

How to cite

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Alas, Ofelia Teresa, et al. "On dense subspaces satisfying stronger separation axioms." Czechoslovak Mathematical Journal 51.1 (2001): 15-28. <http://eudml.org/doc/30610>.

@article{Alas2001,
abstract = {We prove that it is independent of ZFC whether every Hausdorff countable space of weight less than $c$ has a dense regular subspace. Examples are given of countable Hausdorff spaces of weight $c$ which do not have dense Urysohn subspaces. We also construct an example of a countable Urysohn space, which has no dense completely Hausdorff subspace. On the other hand, we establish that every Hausdorff space of $\pi $-weight less than $\mathfrak \{p\}$ has a dense completely Hausdorff (and hence Urysohn) subspace. We show that there exists a Tychonoff space without dense normal subspaces and give other examples of spaces without “good” dense subsets.},
author = {Alas, Ofelia Teresa, Tkachenko, Mihail G., Tkachuk, Vladimir Vladimirovich, Wilson, Richard Gordon, Yashchenko, Ivan V.},
journal = {Czechoslovak Mathematical Journal},
keywords = {Hausdorff space; Urysohn space; completely Hausdorff space; filter of dense sets; Hausdorff space; Urysohn space; completely Hausdorff space; filter of dense sets},
language = {eng},
number = {1},
pages = {15-28},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On dense subspaces satisfying stronger separation axioms},
url = {http://eudml.org/doc/30610},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Alas, Ofelia Teresa
AU - Tkachenko, Mihail G.
AU - Tkachuk, Vladimir Vladimirovich
AU - Wilson, Richard Gordon
AU - Yashchenko, Ivan V.
TI - On dense subspaces satisfying stronger separation axioms
JO - Czechoslovak Mathematical Journal
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 51
IS - 1
SP - 15
EP - 28
AB - We prove that it is independent of ZFC whether every Hausdorff countable space of weight less than $c$ has a dense regular subspace. Examples are given of countable Hausdorff spaces of weight $c$ which do not have dense Urysohn subspaces. We also construct an example of a countable Urysohn space, which has no dense completely Hausdorff subspace. On the other hand, we establish that every Hausdorff space of $\pi $-weight less than $\mathfrak {p}$ has a dense completely Hausdorff (and hence Urysohn) subspace. We show that there exists a Tychonoff space without dense normal subspaces and give other examples of spaces without “good” dense subsets.
LA - eng
KW - Hausdorff space; Urysohn space; completely Hausdorff space; filter of dense sets; Hausdorff space; Urysohn space; completely Hausdorff space; filter of dense sets
UR - http://eudml.org/doc/30610
ER -

References

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