Geodesic
On dense subspaces satisfying stronger separation axioms
Czechoslovak Mathematical Journal, Tome 51 (2001) no. 1, pp. 15-28 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $\pi$-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.
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.
Classification : 22A05, 54C10, 54C25, 54D06, 54D15, 54D25, 54H11
Keywords: Hausdorff space; Urysohn space; completely Hausdorff space; filter of dense sets 
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     author = {Alas, Ofelia T. and Tkachenko, Mikhail G. and Tkachuk, Vladimir V. and Wilson, Richard G. and Yaschenko, Ivan V.},
     title = {On dense subspaces satisfying stronger separation axioms},
     journal = {Czechoslovak Mathematical Journal},
     pages = {15--28},
     year = {2001},
     volume = {51},
     number = {1},
     mrnumber = {1814628},
     zbl = {1079.54518},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2001_51_1_a1/}
}
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%A Tkachuk, Vladimir V.
%A Wilson, Richard G.
%A Yaschenko, Ivan V.
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Alas, Ofelia T.; Tkachenko, Mikhail G.; Tkachuk, Vladimir V.; Wilson, Richard G.; Yaschenko, Ivan V. On dense subspaces satisfying stronger separation axioms. Czechoslovak Mathematical Journal, Tome 51 (2001) no. 1, pp. 15-28. http://geodesic.mathdoc.fr/item/CMJ_2001_51_1_a1/

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