Geodesic
On dense subspaces satisfying stronger separation axioms
Czechoslovak Mathematical Journal, Tome 51 (2001) no. 1, pp. 15-28.

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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.
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},
     publisher = {mathdoc},
     volume = {51},
     number = {1},
     year = {2001},
     mrnumber = {1814628},
     zbl = {1079.54518},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2001__51_1_a1/}
}
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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/