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Keywords:
monotonically normal space; $\sigma$-closed-discrete dense set; $e$-separable space; perfect space; perfectly normal space; point network; perfect images of generalized ordered space
Summary:
A topological space $X$ is said to be $e$-separable if $X$ has a $\sigma$-closed-discrete dense subset. Recently, G. Gruenhage and D. Lutzer showed that $e$-separable PIGO spaces are perfect and asked if $e$-separable monotonically normal spaces are perfect in general. The main purpose of this article is to provide examples of $e$-separable monotonically normal spaces which are not perfect. Extremely normal $e$-separable spaces are shown to be stratifiable.
References:
[1] Balogh Z.: Topological spaces with point networks. Proc. Amer. Math. Soc. 94 (1985), no. 3, 497–501. DOI 10.1090/S0002-9939-1985-0787901-1 | MR 0787901
[2] Borges C. R.: A study of monotonically normal spaces. Proc. Amer. Math. Soc. 38 (1973), 211–214. DOI 10.1090/S0002-9939-1973-0324644-4 | MR 0324644
[3] Cairns P., Junilla H., Nyikos P.: An application of Mary Ellen Rudin's solution to Nikiel's conjecture. Topology Appl. 195 (2015), 26–33. MR 3414872
[4] Collins P. J., Reed G. M., Roscoe A. W., Rudin M. E.: A lattice of conditions on topological spaces. Proc. Amer. Math. Soc. 94 (1985), 487–496. DOI 10.1090/S0002-9939-1985-0787900-X | MR 0787900
[5] Collins P. J., Roscoe A. W.: Criteria for metrisability. Proc. Amer. Math. Soc. 90 (1984), no. 4, 631–640. DOI 10.1090/S0002-9939-1984-0733418-9 | MR 0733418 | Zbl 0541.54034
[6] Dias R. R., Soukup D. T.: On spaces with a $\sigma$-closed discrete dense sets. Topology Proc. 52 (2018), 245–264. MR 3773584
[7] Gruenhage G., Lutzer D.: Perfect images of generalized ordered spaces. Fund. Math. 240 (2018), no. 2, 175–197. DOI 10.4064/fm343-1-2017 | MR 3720923
[8] Gruenhage G., Zenor P.: Proto-metrizable spaces. Houston J. Math. 3 (1977), no. 1, 47–53. MR 0442895
[9] Heath R. W., Lutzer D. J., Zenor P. L.: Monotonically normal spaces. Trans. Amer. Math. Soc. 178 (1973), 481–493. DOI 10.1090/S0002-9947-1973-0372826-2 | MR 0372826
[10] Lutzer D. J.: On generalized ordered spaces. Dissertationes Math. Rozprawy Math. 89 (1971), 32 pages. MR 0324668
[11] Moody P. J., Reed G. M., Roscoe A. W., Collins P. J.: A lattice of conditions on topological spaces II. Fund. Math. 138 (1991), no. 2, 69–81. DOI 10.4064/fm-138-2-69-81 | MR 1124537
[12] Ostaszewski A. J.: Monotone normality and $G_\delta$-diagonals in the class of inductively generated spaces. Topology, Vol. II (Proc. Fourth Colloq., Budapest, 1978), 905–930; Colloq. Math. Soc. János Bolyai, 23, North-Holland, Amsterdam-New York, 1980. MR 0588837
[13] Williams S. W., Zhou H. X.: Strong versions of normality. General Topology and Applications, Lecture Notes on Pure and Appl. Math., Dekker 134 (1991), 379–389. MR 1142815 | Zbl 0797.54011
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