# Article

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Keywords:
metrizable; Lindelöf $p$-space; Lindelöf $\Sigma$-space; remainder; compactification; $\sigma$-space; countable network; countable type; perfect mapping
Summary:
The class of $s$-spaces is studied in detail. It includes, in particular, all Čech-complete spaces, Lindelöf $p$-spaces, metrizable spaces with the weight $\leq 2^\omega$, but countable non-metrizable spaces and some metrizable spaces are not in it. It is shown that $s$-spaces are in a duality with Lindelöf $\Sigma$-spaces: $X$ is an $s$-space if and only if some (every) remainder of $X$ in a compactification is a Lindelöf $\Sigma$-space [Arhangel'skii A.V., Remainders of metrizable and close to metrizable spaces, Fund. Math. {220} (2013), 71--81]. A basic fact is established: the weight and the networkweight coincide for all $s$-spaces. This theorem generalizes the similar statement about Čech-complete spaces. We also study hereditarily $s$-spaces, provide various sufficient conditions for a space to be a hereditarily $s$-space, and establish that every metrizable space has a dense subspace which is a hereditarily $s$-space. It is also shown that every dense-in-itself compact hereditarily $s$-space is metrizable.
References:
[1] Arhangel'skii A.V.: External bases of sets lying in bicompacta. Dokl. Akad. Nauk SSSR 132 (1960), 495–496. English translation: Soviet Math. Dokl. 1 (1960), 573–574. MR 0119188
[2] Arhangel'skii A.V.: On a class of spaces containing all metric spaces and all locally bicompact spaces. Dokl. Akad. Nauk SSSR 151 (1963), 751–754. English translation: Soviet Math. Dokl. 4 (1963), 1051–1055. MR 0152988
[3] Arhangel'skii A.V.: Bicompact sets and the topology of spaces. Dokl. Akad. Nauk SSSR 150 (1963), 9–12. MR 0150733
[4] Arhangel'skii A.V.: Bicompact sets and the topology of spaces. Trudy Moskov. Mat. Obsch. 13 (1965), 3–55 (in Russian). English translation: Trans. Mosc. Math. Soc. 13 (1965), 1–62. MR 0195046
[5] Arhangel'skii A.V.: Perfect maps and injections. Dokl. Akad. Nauk SSSR 176 (1967), 983–986. English translation: Soviet Math. Dokl. 8 (1967), 1217–1220. MR 0238276
[6] Arhangel'skii A.V.: A characterization of very $k$-spaces. Czechoslovak Math. J. 18 (1968), 392–395. MR 0229194
[7] Arhangel'skii A.V.: On a class of spaces containing all metric and all locally compact spaces. Mat. Sb. 67(109) (1965), 55–88. English translation: Amer. Math. Soc. Transl. 92 (1970), 1–39. MR 0190889
[8] Arhangel'skii A.V.: On hereditary properties. General Topology and Appl. 3 (1973), no. 1, 39–46. DOI 10.1016/0016-660X(73)90028-7 | MR 0319142
[9] Arhangelskii A.V.: Relations among the invariants of topological groups and their subspaces. Uspekhi Mat. Nauk 35 (1980), no. 3, 3–22 (in Russian). English translation: Russian Math. Surveys 35 (1980), no. 3, 1–23. MR 0580615
[10] A.V. Arhangel'skii: Remainders in compactifications and generalized metrizability properties. Topology and Appl. 150 (2005), 79-90. DOI 10.1016/j.topol.2004.10.015 | MR 2133669 | Zbl 1075.54012
[11] Arhangel'skii A.V.: Two types of remainders of topological groups. Comment. Math. Univ. Carolin. 49 (2008), no. 1, 119–126. MR 2433629
[12] Arhangel'skii A.V.: Remainders of metrizable spaces and a generalization of Lindelöf $\Sigma$-spaces. Fund. Math. 215 (2011), 87–100. DOI 10.4064/fm215-1-5 | MR 2851703 | Zbl 1236.54006
[13] Arhangel'skii A.V.: Remainders of metrizable and close to metrizable spaces. Fund. Math. 220 (2013), 71–81. DOI 10.4064/fm220-1-4
[14] Arhangel'skii A.V., Bella A.: Cardinal invariants in remainders and variations of tightness. Proc. Amer. Math. Soc. 119 (1993), no. 3, 947–954. DOI 10.2307/2160537 | MR 1185277
[15] Arhangel'skii A.V., Choban M.M.: Some generalizations of the concept of a $p$-space. Topology Appl. 158 (2011), 1381–1389. DOI 10.1016/j.topol.2011.05.012 | MR 2812490 | Zbl 1229.54036
[16] Arhangel'skii A.V., Holsztynski W.: Sur les reseaux dans les espaces topologiques. Bull. Acad. Polon. Sci., Ser. Math. 11 (1963), 493–497 (in French). MR 0159300
[17] Burke D.K.: Covering properties. in: Handbook of Set-theoretic Topology, K. Kunen and J. Vaughan, eds., North-Holland, Amsterdam, 1984, pp. 347–422. MR 0776628 | Zbl 0569.54022
[18] van Douwen E.K., Tall F., Weiss W.: Non-metrizable hereditarily Lindelöf spaces with point-countable bases from CH. Proc. Amer. Math. Soc. 64 (1977), 139–145. MR 0514998 | Zbl 0356.54020
[19] Engelking R.: General Topology. PWN, Warszawa, 1977. MR 0500780 | Zbl 0684.54001
[20] Grabner G., Szymanski A.: Spaces hereditarily of $\kappa$-type and point $\kappa$-type. Rend. Circ. Mat. Palermo (2) 42 (1993), 382–390. DOI 10.1007/BF02844629 | MR 1283352 | Zbl 0802.54019
[21] Henriksen M., Isbell J.R.: Some properties of compactifications. Duke Math. J. 25 (1958), 83–106. DOI 10.1215/S0012-7094-58-02509-2 | MR 0096196 | Zbl 0081.38604
[22] Hodel R.E.: A theorem of Arhangel'skii concerning Lindelöf $p$-spaces. Canad. J. Math. 27 (1975), no. 2, 459–468. DOI 10.4153/CJM-1975-054-8 | MR 0375205 | Zbl 0301.54010
[23] Nagami K.: $\Sigma$-spaces. Fund. Math. 61 (1969), 169–192. MR 0257963 | Zbl 0181.50701
[24] Popov V.: A perfect map needn't preserve a $G_\delta$-diagonal. General Topology and Appl. 7 (1977), 31–33. DOI 10.1016/0016-660X(77)90004-6 | MR 0431093
[25] Pytkeev E.G.: Hereditarily plumed spaces. Math. Notes 28 (1980), no. 4, 603–618. DOI 10.1007/BF01140139 | MR 0594378 | Zbl 0462.54018
[26] Velichko N.V.: Theory of resolvable spaces. Mat. Zametki 19 (1976), no. 1, 19–114. English translation: Math. Notes 19 (1976), no. 1, 65–68. Zbl 0346.54007

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