On Classes of Subcompact Spaces

V. I. Belugin ; A. V. Osipov ; E. G. Pytkeev

Matematičeskie zametki, Tome 109 (2021) no. 6, pp. 810-820.

Résumé

This paper continues the study of P. S. Alexandroff's problem: When can a Hausdorff space $X$ be one-to-one continuously mapped onto a compact Hausdorff space? For a cardinal number $\tau$, the classes of $a_\tau$-spaces and strict $a_\tau$-spaces are defined. A compact space $X$ is called an $a_\tau$-space if, for any $C\in[X]^{\le\tau}$, there exists a one-to-one continuous mapping of $X\setminus C$ onto a compact space. A compact space $X$ is called a strict $a_\tau$-space if, for any $C\in[X]^{\le\tau}$, there exits a one-to-one continuous mapping of $X\setminus C$ onto a compact space $Y$, and this mapping can be continuously extended to the whole space $X$. In this paper, we study properties of the classes of $a_\tau$- and strict $a_\tau$-spaces by using Raukhvarger's method of special continuous paritions.

Détail

Mots-clés : condensation, $a_\tau$-space, dyadic compact space.
Keywords: strict $a_\tau$-space, subcompact space, continuous partition, upper semicontinuous partition, ordered compact space

Bibliographie

[1] M. Katětov, “Über $H$-abgeschlossene und bikompakte Räume.”, Časopis Pěst. Math. Fys., 69 (1940), 36–39 | MR

[2] I. L. Raukhvarger, “Ob uplotneniyakh v kompakty”, Dokl. AN SSSR, 66:13 (1949), 13–15 | MR

[3] V. V. Proizvolov, “O vzaimno-odnoznachnykh nepreryvnykh otobrazheniyakh topologicheskikh prostranstv”, Matem. sb., 68 (110):3 (1965), 417–431 | MR

[4] A. S. Parkhomenko, “O vzaimno odnoznachnykh i nepreryvnykh otobrazheniyakh”, Matem. sb., 5 (47):1 (1939), 197–210 | MR | Zbl

[5] A. S. Parkhomenko, “Ob uplotneniyakh v kompaktnye prostranstva”, Izv. AN SSSR. Ser. matem., 5:3 (1941), 225–232 | MR | Zbl

[6] Yu. M. Smirnov, “Uplotneniya na bikompakty i svyaz s bikompaktnymi rasshireniyami i retraktsiei”, Fundam. math., 63:2 (1968), 199–211 | DOI

[7] N. Khadzhiivanov, “Prodolzhenie otobrazhenii v sfery i problema P. S. Aleksandrova o bikompaktnykh uplotneniyakh”, Dokl. AN SSSR, 194:3 (1970), 525–527 | MR | Zbl

[8] N. Khadzhiivanov, “Ob uplotneniyakh na bikompakty”, Godishn. Sofiisk. un-ta, Mat. fak., 1969–1970, 64 (1971), 7–12

[9] V. K. Belnov, “Ob uplotneniyakh na kompakty”, Dokl. AN SSSR, 193:3 (1970), 506–509 | MR | Zbl

[10] V. K. Belnov, “Ob uplotneniyakh na kompakty nekotorykh klassov metricheskikh prostranstv”, Vestn. Mosk. un-ta. Cer. 1. Matem., mekh., 1972, no. 5, 9–13 | MR

[11] A. V. Arhangel'skii, “On condensations of $C_p$-spaces onto compacta”, Proc. Amer. Math. Soc., 128:6 (2000), 1881–1883 | DOI | MR

[12] A. V. Arhangel'skii, O. Pavlov, “A note on condensations of $C_p(X)$ onto compacta”, Comment. Math. Univ. Carolin., 43:3 (2002), 485–492 | MR

[13] V. I. Belugin, “Uplotneniya na bikompakty”, Dokl. AN SSSR, 207:2 (1972), 259–261 | MR | Zbl

[14] V. I. Belugin, “Uplotnenie na bikompakty”, Dokl. Bolgarskoi AN, 28:11 (1975), 1447–1449 | MR

[15] V. I. Belugin, “Uplotnenie na bikompakty podprostranstv uporyadochennykh bikompaktov”, Topologiya i teoriya mnozhestv, Izhevsk, 1982, 3–8

[16] V. I. Belugin, “Uplotnenie na bikompakty i proizvedeniya prostranstv”, Kardinalnye invarianty i rasshireniya prostranstv, Izhevsk, 1989, 36–43

[17] A. V. Osipov, E. G. Pytkeev, “K voprosu ob uplotnenii na kompakty”, Dokl. AN, 488:2 (2019), 130–132 | DOI

[18] E. G. Pytkeev, “K teorii uplotnenii na kompakty”, Dokl. AN SSSR, 233:6 (1977), 1046–1048 | MR | Zbl

[19] E. G. Pytkeev, “O nasledstvenno peristykh prostranstvakh”, Matem. zametki, 28:4 (1980), 603–618 | MR | Zbl

[20] E. G. Pytkeev, “Ob uplotneniyakh na bikompakty”, Dokl. AN SSSR, 265:4 (1982), 819–823 | MR | Zbl

[21] H. Reiter, “Spaces with compact subtopologies”, Rocky Mountain J. Math., 2:2 (1972), 239–247 | DOI | MR

[22] W. W. Comfort, A. W. Hager, J. van Mill, “Compact condensations and compactifications”, Topology Appl., 259 (2019), 67–79 | DOI | MR

[23] C. Iliadis, “O nekotorykh svoistvakh absolyutov”, Dokl. AN SSSR, 152:4 (1963), 798–800 | MR | Zbl

[24] P. S. Aleksandrov, “O nekotorykh osnovnykh napravleniyakh v obschei topologii”, UMN, 19:6 (120) (1964), 3–46 | MR | Zbl

[25] R. L. Moore, Foundations of Point Set Theory, Amer. Math. Soc., Providence, RI, 1932 | MR

[26] P. Alexandroff H. Hopf, Topologie. I, Springer, Berlin, 1935 | Zbl

[27] P. S. Aleksandrov, P. S. Uryson, Memuar o kompaktnykh topologicheskikh prostranstvakh, Nauka, M., 1971 | MR | Zbl

[28] S. Mrówka, “Mazur theorem and $m$-adic spaces”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 18:5 (1970), 299–305 | MR