Geodesic
Some Properties of Subcompact Spaces
Matematičeskie zametki, Tome 111 (2022) no. 2, pp. 188-201.

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A Hausdorff topological space $X$ is said to be subcompact if it admits a coarser compact Hausdorff topology. P. S. Alexandroff asked the following question: What Hausdorff spaces are subcompact? A compact space $X$ is called a strict $a$-space if, for any $C\in [X]^{\le\omega}$, there exists a one-to-one continuous map of $X\setminus C$ onto a compact space $Y$ which can be continuously extended to the entire space $X$. The paper continues the study of classes of subcompact spaces. It is proved that the product of a compact space and a dyadic compact space without isolated points is a strict $a$-space.
Keywords: continuous bijection, $a$-space, strict $a$-space, subcompact space.
Mots-clés : condensation, dyadic compact space 
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V. I. Belugin; A. V. Osipov; E. G. Pytkeev. Some Properties of Subcompact Spaces. Matematičeskie zametki, Tome 111 (2022) no. 2, pp. 188-201. http://geodesic.mathdoc.fr/item/MZM_2022_111_2_a1/

[1] A. V. Arkhangelskii, “Nekotorye poslednie dostizheniya i otkrytye problemy v obschei topologii”, UMN, 52:5(317) (1997), 45–70 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[17] W. Kulpa, M. Turzański, “Bijections onto compact spaces”, Acta Univ. Carolinae. Math. Phys., 29:2 (1988), 43–49 | Zbl

[18] H. Reiter, “Spaces with compact subtopologies”, Rocky Mount. J. Math., 2 (1972), 239–247 | DOI | Zbl

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

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

[21] V. I. Belugin, A. V. Osipov, E. G. Pytkeev, “O klassakh subkompaktnykh prostranstv”, Matem. zametki, 109:6 (2021), 810–820 | DOI | Zbl

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

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

[24] V. I. Belugin, A. V. Osipov, E. G. Pytkeev, “Compact condensations of Hausdorff spaces”, Acta Math. Hungaria, 164:1 (2021), 15–27 | DOI | Zbl

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

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

[27] R. Engelking, Obschaya topologiya, Mir, M., 1986 | MR | Zbl

[28] R. Engelking, “On closed images of the space of irrationals”, Proc. Amer. Math. Soc., 21 (1969), 583–586 | DOI | Zbl

[29] L. Ivanovskii, “Ob odnoi gipoteze P. S. Aleksandrova”, Dokl. AN, 123 (1958), 785–786 | MR | Zbl

[30] V. I. Kuzminov, “O gipoteze P. S. Aleksandrova v teorii topologicheskikh grupp”, Dokl. AN, 125 (1959), 727–729 | MR | Zbl

[31] B. A. Efimov, “O diadicheskikh bikompaktakh”, Dokl. AN SSSR, 149:5 (1963), 1011–1014

[32] A. V. Arkhangelskii, V. I. Ponomarev, Osnovy obschei topologii v zadachakh i uprazhneniyakh, Nauka, M., 1974 | MR | Zbl

[33] A. V. Osipov, E. G. Pytkeev, “Reshenie problemy Ponomareva ob uplotnenii na kompakty”, Sib. matem. zhurn., 62:1 (2021), 164–172 | DOI | Zbl

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