Some normality-type properties, topological products, and classes of continuous mappings

A. N. Yakivchik

Fundamentalʹnaâ i prikladnaâ matematika, Tome 9 (2003) no. 2, pp. 205-238.

Résumé

Some properties generalizing the normality separation axiom and their behavior under basic topological operations and continuous mappings are discussed. One of the general problems considered in the paper concerns the deduction of “nice” topological properties of spaces from generalized normality of certain topological products, their subspaces, and other related structures; in particular, extensions of the well-known theorems of Katětov and Tamano are suggested.

Bibliographie

[1] Arkhangelskii A. V., “O bikompaktakh, kotorye udovletvoryayut usloviyu Suslina nasledstvenno. Tesnota i svobodnye posledovatelnosti”, DAN SSSR, 199 (1971), 1227–1230 | MR

[2] Arkhangelskii A. V., “Obschaya kontseptsiya rasscheplyaemosti topologicheskikh prostranstv nad klassom prostranstv”, V Tiraspolskii simpozium po obschei topologii i ee prilozheniyam, Shtiintsa, Kishinev, 1985, 8–10

[3] Arkhangelskii A. V., Shakhmatov D. B., “O potochechnoi approksimatsii proizvolnykh funktsii schetnymi semeistvami nepreryvnykh funktsii”, Tr. seminara im. I. G. Petrovskogo, 13, 1988, 206–227 | MR

[4] Banakh T. O., Fedorchuk V. V., O nasledstvennoi subnormalnosti nekotorykh prostranstv, Preprint, 2003

[5] Velichko N. V., “Zametka o peristykh prostranstvakh”, Czechoslovak Math. J., 25:1 (1975), 8–19 | MR | Zbl

[6] Kyunen K., “Kombinatorika”, Spravochnaya kniga po matematicheskoi logike. Ch. II. Teoriya mnozhestv, Nauka, M., 1982, 64–98 | MR

[7] Fedorchuk V. V., “K teoreme Katetova o kube”, Vestnik Mosk. un-ta. Ser. 1, Matematika, mekhanika, 1989, no. 4, 93–96 | MR | Zbl

[8] Shapirovskii B. E., “Kardinalnye invarianty v kompaktakh i otobrazheniya v spetsialnye podmnozhestva tikhonovskikh kubov”, Trudy Bakinskoi mezhdunarodnoi topologicheskoi konferentsii, Baku, 1989, 195–205 | MR

[9] Schepin E. V., “Funktory i neschetnye stepeni kompaktov”, Uspekhi mat. nauk, 36:3 (1981), 3–62 | MR | Zbl

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

[11] Yakivchik A. N., “Subnormalnye prostranstva i problema tipa Daukera”, Fundam. i prikl. mat., 4:1 (1998), 181–186 | MR | Zbl

[12] Arhangelskiĭ A. V., “Divisibility and cleavability of spaces”, Recent Developments of General Topology and its Applications. International Conference in Memory of Felix Hausdorff, Math. Research, 67, Akademie-Verlag, Berlin, 1992, 13–26 | MR

[13] Brandenburg H., “Separating closed sets by continuous mappings into developable spaces”, Canad. J. Math., 33:6 (1981), 1420–1431 | DOI | MR | Zbl

[14] Dowker C. H., “On countably paracompact spaces”, Canad. J. Math., 3:2 (1951), 219–224 | DOI | MR | Zbl

[15] Gartside P. M., Reznichenko E. A., “Katětov revisited”, Topol. Appl., 108:1 (2000), 67–74 | DOI | MR | Zbl

[16] Gruenhage G., Nyikos P., “Normality in $X^2$ for compact $X$”, Trans. Amer. Math. Soc., 340 (1993), 563–586 | DOI | MR | Zbl

[17] Heldermann N. C., “The category of $D$-completely regular spaces is simple”, Trans. Amer. Math. Soc., 262 (1980), 437–446 | DOI | MR | Zbl

[18] Hirata Y., Kemoto N., “Separating by $G_\delta$-sets in finite powers of $\omega_1$”, Fund. Math., 177 (2003), 83–94 | DOI | MR | Zbl

[19] Juhász I., Szentmiklóssy Z., “Convergent free sequences in compact spaces”, Proc. Amer. Math. Soc., 116:4 (1992), 1153–1160 | DOI | MR | Zbl

[20] Katětov M., “Complete normality of Cartesian products”, Fund. Math., 35 (1948), 271–274 | MR | Zbl

[21] Kemoto N., “Subnormality in $\omega_1^2$”, Topol. Appl., 122 (2002), 287–296 | DOI | MR | Zbl

[22] Kramer T. R., “A note on countably subparacompact spaces”, Pacific J. Math., 46 (1973), 209–213 | MR | Zbl

[23] Larson P., Todorčević S., “Katětov's problem”, Trans. Amer. Math. Soc., 354 (2002), 1783–1791 | DOI | MR | Zbl

[24] Michael E., “The product of a normal space and a metric space need not be normal”, Bull. Amer. Math. Soc., 63:3 (1963), 375–376 | DOI | MR

[25] Morita K., “Products of normal spaces with metric spaces”, Math. Ann., 154:4 (1964), 365–382 | DOI | MR | Zbl

[26] Mrówka S., “On completely regular spaces”, Fund. Math., 41 (1954), 105–106 | MR | Zbl

[27] Nyikos P., “A compact nonmetrisable space $P$ such that $P^2$ is completely normal”, Topol. Proc., 2 (1977), 359–363 | MR

[28] Rudin M. E., “A normal space $X$ for which $X \times I$ is not normal”, Fund. Math., 73 (1971), 179–186 | MR | Zbl

[29] Rudin M. E., “Dowker spaces”, Handbook of Set-Theoretic Topology, eds. K. Kunen, J. E. Vaughan, North-Holland, Amsterdam, 1984, 761–780 | MR

[30] Swardson M. A., “A note on the closed character of a topological space”, Topol. Proc., 4 (1979), 601–608 | MR

[31] Szeptycki P. J., “Weak normality in Dowker spaces”, Topol. Proc., 20 (1995), 289–296 | MR | Zbl

[32] Szeptycki P. J., Weiss W. A. R., “Dowker spaces”, The Work of Mary Ellen Rudin, ed. F. D. Tall, New York Acad. Sci., New York, 1993, 119–130 | MR

[33] Tamano H., “On paracompactness”, Pacific J. Math., 10:3 (1960), 1043–1047 | MR | Zbl

[34] Yajima Y., “Analogous results to two classical characterizations of covering properties by products”, Topol. Appl., 84 (1998), 3–7 | DOI | MR | Zbl