# A nice class extracted from ${C}_{p}$-theory

Vladimir Vladimirovich Tkachuk

Commentationes Mathematicae Universitatis Carolinae (2005)

- Volume: 46, Issue: 3, page 503-513
- ISSN: 0010-2628

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topTkachuk, Vladimir Vladimirovich. "A nice class extracted from $C_p$-theory." Commentationes Mathematicae Universitatis Carolinae 46.3 (2005): 503-513. <http://eudml.org/doc/249567>.

@article{Tkachuk2005,

abstract = {We study systematically a class of spaces introduced by Sokolov and call them Sokolov spaces. Their importance can be seen from the fact that every Corson compact space is a Sokolov space. We show that every Sokolov space is collectionwise normal, $\omega $-stable and $\omega $-monolithic. It is also established that any Sokolov compact space $X$ is Fréchet-Urysohn and the space $C_p(X)$ is Lindelöf. We prove that any Sokolov space with a $G_\delta $-diagonal has a countable network and obtain some cardinality restrictions on subsets of small pseudocharacter lying in $\Sigma $-products of cosmic spaces.},

author = {Tkachuk, Vladimir Vladimirovich},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {Corson compact space; Sokolov space; extent; $\omega $-monolithic space; $\Sigma $-products; Corson compact space; Sokolov space; extent; -monolithic space; -products},

language = {eng},

number = {3},

pages = {503-513},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {A nice class extracted from $C_p$-theory},

url = {http://eudml.org/doc/249567},

volume = {46},

year = {2005},

}

TY - JOUR

AU - Tkachuk, Vladimir Vladimirovich

TI - A nice class extracted from $C_p$-theory

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 2005

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 46

IS - 3

SP - 503

EP - 513

AB - We study systematically a class of spaces introduced by Sokolov and call them Sokolov spaces. Their importance can be seen from the fact that every Corson compact space is a Sokolov space. We show that every Sokolov space is collectionwise normal, $\omega $-stable and $\omega $-monolithic. It is also established that any Sokolov compact space $X$ is Fréchet-Urysohn and the space $C_p(X)$ is Lindelöf. We prove that any Sokolov space with a $G_\delta $-diagonal has a countable network and obtain some cardinality restrictions on subsets of small pseudocharacter lying in $\Sigma $-products of cosmic spaces.

LA - eng

KW - Corson compact space; Sokolov space; extent; $\omega $-monolithic space; $\Sigma $-products; Corson compact space; Sokolov space; extent; -monolithic space; -products

UR - http://eudml.org/doc/249567

ER -

## References

top- Arhangel'skii A.V., Topological Function Spaces, Kluwer Acad. Publ., Dordrecht, 1992. MR1485266
- Arhangel'skii A.V., On Lindelöf property and spread in ${C}_{p}$-theory, Topology Appl. 74:1 (1996), 83-90. (1996) MR1425928
- Baturov D.P., On subspaces of function spaces (in Russian), Vestnik Moskov. Univ. Ser. I Mat. Mekh. 42:4 (1987), 66-69. (1987) MR0913076
- Engelking R., General Topology, PWN, Warszawa, 1977. Zbl0684.54001MR0500780
- Fabian M., Gateaux Differentiability of Convex Functions and Topology, Weak Asplund Spaces, Wiley, New York, 1997. Zbl0883.46011MR1461271
- Gruenhage G., Generalized Metric Spaces, Handbook of Set-Theoretic Topology, edited by K. Kunen and J.E. Vaughan, Elsevier Sci. Publ., B.V., Amsterdam, 1984, pp.423-501. Zbl0794.54034MR0776629
- Gruenhage G., Spaces having a small diagonal, preprint. Zbl1028.54025MR1919300
- Gul'ko S.P., On the properties of some function spaces, Soviet Math. Dokl. 19:6 (1978), 1420-1424. (1978) Zbl0421.54013MR0514481
- Gul'ko S.P., On the structure of spaces of continuous functions and their complete paracompactness, Russian Math. Surveys 34:6 (1979), 36-44. (1979) Zbl0446.46014MR0562814
- Juhász I., Cardinal Functions in Topology - Ten Years Later, Mathematical Centre Tracts 123, Amsterdam, 1980. MR0576927
- Shapirovsky B.E., Special types of embeddings in Tychonoff cubes, subspaces of $\Sigma $-products and cardinal invariants, Colloq. Math. Soc. Janos Bolyai 23 (1978), 1055-1086. (1978) MR0588855
- Sokolov G.A., On Lindelöf spaces of continuous functions (in Russian), Matem. Zametki 39:6 (1986), 887-894. (1986) MR0855936
- Sokolov G.A., Lindelöf property and the iterated continuous function spaces, Fund. Math. 143 (1993), 87-95. (1993) Zbl0841.54011MR1234993
- Tkachuk V.V., Calibers of spaces of functions and the metrization problem for compact subsets of ${C}_{p}\left(X\right)$, Vestnik Moskov. Univ. 43:3 (1988), 21-24. (1988) MR0966861
- Tkachuk V.V., Behaviour of the Lindelöf $\Sigma $-property in iterated function spaces, Topology Appl. 107 (2000), 297-305. (2000) MR1779816

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