|
Section
|
Mathematics
|
|
Title
|
On compact $T_1$-spaces
|
|
Author(-s)
|
Voronov M.E.a
|
|
Affiliations
|
Udmurt State Universitya
|
|
Abstract
|
We consider spaces, any subspaces of which are compact. We call such spaces hereditarily compact. The present work covers questions on the existence and methods of constructing hereditarily compact $T_1$-topologies. We prove the existence of $2^{\tau}$ pairwise incomparable hereditarily compact $T_1$-topologies on an infinite set $X$ of power $ \tau $. The characteristics of hereditarily compact spaces are obtained. It is proved that the Tychonoff product of a finite number of hereditarily compact $T_1$-spaces is a hereditarily compact $T_1$-space, but the Tychonoff product of an infinite number of nonsingleton hereditarily compact $T_1$-spaces is not hereditarily compact.
|
|
Keywords
|
compactness, minimal $T_1$-topology, Tychonoff product
|
|
UDC
|
515.122.22, 515.122.252
|
|
MSC
|
54D10, 54D30
|
|
DOI
|
10.20537/vm130302
|
|
Received
|
22 July 2013
|
|
Language
|
Russian
|
|
Citation
|
Voronov M.E. On compact $T_1$-spaces, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, issue 3, pp. 20-27.
|
|
References
|
- de Groot Y. An isomorphism principle in general topology, Bull. Amer. Math. Soc., 1967, vol. 73, no. 3, pp. 465-467.
- Arkhangel'skii A.V. Maps and spaces, Usp. Mat. Nauk, 1966, vol. 21, no. 4, pp. 133-183.
- Arkhangel'skii A.V., Ponomarev V.I. Osnovy obshchei topologii v zadachakh i uprazhneniyakh (Fundamentals of general topology through problems and exercises), Moscow: Nauka, 1974, 424 p.
- Engelking R. Obshchaya topologiya (General topology), Moscow: Mir, 1986, 752 p.
- Fedorchuk V.V., Filippov V.V. Obshchaya topologiya. Osnovnye konstruktsii (General topology. Basic design), Moscow: Fizmatlit, 2006, 336 p.
|
|
Full text
|
|
+7 (3412) 91 60 92
Russian