On Kolmogorov Topological Spaces
Journal of Formalized Mathematics
Volume 6, 1994
University of Bialystok
Copyright (c) 1994
Association of Mizar Users
On Kolmogorov Topological Spaces
-
Zbigniew Karno
-
Warsaw University, Bialystok
Summary.
-
Let $X$ be a topological space.
$X$ is said to be {\em $T_{0}$-space}\/ (or {\em Kolmogorov space})
provided for every pair of distinct points $x,\,y \in X$ there exists
an open subset of $X$ containing exactly one of these points;
equivalently, for every pair of distinct points $x,\,y \in X$ there exists
a closed subset of $X$ containing exactly one of these points
(see [1], [6], [2]).\par
The purpose is to list some of the standard facts on Kolmogorov spaces,
using Mizar formalism.
As a sample we formulate the following characteristics of such spaces:
{\em $X$ is a Kolmogorov space iff for every pair of distinct
points $x,\,y \in X$ the closures $\overline{\{x\}}$ and
$\overline{\{y\}}$ are distinct}.\par
There is also reviewed analogous facts on Kolmogorov subspaces of topological
spaces. In the presented approach $T_{0}$-subsets
are introduced and some of their properties developed.
Presented at Mizar Conference: Mathematics in Mizar (Bia{\l}ystok, September
12--14, 1994).
MML Identifier:
TSP_1
The terminology and notation used in this paper have been
introduced in the following articles
[8]
[10]
[7]
[9]
[3]
[4]
[11]
[5]
-
Subspaces
-
Kolmogorov Spaces
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$T_{0}$-Subsets
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Kolmogorov Subspaces
Bibliography
- [1]
P. Alexandroff and H. H. Hopf.
\em Topologie I.
Springer-Verlag, Berlin, 1935.
- [2]
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- [3]
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Separated and weakly separated subspaces of topological spaces.
Journal of Formalized Mathematics,
4, 1992.
- [4]
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Maximal discrete subspaces of almost discrete topological spaces.
Journal of Formalized Mathematics,
5, 1993.
- [5]
Zbigniew Karno.
Maximal anti-discrete subspaces of topological spaces.
Journal of Formalized Mathematics,
6, 1994.
- [6]
Kazimierz Kuratowski.
\em Topology, volume I.
PWN - Polish Scientific Publishers, Academic Press, Warsaw, New York
and London, 1966.
- [7]
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Topological spaces and continuous functions.
Journal of Formalized Mathematics,
1, 1989.
- [8]
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Axiomatics, 1989.
- [9]
Andrzej Trybulec.
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Journal of Formalized Mathematics,
3, 1991.
- [10]
Zinaida Trybulec.
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Journal of Formalized Mathematics,
1, 1989.
- [11]
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\Tzero\ topological spaces.
Journal of Formalized Mathematics,
6, 1994.
Received July 26, 1994
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