Popov V.V. On projective finite spaces

Popov Vladimir Valеntinovich

Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Computer Science and Experimental Mathematics Volgograd State University
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Prospect Universitetsky, 100, 400062 Volgograd, Russian Federation

Abstract. A.V. Arhangel’skii [1] defines the notion of projective power of a Tychonoff space. In particular, the space X is projectively finite, if for any open continuous mapping f : XY onto metrizable separable space the image Y=f(X) is a finite set. The following results are obtained in this paper:

Теорема 1. There is a projectively finite metrizable space of the weight с=2N0 .

Теорема 2. For every cardinal number τ ≥ 2N0 there exists such a projectively finite Tychonoff space X, that any Tychonoff space of the weight ≤ τ is embeddable in X.

    It follows from theorems 1 and 2 the existance of such a projectively finite space, that contains a non trivial convergent sequence. This is an affirmative answer to one of the question of A.V. Arhangel’skii [1].

Key words: projective finite space, metrizable space, open continuous mapping, connected space, separable space.

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On projective finite spaces by Popov V.V. is licensed under a Creative Commons Attribution 4.0 International License.

Citation in English: Science Journal of Volgograd State University. Mathematics. Physics. №2 (19) 2013 pp. 57-66

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