Some properties of topological spaces of almost continuous maps (in Ukrainian)

B.M.Bokalo; O.P.Malanyuk

A map $f\colon{}X\rightarrow Y$ is called almost continuous if every nonempty subspace $A$ of $X$ contains a point of continuity of the map $f\vert _A \colon{} A \rightarrow f(A)$. By $AC_p(X)$ we denote the space of almost continuous real-valued functions on a space $X$ endowed with the topology of pointwise convergence. It is shown that $AC_p(X)$ is a topological algebra. We investigate properties of spaces $X$, for which the space $AC_p(X)$ is Lindelöf (\v Cech complete).

1. Архангельский А. В. Топологические пространства функций. – М.: Изд. МГУ, 1989.

2. Архангельский А. В., Бокало Б. М. Касание топологий и тангенциальные свойства топологических пространств // ТММО. – 1992. – Т.54. – С.160–185.

3. Энгелькинг Р. Общая топология. – М.: Мир, 1986.

4. Pol R. The Lindelöf property and its analogue in function spaces with weak topology // Topology. 4-th collog. Budapest, 1978. V.2. – Amsterdam, 1980. – P.965—969.

5. Banakh T., Bokalo B. On almost continuous function. – Lviv University, 1999. – preprint.

	Some properties of topological spaces of almost continuous maps (in Ukrainian)
Author	B.M.Bokalo, O.P.Malanyuk Faculty of Mechanics and Mathematics, Lviv National University
Abstract	A map $f\colon{}X\rightarrow Y$ is called almost continuous if every nonempty subspace $A$ of $X$ contains a point of continuity of the map $f\vert _A \colon{} A \rightarrow f(A)$. By $AC_p(X)$ we denote the space of almost continuous real-valued functions on a space $X$ endowed with the topology of pointwise convergence. It is shown that $AC_p(X)$ is a topological algebra. We investigate properties of spaces $X$, for which the space $AC_p(X)$ is Lindelöf (\v Cech complete).
Keywords	almost continuous functions, topological algebra, Lindelöf property, Čech completeness, real-valued functions, function spaces
DOI	doi:10.30970/ms.14.2.197-201
Reference	1. Архангельский А. В. Топологические пространства функций. – М.: Изд. МГУ, 1989. 2. Архангельский А. В., Бокало Б. М. Касание топологий и тангенциальные свойства топологических пространств // ТММО. – 1992. – Т.54. – С.160–185. 3. Энгелькинг Р. Общая топология. – М.: Мир, 1986. 4. Pol R. The Lindelöf property and its analogue in function spaces with weak topology // Topology. 4-th collog. Budapest, 1978. V.2. – Amsterdam, 1980. – P.965—969. 5. Banakh T., Bokalo B. On almost continuous function. – Lviv University, 1999. – preprint.
Pages	197-201
Volume	14
Issue	2
Year	2000
Journal	Matematychni Studii
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