On normality of spaces of scatteredly continuous maps

B. M. Bokalo; N. M. Kolos

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	On normality of spaces of scatteredly continuous maps
Author	B. M. Bokalo, N. M. Kolos Lviv National University
Abstract	A map $f\colon X\rightarrow Y$ between topological spaces is called scatteredly continuous if for each non-empty subspace $A\subset X$ the restriction $f\|_{A}$ has a point of continuity. By $SC_p (X)$ we denote the space of all scatteredly continuous real-valued functions on $X$ endowed with the topology of pointwise convergence. In this paper we focus on the normality of the space $SC_p(X)$ . Particularly, it is proved that if the function space $SC_p(X)$ is normal, then all compact and all scattered subspaces of $X$ are countable.
Keywords	topological spaces; pointwise convergence
Reference	1. Engelking R. General Topology. – PWN, Warzawa, 1977. 2. Bokalo B., Kolos N. When does $SC_p(X)=\mathbb{R}^X$ hold?// Topology – 2009. – V.48. – P. 178–181. 3. Arkhangel’skii A.V. Topological spaces of functions. – M.: MGU, 1989. (in Russian) 4. Arkhangel’skii A.V., Bokalo B.M. The tangency of topologies and tangential properties of topological spaces// Trudy Moskov. Mat. Obshch. – 1992. – V.54. – P. 160–185, 278–279. (in Russian) 5. Banakh T., Bokalo B. On scatteredly continuous maps between topological spaces// Topology and Appl. – 2010. – V.157. – P. 108–122. 6. Aleksandrov P.S., Proskuryakov I.V. On reducible sets// Izv. Akad. Nauk SSSR, Ser. Mat. – 1941. – V.5, №3. – P. 217–224. (in Russian) 7. Choban M.M., Dodon N.K. Theory of P-scattered spaces. – Stiintsa, Kishinev, 1979. (in Russian)
Pages	196-204
Volume	35
Issue	2
Year	2011
Journal	Matematychni Studii
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