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Compactly convex sets in linear topological spaces

Author T. Banakh, M. Mitrofanov, O. Ravsky
tbanakh@yahoo.com
Ivan Franko National University of Lviv (Ukraine) and Jan Kochanowski University, Kielce (Poland); Institute of Applied Problems of Mathematics and Mechanics of Ukrainian Academy of Science, Lviv, Ukraine

Abstract A convex subset X of a linear topological space is called {\em compactly convex} if there is a~continuous compact-valued map Φ:Xexp(X) such that [x,y]Φ(x)Φ(y) for all x,yX. We prove that each convex subset of the plane is compactly convex. On the other hand, the space R3 contains a convex set that is not compactly convex. Each compactly convex subset X of a linear topological space L has locally compact closure ˉX which is metrizable if and only if each compact subset of X is metrizable.
Keywords compactly convex set; linear topological space
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Pages 161-173
Volume 37
Issue 2
Year 2012
Journal Matematychni Studii
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