Weakly discontinuous and resolvable functions between topological spaces

Authors : Taras BANAKH, Bogdan BOKALO

Pages : 103-110

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Publication Date : 2017-02-01

Article Type : Research Paper

Abstract :We prove that a function $f:X\to Y$ from a first-countable insert ignore into journalissuearticles values(more generally, Preiss-Simon); space $X$ to a regular space $Y$ is weakly discontinuous insert ignore into journalissuearticles values(which means that every subspace $A\subset X$ contains an open dense subset $U\subset A$ such that $f|U$ is continuous); if and only if $f$ is open-resolvable insert ignore into journalissuearticles values(in the sense that for every open subset $U\subset Y$ the preimage $f^{-1}insert ignore into journalissuearticles values(U);$ is a resolvable subset of $X$); if and only if $f$ is resolvable insert ignore into journalissuearticles values(in the sense that for every resolvable subset $R\subset Y$ the preimage $f^{-1}insert ignore into journalissuearticles values(R);$ is a resolvable subset of $X$);. For functions on metrizable spaces this characterization was announced insert ignore into journalissuearticles values(without proof); by Vinokurov in 1985.
Keywords : Weakly discontinuous function, resolvable function, Preiss Simon space