2015/12/24 by V. V Mykhaylyuk, Mykhaylyuk, V. V
Mathematics · #FOS: Mathematics #General Topology (math.GN) #math.GN
paper · pdf · doi:10.48550/arxiv.1512.07758
arxiv created 2015/12/24 · arxiv updated 2015/12/25
It is solved a problem of construction of separately continuous functions on the product of compacts with a given discontinuity points set. We obtaine the following results. 1. For arbitrary Čech complete spaces X, Y and a separable compact perfect projectively nowhere dense zero set E⊆ X× Y there exists a separately continuous function f:X× Y→\mathbb R the discontinuity points set of which equals to E. 2. For arbitrary Čech complete spaces X, Y and nowhere dense zero sets A⊆ X and B⊆ Y there exists a separately continuous function f:X× Y→\mathbb R such that the projections of the discontinuity points set of f coincide with A and B respectively. An example of Eberlein compacts X, Y and nowhere dense zero sets A⊆ X and B⊆ Y such that the discontinuity points set of every separately continuous function f:X× Y→\mathbb R does not coincide with A× B, and CH-example of separable Valdivia compacts X, Y and separable nowhere dense zero sets A⊆ X and B⊆ Y such that the discontinuity points set of every separately continuous function f:X× Y→\mathbb R does not coincide with A× B are constructed.