The invariance of the Lindelöf number under some discontinuous functions

Анотація

A function $f:X\to Y$ between topological spaces is called scatteredly continuous (barely continuous) if for each non-empty (closed) subspace $A\subset X$ the restriction $f{|}_A$ has a point of continuity. We show that if $f:X\to Y$ is a scatteredly continuous (barely continuous) surjective function between topological spaces, then for each natural number $n$ we have $hl\left(Y^n\right)\le hl\left(X^n\right)$ ($l\left(Y\right)\le hl\left(X\right)$, respectively).