On scatteredly continuous maps between topological spaces

Details On scatteredly continuous maps between topological spaces On scatteredly continuous maps between topological spaces

2008-01-14

arxiv.org/abs/0801.2131v3

Topology Appl. 157:1 (2010), 108-122

Mathematics

Geometric Topology

We have added a (consistent) example of a $G_\delta$-measurable map which is not piecewise continuous. This answers an old que

Scientific paper

A map $f:X\to Y$ between topological spaces is defined to be {\em scatteredly continuous} if for each subspace $A\subset X$ the restriction $f|A$ has a point of continuity. We show that for a function $f:X\to Y$ from a perfectly paracompact hereditarily Baire Preiss-Simon space $X$ into a regular space $Y$ the scattered continuity of $f$ is equivalent to (i) the weak discontinuity (for each subset $A\subset X$ the set $D(f|A)$ of discontinuity points of $f|A$ is nowhere dense in $A$), (ii) the $\sigma$-continuity ($X$ can be written as a countable union of closed subsets on which $f$ is continuous), (iii) the $G_\delta$-measurability (the preimage of each open set is of type $G_\delta$). Also under Martin Axiom, we construct a $G_\delta$-measurable map $f:X\to Y$ between metrizable separable spaces, which is not piecewise continuous. This answers an old question of V.Vinokurov.

Affiliated with

Also associated with

No associations

Rating

On scatteredly continuous maps between topological spaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On scatteredly continuous maps between topological spaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On scatteredly continuous maps between topological spaces will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-634814