On countable connected Hausdorff spaces in which the intersection of every pair of connected subsets is connected.
We call a function P-preserving if, for every subspace with property P, its image also has property P. Of course, all continuous maps are both compactness- and connectedness-preserving and the natural question about when the converse of this holds, i.e. under what conditions such a map is continuous, has a long history. Our main result is that any nontrivial product function, i.e. one having at least two nonconstant factors, that has connected domain, range, and is connectedness-preserving...
This article gives a short and elementary proof of the fact that the connectedness of the boundary of an open domain in ℝⁿ is equivalent to the connectedness of its complement.