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On maps preserving connectedness and/or compactness

István Juhász, Jan van Mill (2018)

Commentationes Mathematicae Universitatis Carolinae

We call a function f : X Y P-preserving if, for every subspace A X with property P, its image f ( A ) 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, T 1 range, and is connectedness-preserving...

On the connectedness of boundary and complement for domains

Andrzej Czarnecki, Marcin Kulczycki, Wojciech Lubawski (2011)

Annales Polonici Mathematici

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.

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