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Open maps do not preserve Whyburn property. (English) Zbl 1098.54008
A space \(X\) is Whyburn (weakly Whyburn) provided that for each non-closed \(A\subset X\) and for all \(x\in \overline A\setminus A\) (for some \(x\in \overline A\setminus A\), respectively), there is \(F\subset A\) with \(\overline F \setminus F=\{ x\}\). Two topological abelian groups \(G\) and \(H\) are constructed such that \(G\) is Whyburn, \(H\) is not weakly Whyburn and there is an open continuous homomorphism mapping \(G\) onto \(H\). Two more examples in the same direction in the framework of topological spaces are given.
MSC:
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54D55 Sequential spaces
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