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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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