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Homogeneity and \(\sigma\)-discrete sets. (English) Zbl 0767.54013
A space \(X\) is densely homogeneous (DH) provided that \(X\) has a \(\sigma\)- discrete dense subset and that for every two \(\sigma\)-discrete dense subsets \(A\) and \(B\) of \(X\), there is an autohomeomorphism \(h\) on \(X\) such that \(h(A)=B\). If the “\(\sigma\)-discrete” in this definition is replaced by “countable” then the space is called countable dense homogeneous (CDH). This paper continues the study of these properties by first showing that all spaces satisfying DH or CDH are \(T_ 1\)-spaces. Also a characterization is given for when all \(\sigma\)-discrete dense subsets of a metric space are homeomorphic. From this it follows that each two \(\sigma\)-discrete dense subsets of a homogeneous metric space are homeomorphic.

MSC:
54C99 Maps and general types of topological spaces defined by maps
54E99 Topological spaces with richer structures
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