On a problem for spaces of the first category
It is proved that every metric space of first category having the property that the weight of each nonvoid open subset is at least a cardinal k, contains as a closed subspace a homeomorph of the small \(\sigma\)-product of countably many discrete spaces of cardinality k (in fact, a more general result for regular spaces is proved). Another result shows that a metric space is of first category iff it contains a \(\sigma\)-discrete dense \(g_{\delta}\)-set having no isolated point. At the end, several results without proofs are stated dealing with special types of homogenities.
\(\sigma \)- discrete dense \(g_{\delta }\)-set
scientific article; zbMATH DE number 3994530