On properties of \(h\)-homogeneous spaces of first category
A metric space \(X\) is said to be \(h\)-homogeneous if \(X\) is strongly zero-dimensional and each nonempty clopen subset of \(X\) is homeomorphic to \(X\). In the paper under review, the author investigates some properties of \(h\)-homogeneous spaces of the first category. In particular, the author describes how to assign, to any given strongly zero-dimensional metric space \(X\) of weight \(\leq k\), a first category \(h\)-homogeneous space \(h(X,k)\) of weight \(k\). Precisely, the main result of the paper (Theorem 6) states that in the family \(\mathcal H_k(X)\) of first category, \(h\)-homogeneous spaces of weight \(k\) that include a closed copy of \(X\), there exists a unique (up to homeomorphisms, of course) element with the following minimality property: every space in \(\mathcal H_k(X)\) includes a closed copy of \(h(X,k)\). This element \(h(X,k)\) is called the \(h\)-homogeneous extension of the space \(X\) of weight \(k\) with respect to spaces of first category. The proof of Theorem~6 provides a direct construction of \(h(X,k)\) for any space \(X\).
scientific article; zbMATH DE number 5835561
scientific article