Q-compactifications of metric spaces

A. V. Arkhangel'skii

Sbornik. Mathematics, Tome 20 (1973) no. 1, pp. 85-94

Résumé

For $Q$-spaces (also called functionally closed or Hunt spaces) there are defined in this paper two new invariants, the $q$-weight and the $q^*$-weight. With the aid of these the following results are obtained. Theorem 1. {\it If $\tau$ is a nonmeasurable cardinal number and $X$ is a metric space of weight not exceeding $\tau$, then $X$ is homeomorphic to a closed subspace of the product of $\tau^{\aleph_0}$ copies of a real line $R$ $($i.e. X\subset_\mathrm{cl}R^{(\tau^{\aleph_0})})$}. \smallskip Theorem~2. {\it If~$\tau$ is a~nonmeasurable cardinal number and~$X$ is a~complete uniform space whose uniform and topological weights do not exceed~$\tau$, then~$X$ is homeomorphic to a~closed subspace of the product of $\tau^{\aleph_0}$ copies of the real line.} \smallskip Theorem~3. {\it Let~$X$ be paracompact, $bX$~a~Hausdorff compactification of~$X$, and~$\tau$ a~nonmeasurable cardinal number such that the weight of~$X$ does not exceed~$\tau$ and~$X$ is the intersection of a~family of not more than~$\tau$ open subsets of~$bX$. Then~$X$ is homeomorphic to a~closed subspace of the product of $\tau^{\aleph_0}$ copies of the real line.} Bibliography: 8 titles.