Embedding topological spaces into Hausdorff κ-bounded spaces

摘要

Let kappa be an infinite cardinal. A topological space X is kappa-bounded if the closure of any subset of cardinality = kappa in X is compact. We discuss the problem of embeddability of topological spaces into Hausdorff (Urysohn, regular) kappa-bounded spaces, and present a canonical construction of such an embedding. Also we construct a (consistent) example of a sequentially compact separable regular space that cannot be embedded into a Hausdorff omega-bounded space. (C) 2020 Elsevier B.V. All rights reserved.