Embedding topological spaces into Hausdorff \(\kappa\)-bounded spaces (Q776386)

The system ZFC is the set-theoretic framework for this article. Let \(\kappa\) be an infinite cardinal and let \(X\) be a topological space. Then \(X\) is called \(\kappa\)-bounded if, for every subset \(A\) of \(X\) such that \(|A|\leq\kappa\), the closure of \(A\) in \(X\) is compact. For a non-empty family \(\mathcal{F}\) of closed subsets of \(X\), the authors introduce the notions of an \(\mathcal{F}\)-regular, strongly \(\mathcal{F}\)-regular, \(\mathcal{F}\)-Tychonoff, \(\mathcal{F}\)-normal, strongly \(\mathcal{F}\)-normal and a totally \(\mathcal{F}\)-normal space. For the family \(\mathcal{F}\) of closed subsets of the closures of subsets of cardinality \(\leq\kappa\) of \(X\), the space \(X\) is called \(\overline{\kappa}\)-regular (respectively, strongly \(\overline{\kappa}\)-regular and so on) if \(X\) is \(\mathcal{F}\)-regular (respectively, strongly \(\mathcal{F}\)-regular and so on). For the Wallman extension \(WX\) of a \(T_1\)-space \(X\), let \(j_X\) be the canonical embedding of \(X\) into \(WX\). Then the subspace \(W_{\overline{\kappa}}X=\bigcup\{\overline{j_X(C)}: C\subseteq X\wedge |C|\leq\kappa\}\) of \(WX\) is called the Wallman \(\kappa\)-bounded extension of \(X\). For a \(T_1\)-space \(X\) and an arbitrary infinite cardinal \(\kappa\), the authors consider the following statements: (1) \(X\) is \(\overline{\kappa}\)-normal, (2) \(W_{\overline{\kappa}}X\) is Hausdorff, (3) \(X\) is homeomorphic to a subspace of a Hausdorff \(\kappa\)-bounded space, (4) \(X\) is \(\overline{\kappa}\)-regular, (5) \(X\) is strongly \(\overline{\kappa}\)-normal, (6) \(W_{\overline{\kappa}}X\) is Urysohn, (7) \(X\) is homeomorphic to a subspace of a Urysohn \(\kappa\)-bounded space; (8) \(X\) is strongly \(\overline{\kappa}\)-regular, (9) \(X\) is totally \(\overline{\kappa}\)-normal, (10) \(W_{\overline{\kappa}}X\) is regular, (11) \(X\) is homeomorphic to a subspace of a regular \(\kappa\)-bounded space, (12) \(X\) is regular. The authors show that \((1)\leftrightarrow (2)\rightarrow (3)\rightarrow (4)\), \((5)\leftrightarrow (6)\rightarrow (7)\rightarrow (8)\) and \((9)\leftrightarrow (10)\rightarrow (11)\rightarrow (12)\). Furthermore: if each closed subspace of \(X\) of density \(\leq\kappa\) is Lindelöf, then \((4)\rightarrow (1)\); if each closed subspace of \(X\) of density \(\leq\kappa\) is countably paracompact in \(X\) and Lindelöf, then \((8)\rightarrow (5)\); if each closed subspace of \(X\) of density \(\leq\kappa\) is paracompact in \(X\), then \((12)\rightarrow (9)\). By giving suitable examples, the authors show that the following spaces exist in ZFC: (a) a first-countable \(\overline{\omega}\)-normal \(T_3\)-space which is neither functionally Hausdorff nor strongly \(\overline{\omega}\)-normal; (b) a Tychonoff, zero-dimensional, locally compact, locally countable \(\overline{\omega}\)-normal space which is not strongly \(\overline{\omega}\)-normal; (c) a totally \(\overline{\omega}\)-normal, \(\omega\)-bounded \(T_3\)-space which is not functionally Hausdorff; (d) a \(\overline{\kappa}\)-normal, \(\kappa\)-bounded, \(H\)-compact Hausdorff space which is not Urysohn. Finally, the authors prove that it is consistent with ZFC that there exists a separable, sequentially compact scattered \(T_3\)-space which is not \(\overline{\omega}\)-regular, so it cannot be embedded into an \(\omega\)-bounded Hausdorff space. The problem of whether there exists in ZFC a separable, sequentially compact \(T_3\)-space which is not Tychonoff is posed.