Position of subspaces in spaces: Relative versions of compactness, Lindelöf properties, and separation axioms
The authors define relative versions of certain standard topological properties and state certain theorems and examples concerning them. No proofs are given. Let \(X\) be a topological space and let \(Y\) be a subspace of \(X\). The authors say that \(Y\) is regular (superregular) in \(X\) if for each \(y\in Y\) and any set \(A\) which is closed in \(X\) and not containing \(y\) there are disjoint subsets \(U\) and \(V\) open in \(X\) such that \(y\in U\) and \(A\cap Y\subseteq V\) (respectively, \(y\in U\) and \(A\subseteq V\)). They define \(Y\) to be strongly regular in \(X\) if for each \(x\in X\) and each set \(F\) not containing \(x\) and closed in \(X\) there are disjoint open sets \(U\), \(W\) in \(X\) such that \(x\in U\) and \(F\cap Y\subseteq W\). Proposition 1: There exists a \(T_ 2\)-space \(X\) having a countable closed discrete subspace \(Y\) such that \(Y\) is not regular in \(X\). A space \(Y\) is called strongly normal in a space \(X\) if for any two disjoint sets \(A\) and \(B\) closed in \(Y\) there are disjoint open sets \(U\) and \(V\) in \(X\) such that \(A\subseteq U\) and \(B\subseteq V\). Theorem 1: Suppose \(Y\) is regular in \(X\) and \(Y\) is a Lindelöf subspace. Then \(Y\) is strongly normal in \(X\). Propositions 6 and 7: If \(Y\) is compact in a \(T_ 2\)-space \(X\), then \(Y\) is strongly regular in \(X\), but \(Y\) is not necessarily superregular in \(X\). The paper contains other generalizations of classical theorems of general topology and several examples showing the scope and limitations of the results obtained. The paper concludes with a problem.
scientific article; zbMATH DE number 7030