On SAC- P Spaces and Nowhere Densely Reflexive Properties

In this paper, we investigate how the properties of the nowhere dense subspaces reflect those of the entire space. We call a space X an SAC-P space if the set of accumulation points of any infinite family of mutually disjoint open subsets of X is a non-empty set having property P. We prove that for a regular space X, X is SAC-compact precisely when it is compact. In general, for a regular space X, X is SAC-P (here, P is a property which closed subsets inherit) exactly when X is feebly compact and each closed nowhere dense subspace of X has property P. Similar to discrete reflexivity, we introduce the notion of nowhere dense reflexivity. A property P is nowhere densely reflexive inA if each space X∈A has property P precisely when each closed nowhere dense subspace of X has property P. We prove that compactness, linear Lindelöfness and initial κ-compactness are all nowhere densely reflexive both in the class of feebly compact spaces and in that of dense-in-itself spaces. For topological groups, we prove that if G is a non-compact connected feathered topological group, then G is metrizable precisely when each closed nowhere dense subgroup of G is metrizable.