Finite unions of \(D\)-spaces and applications of nearly good relation (Q1956117)

Summary: Some results are obtained on finite unions of \(D\)-spaces. It is proved that if a space is the union of finitely many locally compact \(D\)-subspaces, then it is a \(D\)-space. It follows that a space is a \(D\)-space if it is the union of finitely many locally compact submetacompact subspaces. And a space is a \(D\)-space if it is the union of a \(D\)-subspace with a locally compact \(D\)-subspace. This partially answers one problem raised by Arhangel'skii. At last, some examples are given to exhibit the applications of nearly good relation to discover \(D\)-classes.