Spaces having a compactification which is a \(C\)-space (Q1880710)

The authors characterize those separable metrizable spaces that can be densely embedded into some compact metrizable \(C\)-space. A space \(X\) is called a \(C\)-space provided for every sequence \((\mathcal{C}_n)\) of open covers of \(X\) there exists a sequence \((\mathcal{H}_n)\) of collections of pairwise disjoint open subsets of \(X\) such that \(\bigcup_n\mathcal{H}_n\) covers \(X\) and \(\mathcal{H}_n\) refines \(\mathcal{C}_n\) for each \(n\in\mathbb{N}\). The characterization is in terms of the existence of suitable bases for \(X\).