Spaces having a compactification which is a \(C\)-space. (English) Zbl 1055.54009

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 \(X\).


54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54F45 Dimension theory in general topology
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