High Quality Content by WIKIPEDIA articles! In mathematics, a paracompact space is a topological space in which every open cover admits an open locally finite refinement. Paracompact spaces are often required to be Hausdorff, but we will not make that assumption in this article.As you might guess from the generality of most of the examples above, it is actually harder to think of spaces that are not paracompact than to think of spaces that are. The most famous counterexample is the long line, which is a nonparacompact topological manifold. (The long line is locally compact, but not second countable.) Another counterexample is a product of uncountably many copies of an infinite discrete space. Any infinite set carrying the particular point topology is not paracompact; in fact it is not even metacompact.

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