A paracompact space is a Hausdorff space such that every open cover has a locally finite open refinement. Paracompactness is a very common property that topological spaces satisfy. Paracompactness is similar to the compactness property, but generalized for slightly "bigger" spaces. All manifolds (e.g, second countable and T2-spaces) are paracompact.

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Weisstein, Eric W. "Paracompact Space." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ParacompactSpace.html