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  "body": "\n## Summary\n\nA paracompact space is a [[topological space|0CQHFPGJ45]] in which every [[open cover|049PXYHW1T]] has an open refinement that is a [[locally finite family|DZ297MMGV8]].\nEvery [[compact space|0TVJZYB6SH]] and every [[metric space|0VKD1TP7F3]] is paracompact.\nParacompact Hausdorff spaces are [[normal|0NSR9V8TED]], and they admit partitions of unity subordinate to any open cover.\nThis is why paracompactness is a standard assumption on manifolds: it lets one glue local constructions into global ones.\n\n## Explanations\n\n### L1 human\n\nCompactness asks that every open cover can be cut down to finitely many sets.\nParacompactness asks less: every open cover can be replaced by smaller open sets that are only \"finitely many near each point\".\nOn the real line, any open cover can be refined to small intervals that pile up only finitely often around each point.\nThis local finiteness is enough to add up infinitely many local pieces without trouble.\n",
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