Paracompact space

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Paracompactness is a term from the mathematical branch of topology . It describes a property of topological spaces which plays an essential role in many theorems of topology. The concept of paracompactness was introduced in 1944 by the French mathematician Jean Dieudonné .

In fact, many of the common topological spaces are even paracompact Hausdorff spaces . Some authors always assume the Hausdorff property for paracompact rooms. Paracompact Hausdorff spaces include, in particular, all metric spaces ( Arthur Harold Stone's theorem ) and all manifolds (here paracompactness is part of the usual definition). It is more difficult to find non-paracompact rooms. A common counterexample is the so-called long straight line .

Paracompactness is a weakened form of compactness ; for example, the set of real numbers in the usual topology is paracompact, but not compact.


A topological space M is paracompact if every open cover has a locally finite open refinement .

For comparison: A topological space M is compact if every open cover has a finite partial cover .

It means:

  • open cover of : a family of open sets whose union contains :;
  • Partial coverage : a partial family whose union still contains;
  • Refinement : a new cover , where each set must be included in at least one set of the old cover;
  • locally finite : for each there is an environment that only intersects finitely many sets .


  • Metric spaces are paracompact, the reverse is not true.
  • Differentiable manifolds (which are Hausdorff's by definition and which satisfy the second axiom of countability ) are always paracompact. Paracompactness is often assumed as part of the definition, but it also follows from the Hausdorff condition and the second axiom of countability. In general, non-Hausdorff manifolds need not be paracompact. From the paracompactness follows the existence of a decomposition of the one , which makes the topological property of the paracompactness important for the integration theory on differentiable manifolds.



  • If one demands the defining property only for countable coverages, one speaks of a countable paracompact space . Paracompact spaces are of course countable paracompact, the reverse is not true.
  • If, in the definition of paracompact space, one only demands of the refinement that it is point-finite, that is, each point only contains finitely many sets of refinement, then one speaks of a metacompact space . Paracompact spaces are of course metacompact. The example of the Dieudonné plank shows that the reverse does not apply.


Individual evidence

  1. Guide: General Topology with Applications. 1977, p. 135.
  2. ^ Schubert: Topology. 1975, p. 84.
  3. ^ Schubert: Topology. 1975, p. 90.