# Paracompact space

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.

## definition

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 :;${\ displaystyle M}$ ${\ displaystyle (U_ {i}) _ {i \ in I}}$${\ displaystyle M}$${\ displaystyle M \ subseteq \ bigcup _ {i \ in I} U_ {i}}$
• Partial coverage : a partial family whose union still contains;${\ displaystyle (U_ {i}) _ {i \ in I_ {0}} (I_ {0} \ subseteq I)}$${\ displaystyle M}$
• Refinement : a new cover , where each set must be included in at least one set of the old cover;${\ displaystyle (V_ {j}) _ {j \ in J}}$${\ displaystyle V_ {j}}$${\ displaystyle U_ {i}}$
• locally finite : for each there is an environment that only intersects finitely many sets .${\ displaystyle x \ in M}$${\ displaystyle V_ {j}}$

## Examples

• 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.

## Attenuations

• 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.