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.

properties

• Every paracompact Hausdorff room is normal . The converse does not apply, as the long straight line shows.
• Closed sub-spaces of paracompact spaces are again paracompact.
• Products of paracompact rooms are generally not paracompact again, not even normal, as the Sorgenfrey level shows, see also Tamano's theorem .

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.

literature

• James Dugundji : Topology . 8th printing. Allyn and Bacon, Boston 1973, OCLC 256193625 .
• Lutz Führer : General topology with applications . Vieweg, Braunschweig 1977, ISBN 3-528-03059-3 .
• Gregory Naber: Set-theoretic Topology. With Emphasis on Problems from the Theory of Coverings, Zero dimensionality and cardinal invariants . University Microfilms International, Ann Arbor MI 1977, ISBN 0-8357-0257-X .
• Jun-iti Nagata : Modern General Topology (=  North-Holland Mathematical Library . Volume 33 ). 2nd, revised edition. North Holland, Amsterdam a. a. 1985, ISBN 0-444-87655-3 .
• Boto von Querenburg : Set theoretical topology . 3rd, revised and expanded edition. Springer, Berlin a. a. 2001, ISBN 3-540-67790-9 .
• Horst Schubert : Topology. An introduction . 4th edition. BG Teubner Verlag, Stuttgart 1975, ISBN 3-519-12200-6 .
• Stephen Willard: General Topology . Addison-Wesley, Reading MA et al. a. 1970, ISBN 0-201-08707-3 .

Individual evidence

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