Paracompact space

Para compactness is a term from the mathematical branch of topology. It describes a topological feature spaces that plays an essential role in many sets of the topology. The concept of compactness parameter 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 use for paracompact spaces, the Hausdorff property always with advance. Among the paracompact Hausdorff spaces include in particular all metric spaces ( set of Arthur Harold Stone) and all manifolds. It is more difficult to find non - paracompact spaces is. A common counter-example is the so-called long straight.

Para compactness is an attenuated form of compactness; For example, the set of real numbers in the usual amount topology para compact 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 subcover.

Where is:

• Open cover of: a family of open sets whose union contains :;
• Partial overlap: one part family whose union contains still;
• Refinement: a new overlap, each amount must be included in at least one quantity of the old coverage;
• Locally finite: for each there is a neighborhood that intersects only finitely many sets.

Properties

• Every paracompact Hausdorff space is normal. The converse is not true, as the long straight occupied.
• Closed subspaces para compact spaces are paracompact again.
• Products para compact spaces are generally not re- paracompact, not even normal as the Sorgenfrey - level shows, see also set of Tamano.