Cover (topology)

In mathematics, a coverage is a fundamental concept of the topology. Open overlaps play especially in the compactness of topological spaces play an important role.

• 2.1 Open coverage
• 2.2 compactness
• 2.3 coverage characteristics

Definitions

Coverage

A family of subsets of means coverage of when

Applies. The overlap is called finite (or countable ) if the index set is finite (or countable ) is.

Subcovering

Are and surpluses, so called partial overlap of, if at all one exists with.

Refinement

Are and again two coverings of so called finer than when there is an index to each, so true. The quantity system is then called fining or refining cover of.

Surpluses in topological spaces

Open cover

A cover of a topological space is called open (or closed ) when all are open (or closed).

Compactness

A topological space is called compact if every open cover of a finite subcover contains.

Coverage characteristics

• An eclipse is called point- finite if each point of the space is at most a finite number of coverage amounts.
• An eclipse is called locally finite if every point of the space has a neighborhood that intersects at most finitely many coverage amounts. As is known, a topological space is called paracompact if every open cover has a locally finite covering.
• An eclipse is called locally - finite if it can be written as a countable union of families of sets, so that each point of space to each has a neighborhood that intersects only finitely many sets from.
• An eclipse is called - discrete if it can be written as a countable union of families of sets, so that there is at any point and at any one neighborhood of this point, which intersects at most one of the sets from. The discrete - and - locally finite coverings play an important role in the set of Bing - Nagata - Smirnov.