Relatively compact subspace

A relatively compact subset is a term from the mathematical branch of topology. There is a weakening of the notion of topological compact space.

Definition

A subset of a topological space is called relatively compact if in their topological completion is compact. itself does not need to be compact.

You write it.

Other characterizations

• There is a ( often in applications: open ) subset. A subset is relatively compact if and only in, if is limited and the degree of non- hits in the edge of.
• There are generally a subset of a Hausdorff space and a subset of; continue to be the conclusion of in. Then if and only relatively compact if it is compact and contained in.
• A subset of a metric space is relatively compact if and only if every sequence has a convergent subsequence in.

An example of

As an example, a set of real numbers are used ( with the usual Euclidean topology). Such a set of real numbers is compact, if any infinite sequence of numbers from this set contains an infinite subsequence " is arbitrarily close to " another number, said another number must also belong to this set.

While there are () no boundary points, exists for the set of all positive real numbers, the boundary point ( but not heard ). Because of these statements is true boundary point, the conclusion of in is equal to the set of all real numbers between ( exclusive) and (inclusive). This amount is not compact ( because you re missing the point of accumulation ), so is not relatively compact in.

Applications

The concept of relative compactness is, inter alia, used

• In the definition of a compact operator
• In the set of Arzelà - Ascoli
• In the fixed point theorem of Schauder.