Poincare Project: Open Coverings and Compactness

If you pick a collection of open sets whose union is the space's entire set, then that collection is called an open covering of the space.

For example, consider the set {a, b} with topology { {}, {a}, {b}, {a, b} }. One open covering would be:

{ {a}, {b} }

Another would be

{ {a}, {a, b} }

Clearly it is possible to cover any finite topological space with a finite number of open sets.

It is also possible to cover any infinite topological space with a finite number of open sets. Because X is an open set in any topology on X, a collection consisting of just X itself is an open covering.

If an open covering has a finite subset which still manages to cover the entire set, the covering is said to have a finite subcovering.

Some topological spaces have the property that every open covering has a finite subcovering. Such a space is said to be compact.

Compactness is a topological property. Recall that this means if a topological space is compact, any topological spaces homeomorphic to it will also be compact (and also that a homeomorphism can't exist between a compact topological space and one that is not compact).

UPDATE: next post