Some topological spaces have the property that every open covering has a finite subcovering. 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. Because X is an open set in any topology on X, a collection consisting of just X itself is an open covering. It is also possible to cover any infinite topological space with a finite number of open sets. One open covering would be:Ĭlearly it is possible to cover any finite topological space with a finite number of open sets.

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.įor example, consider the set. Poincare Project: Open Coverings and Compactness

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Some generated some good discussion in the blogosphere at the time others disappointingly didn't generate any response at all. I mentioned in Blog Hits by Age that I would, as others have done recently, list my favourite entries from my blog this year.

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