James Tauber

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James Tauber's Blog 2004/12/30

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

by jtauber : Created on Dec. 30, 2004 : Last modified Aug. 20, 2005 : Categories poincare_project : 0 comments (permalink)

Favourite Posts of 2004

I mentioned in Blog Hits by Age that I would, as others have done recently, list my favourite entries from my blog this year.

Here are the ones that come to mind. Some generated some good discussion in the blogosphere at the time; others disappointingly didn't generate any response at all.

In no particular order...

Conference Reporting

Questions and Observations

Programming Ideas

Little Python Scripts

Typed Citations Meme

Aggregation versus Hosting Meme

XML versus RDF Meme

by jtauber : Created on Dec. 30, 2004 : Last modified Feb. 8, 2005 : (permalink)