Lebesgue's number lemma

A Lebesguezahl is a ( non-unique ) number that you can assign to an open covering of a compact metric space. It was named after the French mathematician Henri Léon Lebesgue.

They are often used as an aid when finiteness conditions are given.

Theorem of the existence

The set of the existence of a Lebesguezahl or the lemma of Lebesgue is a lemma from the field of topology.

He says that is true for every compact metric space with metric:

For every open covering exists a number such that each subset is included with diameter in a coverage amount, ie. Such a number is called Lebesguezahl of coverage for.

Each smaller number is thus of course also a Lebesguezahl to this coverage and this room.

Evidence

If any number can be chosen, since all subsets contained in a coverage amount.

So be now. Since is compact, can be selected from a finite subcover, so be a (finite ) covering of X.

For all, set and define by a function.

For an arbitrary but fixed now, so choose that. Now choose a small enough so that the environment is selected from the coverage amount, so. Now, that is. The function is therefore very positively to.

As defined and continuously on a compact set, it assumes a minimum. This is the sought Lebesguezahl:

Let be a subset of small diameter. For each now in the environment of. Now select any.

Is now chosen so that is a maximum of. Now it is and the environment of and therefore are quite out of the coverage. This is now a so with the Lebesguezahl property found.

Applications

The Lebesguezahl is used in the proof of several basic sets of algebraic topology, as in the proof of the theorem of Seifert -van Kampen or the Mayer -Vietoris sequence and the Ausschneidungsaxioms the singular homology.