Lebesgue number

A Lebesgue number is a (ambiguous) number that can be assigned to an open cover of a compact metric space . It was named after the French mathematician Henri Léon Lebesgue .

It often serves as an aid when finite conditions are given.

Theorem of existence

The theorem of the existence of a Lebesgue number or the Lebesgue lemma is a lemma from the field of topology .

It says that for every compact metric space with metric : ${\displaystyle X}$ ${\displaystyle d}$

For each open covering exists a number so that each subset with diameter in a coverage amount is included, ie . Such a number is called the Lebesgue number of the coverage for . ${\displaystyle {\mathcal {U}}}$ ${\displaystyle \delta >0}$ ${\displaystyle A\subseteq X}$ ${\displaystyle d(A)<\delta }$ ${\displaystyle U\in {\mathcal {U}}}$ ${\displaystyle A\subseteq U}$ ${\displaystyle \delta }$ ${\displaystyle {\mathcal {U}}}$ ${\displaystyle X}$

Every smaller number is of course also a Lebesgue number for this coverage and this space.

proof

If so, any number can be chosen, since all subsets are contained in a coverage set. ${\displaystyle X\in {\mathcal {U}}}$ ${\displaystyle \delta >0}$ ${\displaystyle A\subseteq X}$

So be now . As is compact, can be made choose a finite sub-covering, so be a (finite) covering of X . ${\displaystyle X\not \in {\mathcal {U}}}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {U}}}$ ${\displaystyle \{A_{1},\dots ,A_{n}\}\subseteq {\mathcal {U}}}$

For all , set and define a function by . ${\displaystyle i\in \{1,\dots ,n\}}$ ${\displaystyle C_{i}:=X\setminus A_{i}}$ ${\displaystyle f\colon X\rightarrow \mathbb {R} }$ ${\displaystyle \textstyle f(x):={\frac {1}{n}}\sum _{i=1}^{n}d(x,C_{i})}$

For any, but fixed, choose now so that . Choose now a small enough so that the environment of lies in the selected coverage amount, ie . Well is so is . The function is therefore very positive. ${\displaystyle x\in X}$ ${\displaystyle i}$ ${\displaystyle x\in A_{i}}$ ${\displaystyle \epsilon >0}$ ${\displaystyle \epsilon }$ ${\displaystyle x}$ ${\displaystyle U_{\epsilon }(x)\subseteq A_{i}}$ ${\displaystyle d(x,C_{i})\geq \epsilon }$ ${\displaystyle f(x)\geq {\frac {\epsilon }{n}}}$ ${\displaystyle f}$ ${\displaystyle X}$

Since continuous and is defined on a compact , it assumes a minimum . This is the Lebesgue number you are looking for: ${\displaystyle f}$ ${\displaystyle \delta >0}$

Let a subset with diameter be smaller . For each is now in the neighborhood of . Now choose any one . ${\displaystyle B\subseteq X,d(B)<\delta }$ ${\displaystyle \delta }$ ${\displaystyle x\in B}$ ${\displaystyle B}$ ${\displaystyle \delta }$ ${\displaystyle x}$ ${\displaystyle x_{0}\in B}$

Now be chosen so that for becomes a maximum. Is now , and the environment of , and thus are very out of coverage . So now one with the property of the Lebesgue number is found. ${\displaystyle m}$ ${\displaystyle d(x_{0},C_{i})}$ ${\displaystyle i=m}$ ${\displaystyle \delta \leq f(x_{0})\leq d(x_{0},C_{m})}$ ${\displaystyle \delta }$ ${\displaystyle U_{\delta }(x_{0})}$ ${\displaystyle x_{0}}$ ${\displaystyle B}$ ${\displaystyle A_{m}=X\setminus C_{m}}$ ${\displaystyle {\mathcal {U}}}$ ${\displaystyle \delta }$

Applications

The Lebesgue number is used in the proof of various fundamental theorems of algebraic topology , for example in the proof of the Seifert-van Kampen theorem or the Mayer-Vietoris sequence and the excision axiom of singular homology .

Web links

• Script for Seifert van Kampen's theorem for an application of the Lebesgue number (and a further proof; PDF file; 505 kB)