Lemma from Urysohn

The lemma Urysohn (also Urysohnsches Lemma called) is a fundamental theorem of the mathematical branch of general topology .

The lemma is named after Pavel Urysohn and was published by him in 1925. It is often used to construct continuous functions with certain properties. Its broad application is based on the fact that many of the most important topological spaces such as the metric spaces and the compact Hausdorff spaces have the normality property assumed in the lemma .

Tietze's continuation theorem represents a generalization . In its proof, Urysohn's lemma comes into play in a decisive way.

Formulation of the lemma

The lemma says the following:

Be a normal room i.e. That is, a topological space with the property that every two disjoint closed subsets of have disjoint neighborhoods , and let two such disjoint closed subsets and be given. ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle B}$

Then there is a continuous function for this

${\displaystyle f\colon X\rightarrow [0,1]}$

with for everyone and for everyone . ${\displaystyle f(a)=0}$ ${\displaystyle a\in A}$ ${\displaystyle f(b)=1}$ ${\displaystyle b\in B}$

Remarks

1) Urysohn's lemma says nothing about the values ​​of the continuous function outside the closed subsets and , only that and holds. In the case that disjoint closed and always a continuous with and can be found, one calls a perfectly normal space . ${\displaystyle f}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A\subseteq f^{-1}(0)}$ ${\displaystyle B\subseteq f^{-1}(1)}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle f\colon X\rightarrow [0,1]}$ ${\displaystyle A=f^{-1}(0)}$ ${\displaystyle B=f^{-1}(1)}$ ${\displaystyle X}$

2) For metric spaces, a continuous function of the above type must be specified immediately. To do this, for two given disjoint closed subsets and of the function is defined as follows: ${\displaystyle f}$ ${\displaystyle A\neq \emptyset }$ ${\displaystyle B\neq \emptyset }$ ${\displaystyle X}$ ${\displaystyle f}$

${\displaystyle x\mapsto f(x):={\frac {d(x,A)}{d(x,A)+d(x,B)}}}$     ${\displaystyle (x\in X)}$

Here is the distance from to , so ${\displaystyle d(x,Y)}$ ${\displaystyle x\in X}$ ${\displaystyle Y\subseteq X}$   ${\displaystyle (Y\neq \emptyset )}$

${\displaystyle d(x,Y)=\operatorname {inf} \{d(x,y):y\in Y\}}$.

The function is continuous - even uniformly continuous - and the following applies: ${\displaystyle x\mapsto d(x,Y)}$

${\displaystyle d(x,Y)=0\iff x\in {\overline {Y}}}$.

Metric spaces are therefore always perfectly normal .

Core statement of the lemma

The core of Urysohn's lemma lies in the following statement:

Let be a topological space and be a dense subset of . A set family is given , consisting of open subsets   (   ), which satisfies the following conditions: ${\displaystyle X}$ ${\displaystyle D\subseteq \mathbb {R} }$ ${\displaystyle \mathbb {R} }$   ${\displaystyle (F(d))_{d\in D}}$   ${\displaystyle F(d)\subseteq X}$ ${\displaystyle d\in D}$

1. For and always be    . ${\displaystyle d_{1},d_{2}\in D}$ ${\displaystyle d_{1}<d_{2}}$ ${\displaystyle {\overline {F(d_{1})}}\subseteq F(d_{2})}$
2. ${\displaystyle \bigcap _{d\in D}F(d)=\emptyset }$  .
3. ${\displaystyle \bigcup _{d\in D}F(d)=X}$  .

Finally, let the   following assignment be defined: ${\displaystyle x\in X}$

${\displaystyle x\mapsto f(x):=\operatorname {inf} \{d\in D:x\in F(d)\}}$.

Then a continuous function is given by this assignment . ${\displaystyle f\colon \,X\to \mathbb {R} }$

literature

• GJO Jameson: Topology and normed spaces . Chapman and Hall, London 1974, ISBN 0-412-12880-2 .
• John L. Kelley : General topology . Springer-Verlag, Berlin / Heidelberg / New York 1975, ISBN 3-540-90125-6 (Reprint of the 1955 edition published by Van Nostrand).
• C. Wayne Patty: Foundations of Topology . PWS-Kent Publishing, Boston 1993, ISBN 0-534-93264-9 .
• Boto von Querenburg : Set theoretical topology . 3rd revised and expanded edition. Springer-Verlag, Berlin a. a. 2001, ISBN 3-540-67790-9 .
• Willi Rinow : Textbook of Topology . German Science Publishers, Berlin 1975.
• Horst Schubert : Topology . 4th edition. BG Teubner Verlag, Stuttgart 1975, ISBN 3-519-12200-6 .
• Lutz Führer: General topology with applications . Vieweg, Braunschweig 1977, ISBN 3-528-03059-3 .
• Egbert Harzheim ; Helmut Ratschek: Introduction to General Topology (=  Mathematics. Introductions to the subject matter and results of its sub-areas and related sciences ). Wissenschaftliche Buchgesellschaft, Darmstadt 1978, ISBN 3-534-06355-4 ( MR0380697 ).
• John L. Kelley : General topology. Reprint of the 1955 edition published by Van Nostrand . Springer-Verlag, Berlin / Heidelberg / New York 1975, ISBN 3-540-90125-6 .
• Jun-iti Nagata : Modern General Topology (=  North Holland Mathematical Library . Volume 33 ). 2nd revised edition. North-Holland Publishing, Amsterdam / New York / Oxford 1985, ISBN 0-444-87655-3 ( MR0831659 ).
• Paul Urysohn : About the power of connected sets . In: Math. Ann. tape 94 , 1925, pp. 262-295 .
• Stephen Willard: General Topology . Addison-Wesley, Reading MA et al. a. 1970.

See also

• Katětov's interpolation theorem

Web links

Commons : Urysohn's lemma  - collection of images, videos and audio files

Individual evidence

1. ↑ In H. Schubert, p. 79, the derivation of the lemma is called a remarkable construction . In Jameson, p. 111, it says: Urysohn's 'lemma' is undoubtedly one of the best theorems in General Topology.
2. ↑ Urysohn . In: Mathematical Annals . tape 94 , p. 262 ff .
3. ^ H. Schubert: Topology . 1975, p. 80 .
4. ^ S. Willard: General Topology . 1970, p. 105 .
5. ^ GJO Jameson: Topology and normed spaces . 1974, p. 112 .
6. ^ H. Schubert: Topology . 1975, p. 78 .
7. ^ S. Willard: General Topology . 1970, p. 105 .
8. ^ GJO Jameson: Topology and normed spaces . 1974, p. 111-112 .