Continuation of Tietze's sentence

The continuation set of Tietze ( English Tietze ( 's) extension theorem ), also called extension set of Tietze or as a set of Tietze-Urysohn called, is a set of the mathematical part area of the topology . He relates normal topological spaces with continuous continuations . The sentence was published in 1915 by Heinrich Tietze .

The theorem is a generalization of Urysohn's lemma and can be used in many cases, since all metric spaces and all compact Hausdorff spaces are normal.

Continuation of Tietze's sentence

A topological space is a normal space if and only if to each on a closed subset of defined, continuous functions ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle X}$

${\displaystyle f\colon A\rightarrow \mathbb {R} }$

a continuous function

${\displaystyle F\colon X\rightarrow \mathbb {R} }$

exists with , d. H. for everyone . The function is called the continuous continuation of . ${\displaystyle F|_{A}=f}$ ${\displaystyle F(a)=f(a)}$ ${\displaystyle a\in A}$ ${\displaystyle F}$ ${\displaystyle f}$

This is a pure existence proposition. With a few exceptions, such a continuous continuation is not unambiguous. H. For a given function there can be more than one function with the desired property. ${\displaystyle f}$ ${\displaystyle F}$

Stronger version

Tietze's continuation can be formulated in an even stronger version:

A topological space is then and only then a normal room if at any continuous mapping of the shape with a closed and one of intervals of existing product space is always a steady continuation exists. ${\displaystyle X}$ ${\displaystyle f\colon A\rightarrow \Pi }$ ${\displaystyle A\subseteq X}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \Pi }$ ${\displaystyle F\colon X\rightarrow \Pi }$

The case is particularly important for the applications of the theorem . ${\displaystyle f\colon A\rightarrow \mathbb {R} ^{n}}$

example

In metric spaces , a continuation can be specified explicitly: Let it be closed and nonnegative. Then ${\displaystyle (X,d)}$ ${\displaystyle A\subset X}$ ${\displaystyle f\in C(A)}$

${\displaystyle F(x):={\begin{cases}f(x),&x\in A\\\inf _{a\in A}\{f(a)+{\frac {d(x,a)}{d(x,A)}}-1\},&x\in X\setminus A\end{cases}}}$

a steady continuation of on whole . ${\displaystyle f}$ ${\displaystyle X}$

See also

• Lemma from Urysohn
• Katětov's interpolation theorem

literature

• Graham J. O Jameson: Topology and Normed Spaces . Chapman and Hall, London 1974, ISBN 0-412-12880-2 .
• John L. Kelley : General topology (=  Graduate Texts in Mathematics . Volume 27 ). Springer, New York NY a. a. 1975, ISBN 3-540-90125-6 (Reprint of the 1955 edition published by Van Nostrand, New York).
• C. Wayne Patty: Foundations of Topology . PWS-Kent Publishing, Boston MA 1993, ISBN 0-534-93264-9 .
• Boto von Querenburg : Set theoretical topology . 3rd, revised and expanded edition. Springer, Berlin a. a. 2001, ISBN 3-540-67790-9 .
• Willi Rinow : Textbook of Topology (=  university books for mathematics . Volume 79 ). Deutscher Verlag der Wissenschaften, 1975, ISSN  0073-2842 .
• Horst Schubert : Topology. An introduction . 4th edition. Teubner, Stuttgart 1975, ISBN 3-519-12200-6 .
• Heinrich Tietze : About functions that are continuous on a closed set . In: Journal for pure and applied mathematics . Issue 145, 1915, pp. 9-14. doi : 10.1515 / crll.1915.145.9 . Digitized .
• Paul Urysohn : About the power of connected sets . In: Mathematical Annals . Volume 94, 1925, pp. 262-295. doi : 10.1515 / crll.1915.145.9 . Digitized version (PDF; 1.98 MB) .

Individual evidence

1. ↑ Kelley: General topology. 1975, p. 176.
2. ^ Patty: Foundations of Topology. 1993, p. 176.
3. ↑ Jameson: Topology and normed spaces. 1974, p. 113.
4. ↑ Rinow: Textbook of Topology. 1975, p. 170.
5. ↑ ^{a b} Schubert: Topology. 1975, p. 83.