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
a continuous function
exists with , d. H. for everyone . The function is called the continuous continuation of .
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.
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.
The case is particularly important for the applications of the theorem .
example
In metric spaces , a continuation can be specified explicitly: Let it be closed and nonnegative. Then
a steady continuation of on whole .
See also
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) .