Tietze extension theorem
Encyclopedia
In topology
, the Tietze extension theorem states that, if X is a normal topological space and
is a continuous map from a closed subset A of X into the real number
s carrying the standard topology, then there exists a continuous map
with F(a) = f(a) for all a in A. F is called a continuous extension of f.
The theorem generalizes Urysohn's lemma
and is widely applicable, since all metric space
s and all compact
Hausdorff space
s are normal. It can be generalized by replacing R with RJ for some indexing set J, any retract of RJ, or any normal absolute retract whatsoever.
The theorem is due to Heinrich Franz Friedrich Tietze
.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, the Tietze extension theorem states that, if X is a normal topological space and
is a continuous map from a closed subset A of X into the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s carrying the standard topology, then there exists a continuous map
with F(a) = f(a) for all a in A. F is called a continuous extension of f.
The theorem generalizes Urysohn's lemma
Urysohn's lemma
In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a function....
and is widely applicable, since all metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
s and all compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
Hausdorff space
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
s are normal. It can be generalized by replacing R with RJ for some indexing set J, any retract of RJ, or any normal absolute retract whatsoever.
The theorem is due to Heinrich Franz Friedrich Tietze
Heinrich Franz Friedrich Tietze
Heinrich Franz Friedrich Tietze was an Austrian mathematician, famous for the Tietze extension theorem. He also developed the Tietze transformations for group presentations, and was the first to pose the group isomorphism problem.He was born in Schleinz, Austria and died in Munich,...
.