Tietze's Extension Theorem. A characterization of normal spaces with respect to the definition given by Kelley (1955, p. 112) or Willard (1970, p. 99). It states that the topological space is normal iff, for all closed subsets of , every continuous function , where denotes the real line with the Euclidean topology, can be extended to a ...

The Tietze extension theorem says that continuous functions extend from closed subsets of a normal topological space X to the whole space X. This is a close cousin of Urysohn's lemma with many applications. One implication is that topological vector bundles over a topological space X that trivialize over a closed subspace A are equivalent to ...

A famous, interesting, and old result usually encountered in an introductory topology or analysis course is the Tietze Extension Theorem ([3], p. 149): THEOREM T. Let X be a normal Hausdorff space and A a closed subset of X. If. f E C(A), then f has a continuous extension F E C(X). Furthermore, if I f(x)l < c for each x E A, then F can be ...

Nowadays, Theorem 1.1 is commonly called Tietze's extension theorem. The history of this theorem is fascinating. In 1916, in his book [13], de la Vall´ee Poussin gave a proof of this theorem for X= Rn. The book of Carath´eodory [11] contains another proof of Tietze's extension theorem for Euclidean spaces, it was credited to H. Bohr and ...

The Tietze extension theorem says that continuous functions extend from closed subsets of a normal topological space XX to the whole space XX. For XX a normal topological space and A⊂XA \subset X a closed subspace, there is for every continuous function f:A→ℝf \colon A \to \mathbb{R} to the real line (with its Euclidean metric topology) a continuous function f^:X→ℝ\hat f \colon X \to \mathbb{R} extending it, i.e. such that f^| A\=f\hat f|_A = f: Let XX be a smooth manifold and let {g i∈C ∞(X)} i\=1 n\{g_i \in C^\infty(X)\}_{i = 1}^n be smooth functions that are independent in the sense that at each common zero point x∈Xx\in X, ∀i:g i(x)\=0\forall i : g_i(x)= 0 we have the derivative (dg i):T xX→ℝ n(d g_i) : T_x X \to \mathbb{R}^n is a surjection, then the ideal (g 1,⋯,g n)(g_1, \cdots, g_n) coincides with the ideal of functions that vanish on the zero-set of the g ig_i.

Tietze's Extension Theorem -- from Wolfram MathWorld Topology Barile A characterization of normal spaces with respect to the definition given by Kelley (1955, p. It states that the topological space is normal iff, for all closed subsets of , every continuous function , where denotes the real line with the Euclidean topology, can be extended to a continuous function (Willard 1970, p. With respect to the alternative definition (Cullen 1968, p. (Cullen 1968, p. This entry contributed by Margherita Barile F. Introduction to General Topology. D. "The Tietze Characterization of Normality." §7.44 in Introduction to General Topology. L. General Topology. General Topology. Barile, Margherita. "Tietze's Extension Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. https://mathworld.wolfram.com/TietzesExtensionTheorem.html Topology About MathWorld