Tietze Extension Theorem and Pasting Lemma

Tietze Extension Theorem

If X is a normal topological space and \displaystyle f:A\to\mathbb{R} is a continuous map from a closed subset A\subseteq X, then there exists a continuous map \displaystyle F:X\to\mathbb{R} with F(a)=f(a) for all a in A.

Moreover, F may be chosen such that \sup\{|f(a)|:a\in A\}=\sup\{|F(x)|:x\in X\}, i.e., if f is bounded, F may be chosen to be bounded (with the same bound as f). F is called a continuous extension of f.

Pasting Lemma

Let X, Y be both closed (or both open) subsets of a topological space A such that A=X\cup Y, and let B also be a topological space. If both f|_X: X\to B and f|_Y: Y\to B are continuous, then f is continuous.

Proof:

Let U be a closed subset of B. Then f^{-1}(U)\cap X is closed since it is the preimage of U under the function f|_X:X\to B, which is continuous. Similarly, f^{-1}(U)\cap Y is closed. Then, their union f^{-1}(U) is also closed, being a finite union of closed sets.

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