Tietze extension theorem

In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem) states that continuous functions on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary.

If X is a normal topological space and

is a continuous map from a closed subset A of X into the real numbers carrying the standard topology, then there exists a continuous map

with F(a) = f(a) for all a in A. Moreover, F may be chosen such that ${\displaystyle \sup\{|f(a)|:a\in A\}=\sup\{|F(x)|:x\in X\}}$, 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.

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