Tietze Extension Theorem

The Tietze Extension Theorem deals with the problem of extending a continuous real-valued function that is defined on a subspace of a topological space \$X\$ to a continuous function defined on all of \$X\$.

Definition

Let \$X\$ be a normal space and let \$A\$ be a closed subspace of \$X\$.

1. Any continuous map of \$A\$ into the closed interval \$[a,b]\$ of \$\mathbb{R}\$ may be extended to a continuous map of all of \$X\$ onto \$[a,b]\$.
2. Any continuous map of \$A\$ into \$\mathbb{R}\$ may be extended to a continuous map of all of \$X\$ into \$\mathbb{R}\$.[1]