Urysohn's lemma says that if $X$ is a normal space, then for every two disjoint closed sets $F_{1},F_{2}\in X$, there exists a continuous function $f:X\to [a,b]\in\Bbb{R}$ such that $f(F_{1})=\{a\}$ and $f(F_{2})=\{b\}$.
Tietze Extension theorem says that for every such $f$, there exists a continuous function $f^*:X\to [a,b]$ such that $f^*|F_{1}$ and $f^*| F_{2}=f$.
I don't understand the difference! Why can't $f^*=f$? And if we assume $f^*\neq f$, are we just saying that there are two such continuous functions of the type mentioned in Urysohn's lemma?
Thanks in advance!
Best Answer
One can indeed obtain the result of UL via TET by setting $f|_{F_1} = a$ and $f|_{F_2} = b$. Also, in this Wikipedia article it is explicitly written that TET generalizes UL.