Suppose $X$ is a completely regular space (Definition: all singletons are closed and points and closed sets can be separated by continuous maps) and $K$ be a compact subset of $X$ and $g \in C(K)$ ($:=$ the space of continous fucntions on $K$); then does there exists a continuous extension, $\bar{g}:X \to \Bbb{C}$, of $g$?? It seems that this thing has been used in the following proof (screenshot attached):
Here $X$ is not given to be "normal" so we cannot use "Tietze's exttension theorem". Then how did the author claim the existency of such extension there?
See theorem 2 here: we do have such a theorem for a compact subset in a completely regular space ( so quite close to normal); it’s the best we can achieve.