Tietze extension theorem for complex valued functions Why is this theorem always only stated for real valued functions, and not for complex valued functions? 
Thanks.

The wikipedia article on Tietze's Extension Theorem mentions that one can replace $\mathbb{R}$ with $\mathbb{R}^I$ for any index set $I$.  Taking $\# I = 2$ -- and, of course, using that $\mathbb{C}$ is homeomorphic to $\mathbb{R}^2$! -- we get the result you are asking about.  So to my mind this is a standard reference which includes the version of the theorem you are asking about.
The version for $\mathbb{C}$-valued functions follows immediately from the version for $\mathbb{R}$-valued functions, because a function $f: X \rightarrow \mathbb{C}$ is continuous iff its real and imaginary parts $\Re f, \Im f: X \rightarrow \mathbb{R}$ are continuous: this is the universal property of the direct product.  So you can apply the $\mathbb{R}$-valued Tietze Extension Theorem to each component to get the $\mathbb{C}$-valued version.  (This argument works verbatim with $\mathbb{R}^2$ replaced by $\mathbb{R}^I$ for any set $I$: it is again the universal property of the direct product.)
