I came across this problem while reading about Tietze’s Extension Theorem in my professor’s old lecture notes:
- “Suppose X is $T_4$, A$\subseteq$X is closed, and $f:A \rightarrow D$ is continuous, where $D= {\{(x,y) \in \mathbb(R)^2 : x^2+y^2 \leq 1 }\}$ is the closed unit disk in $\mathbb(R)^2$. Show that there is a continuous $F: X \rightarrow D$ such that the restriction of $F$ to A is $f$.”
My first thought was that this looks like something I need to use Tietze’s extension theorem on, but the issue is that it’s not a real valued function. I know how to handle the case when the co-domain is the unit circle and the range of f is closed (namely there exists an open arc in the unit circle a such that the range is contained in the unit circle minus the arc, a closed arc so it is homeomorphic to [0,1], from there use Tietze’s extension theorem) but in this case I’m at a loss. We don’t know that the range of A is closed in D and I can’t think of some sort of analogous argument to the arc one used above.
Any help would be greatly appreciated!
