State Urysohn's Lemma. State and prove the Tietze Extension Theorem.

Let $X, Y$ be two topological spaces. We say that the extension property holds if, whenever $S \subseteq X$ is a closed subset and $f: S \rightarrow Y$ is a continuous map, there is a continuous function $\tilde{f}: X \rightarrow Y$ with $\left.\tilde{f}\right|_{S}=f$.

For each of the following three statements, say whether it is true or false. Briefly justify your answers.

1. If $X$ is a metric space and $Y=[-1,1]$ then the extension property holds.

2. If $X$ is a compact Hausdorff space and $Y=\mathbb{R}$ then the extension property holds.

3. If $X$ is an arbitrary topological space and $Y=[-1,1]$ then the extension property holds.