$\mathbb{R}^2$ is a Retract of $\mathbb{R}^5$

Question

Looking through some old qualifying exams, I noticed the question:

Let $X\subseteq \mathbb{R}^5$ be homeomorphic to $\mathbb{R}^2$. Prove that $X$ is a retract of $\mathbb{R}^5$.

I'm not even sure where to start. The exam was written for students that have taken a class in elementary singular homology, covering spaces, fundamental groups, and introductory differential topology.

Answer 1

By the Tietze extension theorem, $\Bbb R^2$ has the universal extension property and hence is an absolute retract. It follows that $X$ is a retract of $\Bbb R^5$ as long as it's closed (as Ted points out in the comments).

See this answer for pointers to the definitions and theorems used in this answer.