$F_{1} \times F_{2}$ is a closed subset then $F_{1}$ is a closed subset of $\mathbf{R}^{m}$ and $F_{2}$ is a closed subset of $\mathbf{R}^{n}$

Question

I am reading Axler's MIRA and found the following question:

Suppose $F_{1}$ is a nonempty subset of $\mathbf{R}^{m}$ and $F_{2}$ is a nonempty subset of $\mathbf{R}^{n}$ Prove that $F_{1} \times F_{2}$ is a closed subset of $\mathbf{R}^{m} \times \mathbf{R}^{n}$ if and only if $F_{1}$ is a closed subset of $\mathbf{R}^{m}$ and $F_{2}$ is a closed subset of $\mathbf{R}^{n}$

Answer 1

HINT:

Let $X$, $Y$ be topological spaces. Then the map $$x\mapsto (x, y)$$ from $X$ to $X\times \{y\}\subset X\times Y$ (with subspace topology) is a homeomorphism, with inverse $(x,y) \mapsto x$.

Now consider a section $F_1\times \{y\}$ of $F_1\times F_2$, which equals $(F_1\times F_2) \cap (X\times \{y\})$, so a closed subset of $X\times \{y\}$.

Answer 2

Suppose $F_1 \times F_2$ is closed. Fix any $y \in F_2$. Then $(F_1 \times F_2) \cap (\mathbb R^{m} \times \{y\})$ is closed. This means $F_1 \times \{y\}$ is closed which implies that $F_1$ is closed.

[The projection to the first coordinate becomes a homeomorphism when restricted to $\mathbb R^{m} \times \{y\}$]