Metric subspaces and products of metric spaces

Suppose $G_{1}$ is a nonempty subset of $\mathbb {R}^{m}$ and $G_{2}$ is a nonempty subset of $\mathbb {R}^{n}$. Prove that $G_{1} \times G_{2}$ is an open subset of $\mathbb {R}^{m} \times \mathbb {R}^{n}$ if and only if $G_{1}$ is an open subset of $\mathbb {R}^{m}$ and $G_{2}$ is an open subset of $\mathbb {R}^{n}$.

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

Prove that the set of all rational numbers between $\sqrt{2}$ and $\pi$ is a closed set in the metric space $\mathbb {Q}$ with the distance $d(x, y)=\left|x-y\right|$.

Let $F \subset \mathbb {R}$ be a closed set that is contained in the disjoint union of intervals $(a, b)$ and $(c, d)$. Prove that $F \cap (a, b)$ is a closed set.

Let $A, B$ be closed subsets of $\mathbb {R}$. Prove that $A \times B$ is a closed subset of $\mathbb {R}^{2}$.

Prove Proposition 2.

Proposition 2: Let $X$ be a subspace of the metric space $Y$ and $E$ a subset of $X$. Then $E$ is open in $X$ if and only if $E = X \cap O$, where $O$ is open in $Y$.

Let $X$ be a subspace of the metric space $Y$ and $A$ a subset of $X$. Show that $A$ is closed in $X$ if and only if $A = X \cap F$, where $F$ is closed in $Y$.

Let $X$ be a subspace of the metric space $Y$.

For a metric space ($X, \rho$), show that the metric $\rho : X \times X \rightarrow \mathbb {R}$ is continuous, where $X \times X$ has the product metric.

Show that the product topology on $n$-fold product of $\mathbb {R}$ is the same as the metric topology on $\mathbb {R}^{n}$ induced by the Euclidean metric.

Let ($X, \rho_{1}$) and ($Y, \rho_{2}$) be metric spaces. Show that the product topology on $X \times Y$, where $X$ and $X$ have the topologies induced by their respective metrics, is the same as the topology induced by the product metric

Consider the countable collection of metric spaces $\left\{ \left(X_{n}, \rho_{n}\right)\right\}_{n = 1}^{\infty }$. For the Cartesian product of these sets $X = \prod_{n = 1}^{\infty } X_{n}$, define $\rho : X \times X \rightarrow \mathbb {R}$ by

Use the preceding problem to show that $\rho$ is a metric on $X = \prod_{n = 1}^{\infty } X_{n}$ which induces the product topology on $X$, where each $X_{n}$ has the topology induced by the metric $\rho_{n}$.

Let $X$ be a metric space. Show that the map $\psi :(x, y) \rightarrow d(x, y)$ is continuous. Deduce that $\psi$ is $\mathcal{B}\left(X^{2}\right) \backslash \mathcal{B}(\mathbb {R})$-measurable. If $X$ is separable, deduce that $\psi$ is $\mathcal{B}(X) \otimes \mathcal{B}(X) \backslash \mathcal{B}(\mathbb {R})$-measurable, and hence $\omega \mapsto d(X(\omega ), Y(\omega ))$ is measurable if $X$ and $Y$ are random elements of $X$.