Separability and completeness of Cartesian product of two metric spaces

Question

There are two metric spaces given: $(X_1, \varrho_1)$ and $(X_2, \varrho_2)$.
We define $X = X_1 \times X_2$ and $\varrho = \varrho_1 + \varrho_2$.
It is easy to show that $(X, \varrho)$ is a metric space (I managed to do it).
Now I am to proof that $(X, \varrho)$ is separable space and complete space if and only if $(X_1, \varrho_1)$ and $(X_2, \varrho_2)$ are complete spaces and separable spaces.
How should the proof look like? I'm new to functional analysis and I don't have much intuition yet.

Answer 1

(Ia). Let $A=\{(a_n,b_n):n\in \Bbb N\}$ be any countable subset of $X_1\times X_2.$ If $X_1$ is not separable then $U=X_1\setminus Cl_{X_1}(\{a_n: n\in \Bbb N\})$ is a non-empty open subset of $X_1.$ So if $X_2\ne \emptyset$ then $U\times X_2$ is a non-empty open subset of $X_1\times X_2$ which is disjoint from $A.$

(Ib). Suppose $C_1$ is a countable dense subset of $X_1$ and $C_2$ is a countable dense subset of $X_2.$ Then $C_1\times C_2$ is countable. Now if $U$ is a non-empty open subset of $X_1\times X_2$ then $U\supset U_1\times U_2$ for some non-empty open $U_1\subset X_1$ and some non-empty open $U_2\subset X_2.$ So $U_1\cap C_1\ne \emptyset \ne U_2\cap C_2.$ So $\emptyset \ne (U_1\cap C_1)\times (U_1\times C_2)\subset U\cap (C_1\times C_2).$

(II). If $\rho_1$ is an incomplete metric, let $(q_n)_{n\in \Bbb N}$ be a $\rho_1$-Cauchy sequence with no limit in $X_1.$ For any $z\in X_2$ the sequence $(\;(q_n,z)\;)_{n\in \Bbb N}$ is a $\rho$-Cauchy sequence with no limit in $X_1\times X_2.$

The statements to be shown are valid provided that we assume $X_1$ and $X_2$ are not empty. (The empty set is a metrizable space too.)

Answer 2

I don't know how much of the theory you have encountered so far, but I will give you some hints that I hope will prove useful. For the question on separability, it is useful to know that in metric spaces, this condition is equivalent to:

i) having a countable base,

ii) satisfying the Lindëlof property: every open cover of the space has a countable subcover. These equivalences should allow you to tackle the question more easily.

For the qustion on completeness, the first thing is to remember is that a sequence $(x_n,y_n)$ in $X \times Y$ converges to $(a,b)$ iff $x_n$ converges to $a$ and $y_n$ converges to $b$. In the first case, when $X$ and $Y$ are complete, the idea is to try to obtain Cauchy sequences in these spaces from a Cauchy sequence in $X \times Y$, and a good way to do this is to study what the projection maps do to Cauchy sequences. For the converse, try to obtain a Cauchy sequence in $X \times Y$ from Cauchy sequences in $X$ and $Y$.

Just as a final hint, I think that some of the properties above can be more easily obtained if you work with the maximum metric in the product space: $$ d((x_1,y_1),(x_2,y_2)) = \max\{d_1(x_1,x_2),d_2(y_1,y_2)\}. $$