$Int(A\times B) = Int(A)\times Int(B)$, for which metric? If $A$ and $B$ both have a metric $d$, then which metric should be considered for $A\times B$? I know that this is a simple question, all I have to do is just redefine the metric $d$ for '2 dimensions'. For example:
$d(a,b) = \sqrt{(a-b)²}$
now, how to define it for 
$$d((a,b),(c,d))$$
?
Is there a natural way to do it? I ask because it seems like there are many possibilities.
This question arose when I was trying to prove:
$$Int(A\times B) = Int(A)\times Int(B)$$
If $(x,y)\in A\times B$, then there is an open ball centered in $(x,y)$ and included in $A\times B$. I tried to use some metric to conclude that there is an open ball centered at $x$ and contained in $A$, or an open ball centered in $y$ contained in $B$. But when I try to do it, I feel that I need a metric definition. How do I do that?
ps: I'm intersted in this way of solving this question. I know that might be anothers but solving this way is the main point of the quesiton
 A: Let $A$ be a subset of topological space $X$ and let $B$ be a subset
of topological space $Y$.
If $X\times Y$ is equipped with the product topology then $A^{\circ}\times B^{\circ}$
is open, and secondly it is a subset of $A\times B$.
We conclude that $A^{\circ}\times B^{\circ}\subseteq\left(A\times B\right)^{\circ}$.
Conversely if $\langle a,b\rangle\in\left(A\times B\right)^{\circ}$
then there are sets $U,V$ such that $U$ is open in $X$ and $V$
is open in $Y$ with $\langle a,b\rangle\in U\times V\subseteq\left(A\times B\right)^{\circ}\subseteq A\times B$.
This because sets $U\times V$ with $U$ open in $X$ and $V$ open
in $Y$ form a base of the product topology on $X\times Y$.
We have $a\in U\subseteq A$ and $b\in V\subseteq B$ so that $a\in A^{\circ}$
and $b\in B^{\circ}$.
This is a proof without use of metrics. If $d$ is a metric on $X\times Y$ such that the topology induced by the metric coincides with the product topology then  necessarily for $r>0$ small enough the open disc with radius $r$ and center $\langle a,b\rangle$ is a subset of $U\times V$.
A: In general metric in product should generate a product topology. Usually metric in product is defined by $$ d_{A\times B} ((a,b) , (c,d)) =\max\{ d_{A} (a, c) , d_{B} (b, d) \}$$
A: Let $A,B$ be topological spaces.The Tychonoff product topology $T_{A\times B}$ on $A\times B$ is defined as the weakest topology on $A\times B$ such that the projections $p_A:A\times B\to A$ and $p_B:A\times B\to B$ are continuous.
A sub-base for the product topology is $$ S=\{p_A^{-1} u:u\in T_A\}\cup \{p_B^{-1} v:v\in T_B\}$$ where $T_A$ is the topology on $A$ and $T_B$ is the topology on $B.$
If $d_A, d_B$ are metrics for $T_A,T_B$ respectively, then for $(a_1,b_1), (a_2,b_2)\in A\times B, $ let $$d_{A\times B}(\;(a_1,b_1),(a_2,b_2)\;)=\max (\;d_A(a_1,a_2),d_B(b_1,b_2)\;).$$ Observe that for $r>0,$ if $u,v$ are respectively, open $T_A,T_B$ balls of radius $r$, then $(p_A^{-1}u) \cap (p_B^{-1}v)$ belongs to $S$, and is equal to $u\times v,$ and is an open $T_{A\times B}$ ball of radius $r.$
