Question: Let $A⊂X$ . If d is a metric for the topology of $X$ , show that $d|A×A$ is a metric for the subspace topology on $A$ .

I can not understand the notation $d|A×A$. Can anyone please help me to understand that notation.

Answer: 
The metric is a function $d : X \times X \to \mathbb{R}$. The set $A \times A$ is a subset of $X \times X$, and $d | A \times A$ is merely the restriction of the function $d$ to this subset. This is a usual notation, when you have a function $f : X \to Z$ and a subset $Y \subset X$, then $f | Y$ (sometimes also $f|\_Y$) is the restriction of $f$ to $Y$.
It's the metric $d\_A$ on $A \times A$ defined as follows: take $a\_1,a\_2 \in A$. To compute $d\_A(a\_1,a\_2)$ we just note that in particular $(a\_1,a\_2) \in X \times X$ so that $d(a\_1, a\_2)$ is already defined. So we're lazy and define $d\_A(a\_1, a\_2)$ as that number $d(a\_1, a\_2)$. We don't look at other pairs except those with *both* coordinates in $A$. Note that, for that reason, if $a \in A$ and $r>0$ we have that $B\_{d\_A}(a,r) = B\_d(a,r) \cap A$.