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 $Y$.
A metric $d$ for a space $X$ is a function $d:X\times X \to \mathbb{R}$ that satisfies certain properties. If $A\subseteq X$ we can *restrict* $d$ to points in $A$. This is normally written as $d|\_{A\times A}$ (using a subscript) but it can also be written as $d|A\times A$.