What does $d|A×A$ stand for?

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 1

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 $B_{d_A}(a,r) = B_d(a,r) \cap A$.

Answer 2

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 $Y$.

Answer 3

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 $d|A\times A$.