What does $×$ mean in this context

Question

I have two definitions, from real analysis -

Metric Space:

Given a set $X$, a function $d:X×X→\mathbb{R}$is a metric on $X$ if for all $x, y∈X \dots $

Function:

Let $A,B$ be two sets. A function $F$ from $A$ to $B$ is a subset of $A × B$ such that, for any $x \in A$, there exists one and only one $y \in B$ such that $(x,y) \in F \dots $

Answer 1

They actually mean the same thing. Given two sets $X$ and $Y$, $X \times Y$ is the Cartesian product of the sets. This is simply the set of all ordered pairs $(a, b)$ where $a \in X$ and $b \in Y$.

We use this to create functions that can take multiple arguments; in the case of a metric, we need two points to define a distance. So, in more explicit terms $d(x, y)$ is really $d((x, y))$, where the outer parentheses denote invocation of a function, and the inner set of parentheses denote the formation of an ordered pair.

Answer 2

http://en.wikipedia.org/wiki/Cartesian_product Cartesian product.

Basically what $d\colon X \times X \to\mathbb{R}$ means is that you take two elements from $X$ and get one element from $\mathbb{R}$.