Definition: A set $X$, whose elements we shall call points, is said to be a metric space if with any two points $p$ and $q$ of $X$ there is associated a real number $d(p, q)$, called the distance from $p$ to $q$, such that $$(a) \ \ \ d(p, q) > 0 \text{ if } p \neq q; d(p, p) = 0;$$ $$(b) d(p, q) = d(q, p); $$ $$(c) d(p, q) \le d(p, r) + d(r, q), \text{ for any } r\in X.$$
since $d(p,q)$ is a real valued function can $|X|>|\mathbb{R}|$ ? If there such $X$ can you give me an example of such a metric space ?