[[!redirects Sandbox > history]] [[!redirects Sandbox]] < [[nlab:Sandbox]] Defining the metric on an open interval. Given rational numbers $a:\mathbb{Q}$ and $b:\mathbb{Q}$, we define the open interval $$(a, b) \coloneqq \sum_{c:\mathbb{Q}} (a \lt c) \times (c \lt b)$$ The metric $d_{(a, b)}$ is defined by $$d_{(a, b)}(c, d) =_\mathbb{Q} {| \pi_1(c) - \pi_1(d)|}$$ Rational metric spaces are a set $X$ with a function $d_X:X \times X \to \mathbb{Q}$ such that... category: redirected to nlab