- $(\delta \equiv \{ (x, x) : x \in X \}) \in E$.
- Closed under subsets: $\forall e \in E, f \subset e \implies f \in E$.
- Closed under transpose: if $e \in E$ then $(e^T \equiv \{ (y, x) : (x, y) \in e \}) \in E$.
- Closed under finite unions.
- Closed under composition: $\forall e, f \in E, e \circ f \in E$, where $\circ$is composition of relations.

$(e \subset X \times X) \in E \iff \exists \delta \in \mathbb R, \forall (x, y) \in E, d(x, y) < \delta$

We can check that the functions:
- $f: \mathbb Z \rightarrow \mathbb R, f(x) \equiv x$ and
- $g: \mathbb R \rightarrow \mathbb Z, g(x) \equiv \lfloor x \rfloor$