Definition 5.1.4 (Uniformity). Let $X$ be a set, then a non-empty family $\fU \subset X \times X$ is a uniformity on $X$ if:

1. For any $U \in \fU$ and $V \supset U$, $V \in \fU$.

2. For any $U, V \in \fU$, $U \cap V \in \fU$.

3. For every $U \in \fU$, $U \supset \Delta = \bracs{(x, x)| x \in X}$.

4. For any $U \in \fU$, $U^{-1}\in \fU$.

5. For any $U \in \fU$, there exists $V \in \fU$ such that $V \circ V \subset U$.

The elements of $\fU$ are called the entourages of $\fU$, and the pair $(X, \fU)$ is a uniform space.

For any $x, y \in X$ and $U \in \fU$, $x$ and $y$ are $U$-close if $(x, y) \in U$.