Proposition 6.1.8.label Let $X$ be a set and $\fB \subset 2^{X \times X}$ be a non-empty family of sets such that
- (FB1)
For each $U, V \in \fB$, there exists $W \in \fB$ such that $W \subset U \cap V$.
- (UB1)
For each $V \in \fB$, $\Delta \subset V$.
- (UB2)
For each $V \in \fB$, there exists $W \in \fB$ with $W \subset V^{-1}$.
- (UB3)
For each $V \in \fB$, there exists $W \in \fB$ such that $W \circ W \subset V$.
and let
then $\fU$ is the unique uniformity on $X$ such that $\fB$ is a fundamental system of entourages for $\fU$.
Proof. (F1): By definition of $\fU$.
(F2): For any $U, V \in \fU$, there exists $U_{0}, V_{0} \in \fB$ such that $U_{0} \subset U$ and $V_{0} \subset V$. By (FB1), there exists $W \in \fB$ with $W \subset U_{0} \cap V_{0} \subset U \cap V$. Thus $U \cap V \in \fU$.
(U1), (U2), (U3): For any $U \in \fU$, there exists $U_{0} \in \fB$ with $U_{0} \subset U$. By (UB1), $\Delta \subset U_{0} \subset U$. By (UB2) and (FB1), there exists $V_{0} \in \fB$ with $V_{0} \subset U_{0} \cap U_{0}^{-1}\subset U$. By (UB3), there exists $W_{0} \in \fB \subset \fU$ with $W_{0} \circ W_{0} \subset U_{0} \subset U$.$\square$
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