Corollary 5.26.6.label Let $X$ be a set, $(Y, \fU)$ be a complete Hausdorff uniform space, $\mathcal{B}$ be a base for $Y$ with $\emptyset \in \mathcal{B}$, and $p: \mathcal{B}\to 2^{X}$ be a basic preimage function such that:
- (S’)
For each $x \in X$ and $U \in \fU$, there exists a $U$-small set $V \in \mathcal{B}$ such that $x \in P(V)$.
then there exists a unique $f: X \to Y$ such that $p(U) = f^{-1}(U)$ for all $U \in \mathcal{B}$.
Proof. Let $\topo$ be the topology of $Y$. By Proposition 5.26.4, $p$ extends to a unique open preimage function $P: \topo \to 2^{X}$. Since $\mathcal{B}\subset \topo$, Theorem 5.26.5 implies that there exists a unique $f: X \to Y$ such that $f^{-1}(U) = P^{-1}(U)$ for all $U \in \topo$, and in particular for all $U \in \mathcal{B}$.$\square$
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