Adjusted citation formats. Moved citation off of named theorems if possible.
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Bokuan Li
2026-03-19 23:58:16 -04:00
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so $\fb_S$ is a fundamental system of entourages.
\end{proof}
\begin{definition}[Topology of a Uniform Space, {{\cite[Proposition 2.1.2]{Bourbaki}}}]
\begin{definition}[Topology of a Uniform Space]
\label{definition:uniformtopology}
Let $(X, \fU)$ be a uniform space and
\[
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then there exists a unique topology $\topo \subset 2^X$ such that $\cn_\topo = \cn$, known as the \textbf{topology induced by the uniform structure $\fU$}.
\end{definition}
\begin{proof}
\begin{proof}[Proof {{\cite[Proposition 2.1.2]{Bourbaki}}}. ]
Using \autoref{proposition:neighbourhoodcharacteristic}, it is sufficient to show that $\cn(x)$ is non-empty for all $x \in X$, and that it satisfies (F1), (F2), (V1), and (V2). Firstly, since $\fU \ne \emptyset$, $\cn(x) \ne \emptyset$ for all $x \in X$.
(F1): Let $U \in \fU$ and $V \supset U(x)$, then