Polished A-A and added new lines for broken enumerates.
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Bokuan Li
2026-05-05 01:50:35 -04:00
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@@ -36,7 +36,8 @@
\item $\mathfrak{E}(\sigma, \fU)$ generates a uniformity $\fV$ on $X^T$.
\item The topology induced by $\fV$ is finer than the $\sigma$-open topology on $T^X$.
\item If $\mathfrak{E}(\sigma, \fU)$ forms a fundamental system of entourages for $\fV$.
\end{enumerate}
\end\{enumerate\}
The uniformity $\fV$ is the \textbf{$\sigma$-uniformity}, and the topology induced by $\fV$ is the \textbf{topology of uniform convergence on the sets $\sigma$}/\textbf{$\sigma$-uniform topology} on $X^T$.
\end{definition}
\begin{proof}
@@ -57,7 +58,8 @@
E(S, V) \circ E(S, V) \subset E(S, V \circ V) \subset E(S, U)
\]
\end{enumerate}
\end\{enumerate\}
By \autoref{proposition:fundamental-entourage-criterion}, $\mathfrak{E}$ is a fundamental system of entourages for the uniformity that it generates.
\end{proof}
@@ -111,7 +113,8 @@
\item The product topology on $X^T$.
\item The $\sigma$-open topology, where $\sigma$ is the collection of all finite sets.
\item (If $X$ is a uniform space) The $\mathfrak{F}$-uniform topology, where $\fF = \bracs{F| F \subset X \text{ finite}}$.
\end{enumerate}
\end\{enumerate\}
This topology is the \textbf{topology of pointwise convergence} on $X^T$.
\end{definition}
\begin{proof}