Updated the uniform topology on function spaces.

This commit is contained in:
Bokuan Li
2026-01-05 20:18:26 -05:00
parent a807369ab0
commit badf8ad486

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@@ -33,7 +33,7 @@
\item The topology induced by $\fV$ is finer than the $\mathfrak{S}$-topology on $T^X$.
\item If $\mathfrak{S}$ is upward-directed with respect to inclusion, then $\mathfrak{E}(\mathfrak{S}, \fU)$ is forms a fundamental system of entourages for $\fV$.
\end{enumerate}
and the topology induced by $\fV$ is the \textbf{topology of uniform convergence on the sets $\mathfrak{S}$}, or the $\mathfrak{S}$-topology.
and the topology induced by $\fV$ is the \textbf{topology of uniform convergence on the sets $\mathfrak{S}$}/\textbf{$\mathfrak{S}$-uniform topology} on $X^T$.
\end{definition}
\begin{proof}
(1): Since $\Delta \subset E(S, U)$ for all $S \in \mathfrak{S}$ and $U \in \fU$, $\mathfrak{E}(\mathfrak{S}, \fU)$ generates a uniformity on $X^T$.