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MAJOR RETRACTION IN UNIFORMITY DEFINING PROPOSITION.

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Bokuan Li
2026-06-21 21:56:28 -04:00
parent a57686be8f
commit f107df48bf
8 changed files with 97 additions and 11 deletions
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src/topology/uniform/definition.tex
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@@ -75,25 +75,27 @@
\begin{proposition}
\label{proposition:fundamental-entourage-criterion}
Let $X$ be a set and $\fB \subset 2^{X \times X}$ be a non-empty family of sets. If
Let $X$ be a set and $\fB \subset 2^{X \times X}$ be a non-empty family of sets such that
\begin{enumerate}
\item[(FB1)] For each $U, V \in \fB$, there exists $W \in \fB$ such that $W \subset U \cap V$.
\item[(UB1)] For each $V \in \fB$, $\Delta \subset V$.
\item[(UB2)] For each $V \in \fB$, there exists $W \in \fB$ such that $W \circ W \subset V$.
\item[(UB2)] For each $V \in \fB$, there exists $W \in \fB$ with $W \subset V^{-1}$.
\item[(UB3)] For each $V \in \fB$, there exists $W \in \fB$ such that $W \circ W \subset V$.
\end{enumerate}
then there exists a unique uniformity $\fU \subset 2^{X \times X}$, which is given by
and let
\[
\fU = \bracs{U \subset X \times X| \exists V \in \fB: V \subset U}
\]
then $\fU$ is the unique uniformity on $X$ such that $\fB$ is a fundamental system of entourages for $\fU$.
\end{proposition}
\begin{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) and (U2): 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), there exists $V_0 \in \fB \subset \fU$ with $V_0 \circ V_0 \subset U_0 \subset U$.
(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 $W_0 \circ W_0 \subset U_0 \subset U$.
\end{proof}