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More typo fixes.
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Bokuan Li
2026-06-17 17:21:36 -04:00
parent 66d4a9ef3e
commit f29c3cdeb7
1 changed files with 3 additions and 3 deletions
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src/topology/groups/definition.tex
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@@ -95,7 +95,7 @@
\end{proof} \end{proof}
\begin{proposition}[{{\cite[I.1.1]{SchaeferWolff}}}] \begin{proposition}
\label{proposition:tvs-set-operations} \label{proposition:tvs-set-operations}
Let $G$ be a topological group and $A, B \subset G$, then Let $G$ be a topological group and $A, B \subset G$, then
\begin{enumerate} \begin{enumerate}
@@ -103,7 +103,7 @@
\item If $A$ is closed and $B$ is compact, then $AB$ is closed. \item If $A$ is closed and $B$ is compact, then $AB$ is closed.
\end{enumerate} \end{enumerate}
\end{proposition} \end{proposition}
\begin{proof} \begin{proof}[Proof, {{\cite[I.1.1]{SchaeferWolff}}}. ]
(1): For every $x \in B$, $Ab$ is open by translation invariance, so (1): For every $x \in B$, $Ab$ is open by translation invariance, so
\[ \[
AB = \bigcup_{x \in B}(Ab) AB = \bigcup_{x \in B}(Ab)
@@ -122,7 +122,7 @@
y \in \bigcap_{U \in \fF}\overline{UB^{-1}} \subset \bigcap_{U \in \fF}[UUB^{-1}] y \in \bigcap_{U \in \fF}\overline{UB^{-1}} \subset \bigcap_{U \in \fF}[UUB^{-1}]
\] \]
Since $\fF$ converges to $x$, (TVS1) implies that $\bracs{U + U| U \in \fF}$ contains a neighbourhood base of $x$. Thus Since $\fF$ converges to $x$, (TG1) implies that $\bracs{UU| U \in \fF}$ contains a neighbourhood base of $x$. Thus
\[ \[
y \in \bigcap_{U \in \fF}[UUB^{-1}] \subset \bigcap_{V \in \cn_G(1)}[xVB^{-1}] = \overline{xB^{-1}} = xB^{-1} y \in \bigcap_{U \in \fF}[UUB^{-1}] \subset \bigcap_{V \in \cn_G(1)}[xVB^{-1}] = \overline{xB^{-1}} = xB^{-1}
\] \]