diff --git a/src/topology/groups/definition.tex b/src/topology/groups/definition.tex index f607161..77d8ca3 100644 --- a/src/topology/groups/definition.tex +++ b/src/topology/groups/definition.tex @@ -95,7 +95,7 @@ \end{proof} -\begin{proposition}[{{\cite[I.1.1]{SchaeferWolff}}}] +\begin{proposition} \label{proposition:tvs-set-operations} Let $G$ be a topological group and $A, B \subset G$, then \begin{enumerate} @@ -103,7 +103,7 @@ \item If $A$ is closed and $B$ is compact, then $AB$ is closed. \end{enumerate} \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 \[ AB = \bigcup_{x \in B}(Ab) @@ -122,7 +122,7 @@ y \in \bigcap_{U \in \fF}\overline{UB^{-1}} \subset \bigcap_{U \in \fF}[UUB^{-1}] \] - 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 \[ y \in \bigcap_{U \in \fF}[UUB^{-1}] \subset \bigcap_{V \in \cn_G(1)}[xVB^{-1}] = \overline{xB^{-1}} = xB^{-1} \]