From 50d4a326db9d11a8ab889fba77e9352715dcf36f Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Wed, 17 Jun 2026 17:53:04 -0400 Subject: [PATCH] Slight adjustment to topological groups. --- src/topology/groups/definition.tex | 21 +++++++++++++++++++-- 1 file changed, 19 insertions(+), 2 deletions(-) diff --git a/src/topology/groups/definition.tex b/src/topology/groups/definition.tex index 77d8ca3..e9c8d31 100644 --- a/src/topology/groups/definition.tex +++ b/src/topology/groups/definition.tex @@ -10,7 +10,7 @@ \item The inversion map $G \to G$ with $g \mapsto g^{-1}$ is continuous. \end{enumerate} - then the pair $(E, \mathcal{T})$ is a \textbf{topological group}. + then the pair $(G, \mathcal{T})$ is a \textbf{topological group}. \end{definition} \begin{definition}[Translation-Invariant Topology] @@ -128,4 +128,21 @@ \] so $x \in yB \subset AB$. -\end{proof} \ No newline at end of file +\end{proof} + +\begin{definition}[Symmetric Neighbourhood] +\label{definition:tg-symmetric-neighbourhood} + Let $G$ be a topological group and $U \in \cn_G(1)$, then $U$ is \textbf{symmetric} if $U = U^{-1}$. +\end{definition} + +\begin{proposition} +\label{proposition:tg-good-neighbourhood-base} + Let $G$ be a topological group, then + \begin{enumerate} + \item $G$ admits a fundamental system of neighbourhoods at $0$ consisting of symmetric sets. + \item The system in (1) may be taken to be open or closed. + \end{enumerate} +\end{proposition} +% Obvious so proof omitted. + +