In Jacques Faraut's book "Analysis of Lie Groups", the author says that the topology of a topological group $G$ is determined by the collection of neighborhoods of the identity element of the group, i.e. is determined by the collection
$\Omega = \lbrace{U\subseteq{G} : U\,\, \mbox{is a neighborhood of}\,\,{e} }\rbrace$
Could someone explain to me what this means?
Note: The neighborhoods of $e$ are not necessarily open
Suppose that we have a topological group $(G, \ast, e, \mathcal{T})$. Then for any $g \in G$, the translation $t_g: G \to G$ defined by $t_g(x) = g\ast x$ is continuous (as the operation $m: G \times G \to G, m(g,h) = g \ast h$ is continuous by definition of a topological group and $t_g = m \circ e_g$, where $e_g: G \to G \times G, e_g(x) = (g,x)$ is also continuous).
As $t_g$ and $t_{g^{-1}}$ are each other's continuous inverse, translations are homeomorphisms of $G$.
This implies that $U$ is a neighourhood of $g$ iff $t_{g^{-1}}[U]$ is a neighbourhood of $e$, so we only have to specify the neighbourhoods of $e$ to know all neighbourhoods of all points of $G$.
