9.1 Group Topologies
Definition 9.1.1 (Topological Group).label Let $G$ be a group and $\mathcal{T}\subset 2^{G}$ be a topology. If
- (TG1)
The composition map $G \times G \to G$ with $(g, h) \mapsto gh$ is continuous.
- (TG2)
The inversion map $G \to G$ with $g \mapsto g^{-1}$ is continuous.
then the pair $(G, \mathcal{T})$ is a topological group.
Definition 9.1.2 (Translation-Invariant Topology).label Let $G$ be a group and $\topo \subset 2^{G}$ be a topology, then $\topo$ is left translation-invariant if for every $U \in \topo$ and $g \in G$, $gU \in \topo$, and right translation-invariant if for every $U \in \topo$ and $g \in G$, $Ug \in \topo$. If $\topo$ is both left and right translation-invariant, then $\topo$ is translation-invariant.
Definition 9.1.3 (Translation-Invariant Uniformity).label Let $G$ be a group, $\fU$ be a uniformity on $G$, and $U \in \fU$, then $U$ is left translation-invariant if for every $z \in G$,
and right translation-invariant if for every $z \in G$,
The uniformity $\fU$ is left/right translation-invariant if it admits a fundamental system of left/right translation-invariant entourages.
Lemma 9.1.4.label Let $G$ be a group and $\fU \subset 2^{G \times G}$ be a left/right translation-invariant uniformity, then $\fU$ admits a fundamental system of symmetric, left/right translation-invariant entourages.
Proof. Let $z \in G$, then the map $(x, y) \mapsto (zx, zy)$ is a bijection. Thus for any translation-invariant entourages $U, V \in \fU$, $(U \cap V) = zU \cap zV$, and $U \cap V$ is left translation-invariant. By Lemma 6.1.9, $\fU$ admits a left fundamental system of symmetric, translation-invariant entourages.$\square$
Definition 9.1.5.label Let $G$ be a topological group, then:
- (L1)
There exists a unique left translation-invariant uniformity $\fU_{L}$ on $G$ that induces its topology.
- (L2)
For each $U \in \cn_{G}(1)$, let $U_{L, V}= \bracsn{(x, y) \in G^2|x^{-1}y \in V}$, then $\fB_{L} = \bracs{U_{L, V}|V \in \cn_G(1)}$ is a fundamental system of entourages for $\fU_{L}$.
and
- (R1)
There exists a unique right translation-invariant uniformity $\fU_{R}$ on $G$ that induces its topology.
- (R2)
For each $U \in \cn_{G}(1)$, let $U_{R, V}= \bracsn{(x, y) \in G^2|yx^{-1} \in V}$, then $\fB_{R} = \bracs{U_{R, V}|V \in \cn_G(1)}$ is a fundamental system of entourages for $\fU_{R}$.
The uniformities $\fU_{L}$ and $\fU_{R}$ are the left and right uniformities of $G$, respectively. The inversion map $G \to G$ with $g \mapsto g^{-1}$ is an isomorphism of the left and right uniformities.
Proof, [I.1.4, SW99]. (L2): For each $V \in \cn_{G}(0)$, $U_{L, V}$ is left translation-invariant.
- (FB1)
For each $V, V' \in \cn_{G}(1)$, $V \cap V' \in \cn_{G}(0)$, so $U_{L, V \cap V'}= U_{L, V}\cap U_{L, V'}$.
- (UB1)
For each $V \in \cn_{G}(1)$, $1 \in V$, so $\Delta \subset U_{L, V}$.
- (UB2)
For each $V \in \cn_{G}(1)$, by (TG1), there exists $W \in \cn_{G}(1)$ such that $WW \subset V$. In which case, $U_{L, W}\circ U_{L, W}\subset U_{L, V}$.
By Proposition 6.1.8, $\fB_{L}$ forms a fundamental system of entourages for a left translation-invariant uniformity $\fU_{L}$ on $G$.
(L1): Let $\fV$ be a left translation-invariant uniformity on $G$. For each symmetric, left translation-invariant entourage $V \in \fV$, and $g \in G$,
so $V = U_{L, V(1)}$. Therefore $\fV \subset \fU_{L}$ by Lemma 9.1.4.
On the other hand, for each $W_{0} \in \cn_{G}(1)$, there exists a symmetric, left translation-invariant entourage $W \in \fV$ such that $W(1) \subset W_{0}$. In which case, $W = U_{L, W(1)}\subset U_{L, W_0(1)}$, and $\fV \supset \fU_{L}$.$\square$
Definition 9.1.6 (Left/Right Continuous).label Let $G$ be a topological group, $Y$ be a uniform space, and $f: G \to Y$, then $f$ is left/right uniformly continuous if it is uniformly continuous with respect to the left/right uniformity of $G$.
Proposition 9.1.7.label Let $G$ be a topological group, $A \subset G$, and $\fB \subset \cn_{G}(1)$ be a fundamental system of neighbourhoods, then
Proof. By Proposition 6.1.13.$\square$
Proposition 9.1.8.label Let $G$ be a topological group and $A, B \subset G$, then
- (1)
If $A$ is open, then $AB$ is open.
- (2)
If $A$ is closed and $B$ is compact, then $AB$ is closed.
Proof, [I.1.1, SW99]. (1): For every $x \in B$, $Ab$ is open by translation invariance, so
is open.
(2): Let $x \in \overline{AB}$, then there exists a filter $\fF \subset 2^{A \cup B}$ converging to $x$. For any $U \in \fF$, $U \cap AB \ne \emptyset$, so $UB^{-1}\cap A \ne \emptyset$, and $\fB = \bracsn{UB^{-1}| U \in \fF}$ is a filter base in $A$. By compactness of $A$, there exists $y \in A$ such that
By Proposition 9.1.7, $\overline{UB^{-1}}\subset UUB^{-1}$, so
Since $\fF$ converges to $x$, (TG1) implies that $\bracs{UU| U \in \fF}$ contains a neighbourhood base of $x$. Thus
so $x \in yB \subset AB$.$\square$
Definition 9.1.9 (Symmetric Neighbourhood).label Let $G$ be a topological group and $U \in \cn_{G}(1)$, then $U$ is symmetric if $U = U^{-1}$.
Proposition 9.1.10.label Let $G$ be a topological group, then
- (1)
$G$ admits a fundamental system of neighbourhoods at $0$ consisting of symmetric sets.
- (2)
The system in (1) may be taken to be open or closed.
Proposition 9.1.11.label Let $G$ be a topological group and $H \subset G$ be a subgroup, then $\ol H$ is also a subgroup.
Proof. By Proposition 5.5.3, for each $g \in G$, $g\ol H \subset \ol{gH}\subset \ol H$. Similarly, $\ol{H}^{-1}\subset \ol{H^{-1}}\subset \ol H$.$\square$
Definition 9.1.12 (Left and Right Translations).label Let $G$ be a group, $g \in G$, and $Y$ be a set, then
is the left translation map by $g$, and
is the right translation map by $g$.
Proposition 9.1.13.label Let $G$ be a group, $Y$ be a set, and $x, y \in G$, then:
- (1)
$L_{xy}= L_{x}L_{y}$.
- (2)
$R_{xy}= R_{x}R_{y}$.
Proof. (1): For any $f \in Y^{G}$ and $z \in G$,
$\square$
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