CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A topological group is a topological space with a continuous group structure: a group object in the category Top.
A topological group is an internal group object in the category of topological spaces.
More explicitly, it is a group equipped with a topology such that the multiplication and inversion maps are continuous.
A topological group $G$ carries two canonical uniformities: a right and left uniformity. The right uniformity consists of entourages $\sim_{l, U}$ where $x \sim_{l, U} y$ if $x y^{-1} \in U$; here $U$ ranges over neighborhoods of the identity that are symmetric: $g \in U \Leftrightarrow g^{-1} \in U$. The left uniformity similarly consists of entourages $\sim_{r, U}$ where $x \sim_{r, U} y$ if $x^{-1} y \in U$. The uniform topology for either coincides with the topology of $G$.
Obviously when $G$ is commutative, the left and right uniformities coincide. They also coincide if $G$ is compact Hausdorff, since in that case there is only one uniformity whose uniform topology reproduces the given topology.
Let $G$, $H$ be topological groups, and equip each with their left uniformities. Let $f: G \to H$ be a group homomorphism.
The following are equivalent:
The map $f$ is continuous at a point of $G$;
The map $f$ is continuous;
The map $f$ is uniformly continuous.
Suppose $f$ is continuous at $g \in G$. Since neighborhoods of a point $x$ are $x$-translates of neighborhoods of the identity $e$, continuity at $g$ means that for all neighborhoods $V$ of $e \in H$, there exists a neighborhood $U$ of $e \in G$ such that
Since $f$ is a homomorphism, it follows immediately from cancellation that $f(U) \subseteq V$. Therefore, for every neighborhood $V$ of $e \in H$, there exists a neighborhood $U$ of $e \in G$ such that
in other words such that $x \sim_U y \Rightarrow f(x) \sim_V f(y)$. Hence $f$ is uniformly continuous with respect to the right uniformity. By similar reasoning, $f$ is uniformly continuous with respect to the right uniformity.
A unitary representation $R$ of a topological group $G$ in a Hilbert space $\mathcal{H}$ is a continuous group homomorphism
where $\mathcal{U}(\mathcal{H})$ is the group of unitary operators on $\mathcal{H}$ with respect to the strong topology.
Here $\mathcal{U}(\mathcal{H})$ is a complete, metrizable topological group in the strong topology, see (Schottenloher, prop. 3.11).
In physics, when a classical mechanical system is symmetric, i.e. invariant in a proper sense, with respect to the action of a topological group $G$, then an unitary representation of $G$ is sometimes called a quantization of $G$. See at geometric quantization and orbit method for more on this.
The reason that in the definition of a unitary representation, the strong operator topology on $\mathcal{U}(\mathcal{H})$ is used and not the norm topology, is that only few homomorphisms turn out to be continuous in the norm topology.
Example: let $G$ be a compact Lie group and $L^2(G)$ be the Hilbert space of square integrable measurable functions with respect to its Haar measure. The right regular representation of $G$ on $L^2(G)$ is defined as
and this will generally not be continuous in the norm topology, but is always continuous in the strong topology.
The category TopGrp of topological groups and continuous group homomorphisms between them is a protomodular category.
A proof is spelled out by Todd Trimble here on MO.
topological group,
The following monograph is not particulary about group representations, but some content of this page is based on it: