If $X$ is a topological space that is first-countable, then a function $f: X \to Y$ into another topological space $Y$ is continuous if and only if $f$ is sequential continuous. Only the implication from left to right is true in general for arbitrary $X$.
1) What are examples of locally compact spaces and of locally compact groups that are not first-countable?
2) Is my understanding correct that every Lie group is first countable, since it is locally isomorphic to an Euclidean space?
3) I am interest in understanding the continuity condition for infinite-dimensional representations of groups If $\Phi: G \to L(V)$ is a representation of $G$ and $G$ is a Lie group and $V$ is a topological vector space, one usually demands continuity of the map, more precisely on the right side one equips $L(V)$, the group of bounded linear operators on $V$, with the strong operator topology. This is equivalent to: the map $G \to V, g \mapsto \Phi(g)v$ is continuous for all $v \in V$. Now, if i am correct in 2), given a representation, one usually checks the continuoity of these maps with sequences. I wonder if checking the continuity for locally compact groups that are not first-countable becomes much more difficult then. Are there illustrative examples?
4) Often one reads that a representation is a map $\Phi: G \to GL(V)$ that is a group homomorphism and continuous in the above sense. But one should add that the operators $\Phi(g)$ are bounded, right? Or does one also consider 'unbounded' representations? And if not, why not? Why doesnt one then usually write: a representation is a map $\Phi: G \to L(V)$ that is a group homormorphism and remarks that by the group homomorphism property, $\Phi(g)$ is automatically invertible for all $g$?