could someone please clarify the definitions of extensions of topological groups and Lie groups.
For topological groups, what I see in most papers is as follows: An extension of topological groups $0 \to N \stackrel{i}{\rightarrow} G \stackrel{\pi}{\rightarrow} Q \to 0$ is an algebraically exact sequence of topological groups such that $i$ is closed and continuous and $\pi$ is open and continuous.
However, for Lie groups, it is defined as follows: An extension of Lie groups $0 \to N \stackrel{i}{\rightarrow} G \stackrel{\pi}{\rightarrow} Q \to 0$ is an algebraically exact sequence of Lie groups such that both $i$ and $\pi$ are smooth and $\pi$ has a local smooth section in a neighbourhood of identity in $G$.
Are these the standard definitions? Does these definitions imply that an extension of Lie groups give an extension of topological groups by restricting to the topological structure?
Kindly provide any references.