I am looking for a reference for the assertion in the title. This assertion is proved in a comment of user nfdc23 to [this question](https://mathoverflow.net/q/262185/4149). Has any proof of this assertion been published?



Reference request: Any connected Lie group has a countable base for its topology
To summarize the easy argument given in MathSE by Cronus: if $V$ is a neighborhood of the unit homeomorphic to an open ball, $(B\_n)$ a countable basis of $V$ and $D$ a dense countable subset of $V$, then the $(g\_1\dots g\_kB\_n)$ when $k,n$ range over integers and $g\_i$ over $D$, form a countable basis for $G$.


