I wonder why one requires that the base manifold of a Lie groupoid is second-countable? 



Why is the base manifold of a Lie groupoid required to be second-countable?
**Answer #1:**  
There is *no real reason* for imposing that the base manifold of a groupoid be second countable.


**Answer #2:**  

You lose some desirable properties if you don't impose second countability:
For example, without it, 
the homotopy type of the geometric realisation of the nerve
will no longer be an invariant of the Morita equivalence class of the groupoid.


