Well, so far, I have noticed that whenever a matrix lie group is connected it is path connected, so is it true that in matrix lie group connected $\Rightarrow$ path connected?If yes, could anyone tell me where I can get the proof?or if some one tell me the sketch of the proof. Thank you.
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This is true for any locally path-connected space (this is the crucial property to use in the proof), in particular any manifold. Slightly more generally, for locally path-connected spaces, components and path components coincide.