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While studying differential geometry I often come across propositions with $M$ being a connected surface as their hypothesis. They then often take paths between arbitrary points, which to me suggests that they actually mean that $M$ is path-connected.

Is this true? Is it some sort of convention in differential geometry to not distinguish between normal- and path-connectedness?

user2520938
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    Elementary topology says that connected plus locally path-connected implies path-connected. – David C. Ullrich Oct 05 '15 at 14:49
  • See here: http://math.stackexchange.com/questions/332108/showing-that-every-connected-open-set-in-a-locally-path-connected-space-is-path –  Oct 05 '15 at 14:50
  • Thank you both, that clears it up. Didnt quite think it through far enough... – user2520938 Oct 05 '15 at 14:53
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    Seems like this question has been answered in the comments. Perhaps one of you would like to submit an answer that @user2520938 can accept? – treble Oct 05 '15 at 14:56

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(The question is basically answered in the comment)

As a topological manifold is locally path connected, a connected manifold is automatically path connected, as a connected locally path connected topological space is path connected

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