All spaces I have seen which are connected are also path connected (apart from examples to show that the two are not equivalent). Is there a reason for using the weaker definition of connectedness when all applications (that I have seen at least) will involve the stronger concept of path connected spaces?
2026-04-06 14:34:53.1775486093
Why Not Define Connectedness to Mean Path Connected?
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Even in the setting where you only consider path-connected spaces, connected is still a technically simpler concept, and is often easier to prove things about connectedness than it is about path-connectedness, and conversely, it is often easier to use connectedness to prove things than it is to use path-connectedness.
(assuming, of course, that you're comfortable reasoning with open sets and such)