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This question already has an answer here:

Wikipedia says that path-connectedness is a stronger property than connectedness.

My intuition cannot seem to come up with an example of an object that is connected but not path-connected. Are there any examples?

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marked as duplicate by Dietrich Burde, user296602, Forever Mozart, Daniel W. Farlow, John B Apr 25 '16 at 0:49

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    $\begingroup$ Did you do any research before posting, such as googling the phrase "connected not path connected"? Or reading the part of the Wikipedia article that gives an explicit counterexample? $\endgroup$ – user296602 Apr 24 '16 at 19:09
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Consider the space $$ X=\{(x,\sin x^{-1} ) : x>0\}\cup (\{0\}\times [-1,1])$$ with topology induced by euclidean metric.

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Topologist's sine curve - see this Wikipedia page: https://en.wikipedia.org/wiki/Topologist%27s_sine_curve

That's the best known counterexample.

And you're right - it's definitely not intuitive at all.

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