Example of a space that is connected but not path connected?

Question

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?

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? — , Apr 24 '16 at 19:09

score 2 · Answer 1 · answered Apr 24 '16 at 19:10

2

Consider the space $$ X=\{(x,\sin x^{-1} ) : x>0\}\cup (\{0\}\times [-1,1])$$ with topology induced by euclidean metric.

answered Apr 24 '16 at 19:10

score 1 · Answer 2 · answered Apr 24 '16 at 19:09

1

That's the best known counterexample.

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

answered Apr 24 '16 at 19:09

Chill2Macht

2 Answers2