Space which is connected but not path-connected

Others have mentioned the topologist's sine curve, that's the canonical example. But I like this one, though it is the same idea:

enter image description here

This is the image of the parametric curve $\gamma(t)=\langle (1+1/t)\cos t,(1+1/t)\sin t\rangle,$ along with the unit circle. There is no path joining a point on the curve with a point on the circle. Yet, the space is the closure of the connected set $\gamma((1,\infty))$, so it is connected.


This picture of the comb space from the Wikipedia article on it may help: Comb space

This answer contains a pretty extensive discussion of why the deleted comb space, though connected, is not path-connected. (The deleted comb space has only two points on the $y$-axis, the origin, and $\langle 0,1\rangle$. There is no path from $\langle 0,1\rangle$ to any other point of the space.)

Tags:

General Topology

Path Connected

Connectedness

Examples Counterexamples

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