Space which is path connected but not locally path connected

Question

Can you give an example of a topological space which is path connected but not locally path connected, besides the graph of $\sin(1/x)$?

Note that the "topologist's sine curve" is not path-connected in its standard form. It's connected but not locally connected. — Henno Brandsma, Jun 23 '20 at 05:56

score 2 · Answer 1 · edited Jun 23 '20 at 02:16

2

An example is the comb space,

which is $(\{0\}\times[0,1])\cup(K\times[0,1])\cup([0,1]\times\{0\}),$ where $K=\{\frac1n\mid n\in\mathbb N\}$,

with the subspace topology in $\mathbb R^2$.

edited Jun 23 '20 at 02:16

Brian M. Scott

answered Jun 23 '20 at 02:10

J. W. Tanner

1 Answers1