Is there a topological space which is simply connected but not local path connected Is there a topological space which is simply connected but not local path connected?
I think these two properties have no relation at all but I can't find a example.
 A: The Cantor-Knaster-Kuratowski Fan is constructed by taking the Cantor set $\mathcal{C}$ in $[0,1]\times\{0\}\subseteq\Bbb R^2$ and drawing a line to $p=\left(\frac12,\frac12\right)$ from each $c\in \mathcal{C}$. This is homotopy-equivalent to a point, hence is simply connected, but as each point in the Cantor set is a limit point of itself, any open set not containing $p$ will not be path connected.
A: Yes.  The comb space works.  It is the subspace of $\mathbb{R}^2$ 
$$(\{0\} \times [0,1])  \cup   ([0,1] \times \{0\}) \cup_{n \in \mathbb{Z}^+} (\{1/n\} \times [0,1] )$$
It is easy to see that this space is contractible (collapse the tines of the comb to the $x$-axis, then contract the interval) and hence simply connected, but it is not locally path connected.  There is no open neighborhood of a point $(0, y)$ with $0 < y \leq 1$ that is path connected.   
A: quasi-circle would be another useful counter example, this is not locally path connected. See here ex1.3.7  that it is simple connected (http://www.math.ku.dk/~moller/f03/algtop/opg/S1.3.pdf) .
