Does semi-locally simply connected imply locally path connected?

Question

I'm reading about the classification of covering maps, and getting a little confused about the concepts. Does semi-locally simply connected imply locally path connected?

Answer 1

No. Take one of the usual non-locally path connected spaces such as a comb with rational teeth $$X=[0,1]\cup\{(q,x)\,|\, q\in\mathbb Q\cap[0,1]\text{ and } x\in[0,1]\}.$$ This is semi-locally simply connected but not locally path connected. Small loops can be contracted without any problem.