Local path-connectedness v. path-connectedness

cogito²

Can a space be path-connected and not locally path-connected? (To be clear, "locally path-connected" just means that there is a basis of path-connected of sets.)

My general intuition says no, but my intuition seems to usually be wrong...and this would explain why Hatcher keeps referring to spaces that are both p.c. and l.p.c. in his Algebraic Topology....

cogito²

My first time around google didn't bring me to this site which claims to contradict my intuition. So I guess this thread can be ignored...

mathwonk

what is the definition of lcp?

mathwonk

well planet maths ays lcp means every nbhd of p conrtains a pc nbhd.

so just take any silly space like say the union of the y axis, the x axis and the horizontal lines at ordinates y = 1/n.

thAT SHOULD BE PATH CONNECTED AND NOT LOCALLY SO at any point of the x axis except (0,0).

cogito²

That's a pretty nice example of such a space.

Well I guess it's time to fine-tune my intuition....