Local path-connectedness v. path-connectedness

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A space can indeed be path-connected without being locally path-connected, as demonstrated by examples such as the union of the x-axis and horizontal lines at y = 1/n. This configuration is path-connected but fails to be locally path-connected at points on the x-axis, excluding the origin. The discussion highlights the distinction between path-connectedness and local path-connectedness, emphasizing the importance of understanding neighborhood structures. The reference to Hatcher's work suggests that many spaces exhibit both properties, but counterexamples exist. Overall, the conversation serves to refine the understanding of these topological concepts.
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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...
 
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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...
 
what is the definition of lcp?
 
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).
 
That's a pretty nice example of such a space.

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

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