Local path-connectedness v. path-connectedness

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In summary, the conversation discusses the concept of path-connected and locally path-connected spaces. The speaker's intuition suggests that a space cannot be path-connected and not locally path-connected, but they have come across a website that contradicts this intuition. The definition of locally path-connected is mentioned, and an example of a space that is path-connected but not locally path-connected is given. The speaker acknowledges the need to refine their intuition on this topic.
  • #1
cogito²
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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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  • #2
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...
 
  • #3
what is the definition of lcp?
 
  • #4
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).
 
  • #5
That's a pretty nice example of such a space.

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

Related to Local path-connectedness v. path-connectedness

1. What is the difference between local path-connectedness and path-connectedness?

Local path-connectedness refers to the property of a topological space where every point has a neighborhood that is path-connected. This means that for any given point in the space, there exists a path connecting that point to any other point in its neighborhood. On the other hand, path-connectedness refers to the property of a space where every two points can be connected by a continuous path, regardless of their distance.

2. Can a space be locally path-connected but not path-connected?

Yes, it is possible for a space to be locally path-connected but not path-connected. This can happen when a space has disjoint path-connected components, where each component is locally path-connected but they cannot be connected by a continuous path.

3. How do you prove that a space is locally path-connected?

To prove that a space is locally path-connected, you need to show that for any point in the space, there exists a neighborhood around that point that is path-connected. This can be done by explicitly constructing a path between any two points in the neighborhood, or by utilizing the concept of path-connectedness in a more general proof.

4. What are the implications of a space being locally path-connected?

A space being locally path-connected has several implications. It guarantees that the space is path-connected around each point, which makes it easier to study the topology of the space. It also allows for the use of certain theorems and techniques that rely on local path-connectedness, such as the fundamental group and covering spaces.

5. Is local path-connectedness a stronger or weaker property than path-connectedness?

Local path-connectedness can be considered a stronger property than path-connectedness because it implies path-connectedness. However, it is not necessarily a stronger or weaker property in all cases, as a space can be locally path-connected but not path-connected, as mentioned in question 2. Therefore, the strength of these properties depends on the specific space being studied.

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