Local path-connectedness v. path-connectedness

  • Thread starter cogito²
  • Start date
  • #1
98
0

Main Question or Discussion Point

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....
 

Answers and Replies

  • #2
98
0
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
mathwonk
Science Advisor
Homework Helper
10,780
950
what is the definition of lcp?
 
  • #4
mathwonk
Science Advisor
Homework Helper
10,780
950
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
98
0
That's a pretty nice example of such a space.

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

Related Threads for: Local path-connectedness v. path-connectedness

  • Last Post
Replies
10
Views
3K
Replies
1
Views
1K
Replies
1
Views
1K
  • Last Post
Replies
1
Views
674
  • Last Post
Replies
5
Views
3K
  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
21
Views
6K
  • Last Post
Replies
5
Views
3K
Top