• Support PF! Buy your school textbooks, materials and every day products Here!

Closure of Path-Connected Set

  • Thread starter Bashyboy
  • Start date
  • #1
1,421
5

Homework Statement


I am trying to determine whether the closure of a path-connected set is path-connected.

Homework Equations




The Attempt at a Solution


Let ##S = \{(x, \sin(1/x) ~|~ x \in (0,1] \}##. Then the the closure of ##S## is the Topologist's Sine Curve, which is known not to be path-connected. However, recalling that the image of a path-connected space under a continuous function is path-connected, and defining ##g : (0,1] \rightarrow \mathbb{R}## as ##g(x) = (x, \sin (1/x))##, we see that ##S = g((0,1])## must be a path-connected space.

My question is, would this constitute a counterexample to the claim, or have I made some error?
 

Answers and Replies

  • #2
334
61
there is no error I guess
 

Related Threads on Closure of Path-Connected Set

Replies
2
Views
4K
Replies
3
Views
3K
Replies
4
Views
834
  • Last Post
Replies
19
Views
3K
Replies
1
Views
1K
Replies
9
Views
694
  • Last Post
Replies
4
Views
2K
Replies
1
Views
5K
  • Last Post
Replies
5
Views
5K
Top