1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Closure of Path-Connected Set

  1. May 8, 2017 #1
    1. The problem statement, all variables and given/known data
    I am trying to determine whether the closure of a path-connected set is path-connected.

    2. Relevant equations


    3. 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?
     
  2. jcsd
  3. May 8, 2017 #2
    there is no error I guess
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted