# Closure of Path-Connected Set

1. May 8, 2017

### Bashyboy

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. May 8, 2017

### zwierz

there is no error I guess