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## 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?