Is $\overline{A}$ path-connected?

Question

True/False

If $A\subset X$ is path-connected, then $\overline{A}$ is path-connected.

Any hints would be much appreciated; I'm not looking for a full solution.

Answer 1

Hint: Take the graph of $\sin\frac1x, \ x>0$.

Answer 2

It's false, the famous closed topologist's sine curve shows why.

https://en.wikipedia.org/wiki/Topologist%27s_sine_curve#Variants .