$\begingroup$

We know path connectedness implies connectedness . Is the other direction true or false? I ve been trying to prove it but cannot do it. I cannot find a counter-example either. math is hard.

$\endgroup$ 3

2 Answers

$\begingroup$

My favorite counterexample is Cantor's leaky tent.

Not only is it connected, but not path connected, removing a single point renders it completely disconnected! (Since I haven't taken topology yet, this is what prompted me to look up what "connected" really means)

$\endgroup$ 1 $\begingroup$

Look up the topologist' sine. It is not true. That is why you cannot prove it :p.

Another funky one is $\mathbb{R}_{\text{cocountable}}$.

$\endgroup$