counterexample in topology

Question

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.

You know, there is a book called, emmm, Counterexamples in Topology ;-) — Matemáticos Chibchas, Dec 09 '13 at 23:36
This isn't the place for a (potentially heated) semantics discussion. :\
Also, is there a university library you could check books out of? — Henry Swanson, Dec 10 '13 at 00:00

Henry Swanson · Accepted Answer · 2013-12-09T23:37:13.543

3

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)

edited Dec 09 '13 at 23:37

answered Dec 09 '13 at 23:31

Henry Swanson

12,972

Great example!! – M.B. Dec 09 '13 at 23:35

score 0 · Answer 2 · answered Dec 09 '13 at 23:30

0

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

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

answered Dec 09 '13 at 23:30

LASV

3,108

counterexample in topology

2 Answers2