Show the following are not connected in $\mathbb{R}^n$

Question

Is the interior, boundary and closure of a connected set in $\mathbb{R}^n$ connected?

I know the interior is not connected we can show it by a counterexample but I am not quite sure for the closure and boundary

Have you tried to find a proof or counterexample of the closure or boundary? — Mike Pierce, Jul 01 '15 at 20:53

mathcounterexamples.net · Answer 1 · 2015-07-01T20:59:17.573

5

Not for the interior. Consider two externally tangent closed balls.

Not for the boundary. Consider the end points of a closed segment in $\mathbb R$

Yes for the closure. That is a theorem that you can prove considering continuous constant maps.

edited Jul 01 '15 at 20:59

answered Jul 01 '15 at 20:54

1 Answers1