I am interested in the properties of convex set in $\mathbb R^n$ and want to clarify the three statements below
$A$ is an open convex set. Can we get the conclusion that $\bar{A}$ is convex?
Conversely, if $A$ is a closed convex set in $\mathbb R^n$, is that $\text{Int}A$ is convex always right?
More generally, we assume $A$ is convex. Then can we claim $\bar{A}$ or $\text{Int}A$ is convex?
Counterexamples or proofs are welcome. Thanks.
(1) Yes. If $a,b\in\bar A$, say $a_n\to a, b_n\to b$ with $a_n,b_b\in A$, and $t\in[0,1]$ then $ta_n+(1-t)b_n\to ta+(1-t)b$.
(2) Yes. If $a,b$ are interior points with $\epsilon$-balls around them still belonging to $A$, then a convex combination $c=ta+(1-t)b$ also has an $\epsilon$-ball around it belonging to $A$: If $|c'-c|<\epsilon$ then $c'=t(a+c'-c)+(1-t(b+c'-c)$.
(3) Yes because of (1) and (2)