If $A$ is a connected set, then its closure $A'$ is connected.
What does this mean?
My professor gave use the definition of a connected set to be:
A set $\Omega$ is said to be connected if $\Omega$ is not the union of nonempty separated subsets of a metric space X.
Does this mean that $A \cap B' = 0$ $A' \cap B = 0$, where $A'$ and $B'$ are the closures of A and B, respectively.
Definition of separated: Let $A$ and $B$ be two subsets of a metric space X. $A$ and $B$ are said to be separated if $A \cap B' = 0$ and $A' \cap B = 0$
This is proven by contradiction.
Assume to the contrary that $\overline{A}=U\cup V$ where $U$ and $V$ are nonempty and separated. Show that $A\cap U$ and $A\cap V$ must also be separated in contradiction to the assumption.
This is an outline without giving the whole proof.