Relation between the notions connected and disconnect, confused In the textbook "Topology without tears" I found the definition.
$(X, \tau)$ is diconnected iff there exists open sets $A,B$ with $X = A \cup B$ and $A \cap B = \emptyset$.
In Walter Rudin: Principles of Analysis, I found.
$E \subseteq X$ is connected iff it is not the union of two nonempty separated sets. Where two sets $A,B$ are separeted iff $A \cap \overline{B} = \emptyset$ and $\overline{A} \cap B = \emptyset$.
So I am confused, why is in the first defintion nothing about the concept of separted sets said, for these two definitions are not logical negates of one another????

First, note that one should (in both versions) add that $A,B$ should be nonempty.
If $A,B$ are open and disjoint, then also $\overline A$ and $B$ are disjoint as $\overline A$ is the intersection of all closed sets containing $A$, thus $\overline A$ is a subset of the closed set $X\setminus B$.
