I have seen two different definitions of disconnected sets. Here (X,T) is a topological space.
A subset $A$ of X is disconnected if there are two open sets $U,V$ of X such that $U \cap A \neq \emptyset $ , $V \cap A \neq \emptyset $, $U \cap A \cap V = \emptyset $ and $A\subset U\cup V$.
Two subsets $A,B$ of X are separated if no point of A lies in the closure of B and vice versa. A subset E of X is disconnected if E is a union of two nonempty separated sets.
I am having trouble understanding the connection between these two definitons. Why do they mean the same thing?
Thank you kindly.