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**Plato** Well that approach is used by some authors. It is not as instructive as is the separated sets approach. But they are equivalent.

Stated properly your text’s approach ought to say:

A set $\displaystyle X$ is not connected if and only if there exists two disjoint open sets $\displaystyle U~\&~V$ each having non-empty intersection with $\displaystyle X$ and $\displaystyle X\subseteq U\cup V$

In you problem if that is true for $\displaystyle A\cup B$ then because each of $\displaystyle A~\&~B$ is connected then we can say that $\displaystyle A\subseteq U~\&~ B\subseteq V $ or visa versa. WHY?

**I'm unsure of this, I get because U union V is an open cover that is disjoint for both A and B then either U or V is empty so that either A,B is a subset of U or A,B is a subset of V**

But the given tells us that $\displaystyle A\cap \overline{B}\ne \emptyset$.

So you have a contradiction. HOW?

Hence $\displaystyle A\cup B$ is connected.