How to show "If A and B connected, is $A\cup B$ connected"?

Question

If A and B connected, is $A\cup B$ connected? or give a counterexample.

I'd say no because when we take $A=[1,2]$, $B=[3,4]$, these closed intervals are connected. But when we take $U=]\frac 12,\frac 52[ $ and $V=]\frac 52,\frac92[$

$$(A\cup B)\subset (U\cup V), (A\cup B)\cap V\neq\emptyset,(A\cup B)\cap U\neq\emptyset,(A\cup B)\cap V \cap U =\emptyset$$

Is this true and sufficient?

Answer 1

Of course if $A,B$ are connected, there is not reason $A\cup B$ is, as you show. However, if $\{A_i\}$ is a family of connected sets and $\bigcap A_i\neq\varnothing$, $\bigcup A_i$ is connected. Can you prove this?