Let $X$ be a topological space. Let $A$ and $B$ be sets in $X$ such that the intersection of $A$ and $B$ is empty. Suppose the union of $A$ and $B$ is open. Does it follow that both $A$ and $B$ are open?
unions of sets with a trivial intersection
3
$\begingroup$
general-topology
-
0You can consider a general set $A$ and its complement. – 2012-03-02
2 Answers
7
No. For example, let $X$ be $\mathbb{R}$ with the usual topology, $A$ be $(0,1]$, and $B$ be $(1,2)$.
4
Not even in the reals. Let $A$ be the rationals and $B$ the irrationals. There are many other examples.
In general, $A$ and $B$ can be more or less arbitrarily badly (or well) behaved.