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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?

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You can consider a general set $A$ and its complement. – David Mitra Mar 2 '12 at 19:59
up vote 7 down vote accepted

No. For example, let $X$ be $\mathbb{R}$ with the usual topology, $A$ be $(0,1]$, and $B$ be $(1,2)$.

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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.

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