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Let $X$ be a topological space containing at least 2 elements. Is it true, in general, that there exist 2 disjoint closed nonempty subsets $A$ and $B$ of $X$, such that $$X = A \cup B?$$ Is it false, in general?

Andrey Portnoy
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    Definitely no: connected spaces not only exist, but they are quite important. –  Feb 05 '18 at 07:52
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    This question is not at all obvious, if you are just considering the abstract definition of a topological space. But if you consider the example of the real numbers with the ordinary (metric) topology, you intuition should take you to the correct answer. – Eric Wilson Feb 05 '18 at 14:46

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It is true if and only if $X$ is not connected.

Robert Israel
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