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In a book on beginning measure theory, the following statement is made: "It is clear that any intersections and finite unions of closed sets are closed." However the intersection of two disjoint closed sets is the empty set which is open by definition. Is there something wrong with my understanding, or was the statement false?

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The empty set is also closed, being the complement of the whole space, which is open. – enzotib Jul 29 '12 at 13:07
The empty set is both open and closed. There's no rule that says a set can't be both! – Zhen Lin Jul 29 '12 at 13:11
There are topological spaces whose subsets are not only open, but also closed, for example, unconnected spaces. – Paul Jul 29 '12 at 13:22
To a layman a door is either open or closed (but not both). To a topologist a set can be either or neither!! – BenjaLim Jul 29 '12 at 13:40

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You seem to believe that if a set is open then it is not closed. This is false. In particular, the empty set and the whole space are always both open and closed.

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