Question

Is the set {1} open or closed in the cofinite topology on the set of integers ?​

Answer 1

Answer:

Step-by-step explanation:A closed set in Z is finite (or the whole set Z). Thus the only closed set containing E is Z and therefore the closure of E is Z: the closure of a set is the intersection of all closed sets containing it.

The interior of a set is the largest open set contained in it. So, let U be an open set, U⊆E; then Z−U is closed, but it's clearly infinite, because Z−U⊇Z−E. The only infinite closed set in Z is Z; therefore Z−U=Z and U=∅.

Similarly, A contains no nonempty open set and therefore the interior of A is again empty. Of course A is closed, by definition of cofinite topology.