Proposition 2.13

Proposition 2.13: If $$X$$ is a topological space, then (i) the intersection of any collection of closed sets is a closed set; (ii) the union of finitely many closed sets is a closed set.

Jamie
i) Consider the complement of some intersection of closed sets. By De Morgan's Law, this is equivalent to the union of the complements of each of the sets. Because each set is closed, all of the complement sets are open. By Definition 2.1, we know that the union of the complement sets is open. Therefore, the complement of the union of complements of closed sets is open, and the intersection of closed sets is closed.

ii) By De Morgan's Law, this is equivalent to showing that the intersection of a finite number of open sets is open. This must be true by Definition 2.1.

203.173.190.104 09:44, March 15, 2011 (UTC)

'''Corrected Proposition 1.12 dealing with metric spaces to Definition 2.1 for topological spaces. Also linked to De Morgan's Law. Nice work. --Steven.clontz 20:04, March 17, 2011 (UTC)'''