For a class in advanced calculus and proofs Let R be a set o

For a class in advanced calculus and proofs

Let R be a set of sets. Assume, for all P, Q R, that Show that P Q R. Show that ®_Fin = R. Let B be a set of sets. Assume, for all that B, C B, that B C (B) Show that (B) is a topology.

Solution

A)Consider the collection T which consist of unions of finite intersections of sets from C and also includes the whole space and the empty set. By properties in any topology in which sets from C are open the sets from T are also open. It follows immediately from the properties of intersections PQR. Both closed and open interval as topological spaces have the property that the only sets which are open and closed at the same time are the space itself and the empty set.

B) It suffices to show that in a compact space every collection ofclosed sets with the finite intersection property has nonempty intersection. Arguing by contradiction, suppose there is a collection of closed subsets in a compact space K with empty intersection. Then their complements form an open cover of K. Since it has a finite subcover, the finite intersection property does not hold.

Euclidean space Rⁿwith the standard topology (the usual open and closed sets) has bases consisting of all open balls, open balls of rational radius, open balls of rational center and radius. The latter is a countable base.