Question

Prove that set of all open subsets of a metric space is a topology

Answer 1

Solution:

It is the definition. We define as following:

A topology on a nonempty set X is a collection of subsets of called open sets, such that:

(a) the empty set $\emptyset$ and the set X are open;

(b) the union of an arbitrary collection of open sets is open;

(c) the intersection of a finite number of open sets is open.