Definition 1 (Topological space). Let X be a set. A collection Oof subsets of X is called a topology on the set X if the followingproperties are satisfied:
(1) emptyset ? O and X ? O.
(2) For all A,B ? O, we have A?B ? O (stability underintersection).
(3) For all index sets I, and for all collections {Ui}i?I ofelements of O (i.e., Ui ? O for all i ? I), we have U i?I Ui ? O(stability under arbitrary unions). A set X equipped with atopology O is called a topological space and the sets in O arecalled open sets.
Exercise 1. Let X be a set. (1) Consider O_trivial ={emptyset,X}. Prove that O_trivial is a topology on X. (2) ConsiderO_discrete = P(X). Is O_discrete is a topology on X? Justifybriefly your answer. Hint. You have to verify whether thecollections O_trivial and O_discrete satisfy the three propertiesin Definition 1.
