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  3. definition 1 (topological space). let x be a set.

Definition 1 (Topological space). Let X be a set. A collection O of subsets of X...

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Advance Math

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.

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