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

Definition 1 (Topological space). Let X be a set. A collection O of subsets of X is called a topology on the set X if the following properties are satisfied:

(1) emptyset ∈ O and X ∈ O.

(2) For all A,B ∈ O, we have A∩B ∈ O (stability under intersection).

(3) For all index sets I, and for all collections {Ui}i∈I of elements 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 a topology O is called a topological space and the sets in O are called 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) Consider O_discrete = P(X). Is O_discrete is a topology on X? Justify briefly your answer. Hint. You have to verify whether the collections O_trivial and O_discrete satisfy the three properties in Definition 1.

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Colby Messinger answered 1 year ago

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