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Week 2 Exercises

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  • Exercise 2.1. (a) Prove that the collection \(\mathscr T = \{\emptyset , \RR \}\cup \{ (x, +\infty ) : x \in \RR \}\) is a topology on the set \(\RR \) of real numbers.

    (b) Prove that the collection \(\mathscr N = \{ \emptyset , \RR \} \cup \{ [x, +\infty ) : x \in \RR \}\) is not a topology on the set \(\RR .\) Which axiom(s) of topology is/are not satisfied?

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  • Exercise 2.2. Consider the set \(X = \{1,2\}\) with two points. Describe all possible topologies \(\mathscr T\) on \(X.\) Among the topologies that you describe, identify the discrete topology, the antidiscrete topology and the cofinite topology.

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  • Exercise 2.3. Call a subset \(A\) of \(\mathbb R\) “cocountable” if \(A=\emptyset \) or \(\mathbb R \setminus A\) is finite or countably infinite.

    (a) Show that the collection of all cocountable subsets of \(\mathbb R\) is a topology on \(\mathbb R.\)

    (b) Is this topology the same as discrete topology? Antidiscrete? Cofinite topology?

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Version 2024/11/02. These exercises in PDF To other course material