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?

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.

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?