Exercise 2.1. (a) Prove that the collection $\mathcal{T}=\{\mathrm{\varnothing },\mathbb{R}\}\cup \{(x,+\mathrm{\infty }):x\in \mathbb{R}\}$ is a topology on the set $\mathbb{R}$ of real numbers.

(b) Prove that the collection $\mathcal{N}=\{\mathrm{\varnothing },\mathbb{R}\}\cup \{[x,+\mathrm{\infty }):x\in \mathbb{R}\}$ is not a topology on the set $\mathbb{R}.$ 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 $\mathcal{T}$ 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 $A=\mathrm{\varnothing }$ 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?