Question:
Let $T = \{U \subseteq \mathbb{R} : U = \emptyset \text{ or } U = \mathbb{R}\text{ or } U = (−\infty, a) \text{ for some } a \in\mathbb{R}\}$.
Prove that $T$ is a topology on $\mathbb{R}$.
I know the axioms are:
- The empty set and $X$ are in $T$.
- The intersection of any finite collection of subsets of $X$ in $T$ is also in $T$.
- The union of any collection of sets in $T$ is also in $T$.
and can prove the 1st axiom easily but I am struggling to understand how to go about actually showing that the 2nd and 3rd axioms of topologies hold for any example question I attempt, not just this one.
I would appreciate if someone could give me the basics of the process involved in proving axiom 2 and 3 hold.