# Topologies

**Geometry**Level 4

Which of the following collections \( \mathcal T\) of subsets of \(\mathbb R\) is a topology on \(\mathbb R\)?

I. \( \mathcal T = \) the empty set, \(\mathbb R\), and all intervals of the form \( [a,\infty) \) for any \( a \in \mathbb R.\)

II. \(\mathcal T = \) the empty set, plus all subsets \( Y \subseteq \mathbb R\) such that the complement \({\mathbb R} \setminus Y \) is finite.

III. \(\mathcal T = \) the empty set, plus all infinite subsets \(Y \subseteq \mathbb R.\)