# 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 $$Y \subseteq \mathbb R$$

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