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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