1. Which of the following collections are topologies for \( \mathbb{R} \) ? If a collection is not a topology for \( \mathbb{R} \), explain why it is not.
(a) \( \{\mathbb{R}, \phi,(-\infty, 0],(0,+\infty)\} \)
(b) \( \{\mathbb{R}, \phi,(-\infty, 1),(0,+\infty)\} \)
(c) \( \{\mathbb{R},\{1\}, \phi\} \)
(d) \( \{\mathbb{R}, \phi,(-\infty, 0),[0,+\infty)\} \)
(c) \( \{\mathbb{R}, \phi,(1,3),(2,4)\} \)
(f) \( \{\mathbb{R}, \phi,(1,4),(2,5),(1,5)\} \)
(g) \( \{U: U=\mathbb{R} \), or \( U=\phi \) or \( U=(a,+\infty) \) for some \( \alpha \in \mathbb{R}\} \)
