Which of the following collections are topologies for $\mathbb R$? Am I correct? Which of the following collections are topologies for $\mathbb R$?
I think I have these correct I just want to double check my answers.
(a) $\{\mathbb R, \emptyset, (-\infty, 0), (0,\infty)\}$- Yes, empty set and whole line are included, intersections and unions are as well.
(b) $\{\mathbb R, \emptyset, (1,4), (2,5)\}$ no as the intersection of $(1,4)$ and $(2,5)$ is not included in the topology.
(c) $\{U : U = \emptyset$ or $U = \mathbb R$ or $U = (-\infty, b]$ for some $b \in \mathbb R\}$- I originally thought this was no now I am going back and saying yes.  The other 2 I feel confident about this is the one I am shaky on. 

You are correct in thinking that (b) is not a topology, for the reason that you gave. However, (a) is also not a topology: it doesn’t contain $(-\infty,0)\cup(0,\infty)=\Bbb R\setminus\{0\}$. Finally, (c) also fails to be a topology:
$$\bigcup_{b<0}(-\infty,b]=(-\infty,0)\;,$$
which is not in the family.
