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Which of the followings are topology of $\mathbb R$? enter image description here

I believe that 7,8 are while topology, which I think I can understand why. While for 9,10, I believe there're trap, preventing them from being a topology, but I have no clue why are they not. I suspect they went wrong when considering infinite union of them. And maybe, I'm completely wrong. Please help me distinguishing.

JSCB
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2 Answers2

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What is $\bigcup_{0<r<1}[-r,r)$?

Hagen von Eitzen
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  • AHHHHHH.. I'm utterly defeated... – JSCB Jul 14 '14 at 14:09
  • @ᴊᴀsᴏɴ Try picking some example points. Let $S = \bigcup_{0<r<1}[-r,r)$. Then ask: is $17\in S$? Why, or why not? Is $\frac12\in S$? Is $1\in S$? Is $0\in S$? Then one you have figured those out, try to generalize the arguments you made. – MJD Jul 14 '14 at 14:13
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All of the "topologies" contain $\mathbb{R}$ and $\phi$, so what you need to check is that if you take an arbitrary union of sets in each given $\tau$, the result is also in $\tau$, and the same for finite intersections.

As a clue on how to approach the question, I will tell you that the only actual topologies are $9$ and $10$. So you must show that $\tau_9,\tau_{10}$ are closed under arbitrary unions and finite intersections, and for each of the others, try to find sets whose union (or finite intersection) doesn't belong to the $\tau_x$

Edit: Upon request, here's some more details for $9)$:

Take any collection of sets $(-r_i,r_i),[-r_j,r_j]$ for $i \in I$, $j \in J$ some indexing sets. We aim to show that $U = \cup_{i\in I} (-r_i,r_i)\cup_{j\in J} [-r_j,r_j]$ is contained in $\tau_9$. What happens if one of $\{r_i : i \in I\},\{r_j:j \in J\}$ is unbounded? (Easy.) What happens otherwise? (Harder.)

Tom Oldfield
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