Topologies on the Real Line

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• Oct 31st 2013, 09:34 PM
zhandele
Topologies on the Real Line
My book lists ten collections of subsets of R (the set of all real numbers), and tells me that exactly three of them are topologies on R. My problem is that nine of the ten look like topologies to me. Can somebody tell me what I’m missing?

$\ \tau _1 \ \text{ consists of } \ \mathbb{R} \text{, } \emptyset \ \text{ and every interval }$ $(a, b) \text{ for } a \text{ and } b \text{ any real numbers with } a < b$

This one is not a topology, as it seems to me, because for example

$\ \left( {2,3} \right) \cup \left( {4,5} \right) \notin \tau _1$

(it’s a disjoint set). But what about the other nine (I will list them below)? I suspect that the three I’m supposed to pick are ii, iii and v, which involve only open intervals, but I don’t see why a collection of closed or half-open sets couldn’t still meet the definition of a topology.

Here are the candidates:

$\ \tau _2 \ \text{ consists of } \ \mathbb{R} \text{, } \emptyset \text{ and every interval } \left( { - r,r} \right){\text{ for }}r{\text{ any positive real number}$

$\ \tau _3 \ \text{ consists of } \ \mathbb{R} \text{, } \emptyset \text{ and every interval } \left( { - r,r} \right){\text{ for }}r{\text{ any positive rational number}$

$\ \tau _4 \ \text{ consists of } \ \mathbb{R} \text{, } \emptyset \text{ and every interval } \left[ { - r,r} \right] {\text{ for }}r{\text{ any positive rational number}$

$\ \tau _5 \ \text{ consists of } \ \mathbb{R} \text{, } \emptyset \text{ and every interval } \left( { - r,r} \right){\text{ for }}r{\text{ any positive irrational number}$

$\ \tau _6 \ \text{ consists of } \ \mathbb{R} \text{, } \emptyset \text{ and every interval } \left[ { - r,r} \right] {\text{ for }}r{\text{ any positive irrational number}$

$\ \tau _7 \ \text{ consists of } \ \mathbb{R} \text{, } \emptyset \text{ and every interval } \left[ { - r,r} \right) {\text{ for }}r{\text{ any positive real number}$

$\ \tau _8 \ \text{ consists of } \ \mathbb{R} \text{, } \emptyset \text{ and every interval } \left( { - r,r} \right] {\text{ for }}r{\text{ any positive real number}$

$\ \tau _9 \ \text{ consists of } \ \mathbb{R} \text{, } \emptyset \text{, every interval } \left[ { - r,r} \right] \text{ and every interval } \left( { - r,r} \right) {\text{ for }}r{\text{ any positive real number}$

$\ \tau _{10} \ \text{ consists of } \ \mathbb{R} \text{, } \emptyset \text{, every interval } \left[ { -n,n} \right] \text{ and every interval } \left( { - r,r} \right) {\text{ for }}n{\text{ any positive integer and }r{\text{ any positive real number}$
• Oct 31st 2013, 09:55 PM
SlipEternal
Re: Topologies on the Real Line
Consider the geometric sum $\sum_{k\ge 0}\dfrac{1}{2^k} = 2$. Let $a_n = \sum_{k = 0}^n \dfrac{1}{2^k}$. Then the interval $\left[-a_n,a_n\right] \in \tau_6$ for all $n$. Consider $\bigcup_{n\ge 0}\left[-a_n,a_n\right]$. Since $-2,2\notin \left[-a_n,a_n\right]$ for any $n$, the union must be the set $(-2,2) \notin \tau_6$. This is why closed and half-closed intervals don't work for topologies. Unions are not closed.

For $\tau_3$, let $b_n$ be the sequence $b_0 = 3, b_1 = 3.1, \ldots, b_5 = 3.14159, \ldots$. Hence, $b_n$ is a strictly increasing sequence of rational numbers. The union $\bigcup_{n\ge 0}(-b_n,b_n) = (-\pi,\pi)\notin \tau_3$. So, that doesn't work, either.

The three that are topologies are $\tau_2, \tau_9,\tau_{10}$ (I think)