I'm working through the book Topology without Tears by Sidney A. Morris, and I've come to a roadblock while trying to solve exercise $1.1.9$, specifically with part iii. My background is primarily in mechanical and software engineering, and am not yet very comfortable with pure mathematics and set theory, but I'm hoping to get there.
The problem is stated as:
"Let $\mathbb{R}$ be the set of all real numbers. Precisely three of the following ten collections of subsets of $\mathbb{R}$ are topologies? Identify these and justify your answer".
It then continues to list 8 different sets that could potentially be topologies in $\mathbb{R}$. I'm specifically concerned with the third one being listed:
$\tau_3$ consists of $\mathbb{R}, \emptyset$, and every interval $(-r,r)$, for $r$ any positive rational number.
My intuition tells me that this should be a topology. After all, any pair of positive rational numbers $r_1$ and $r_2$ must be such that either $r_1>r_2$, $r_1<r_2$, or $r_1=r_2$. In any case, the intervals $(-r,r)$ formed by $r_1$ and $r_2$ would result in one interval being a subset of the other. Meaning that the union of the two intervals would simply be the larger of the two intervals and thus remain an open set, and the intersection of the two intervals would simply be the smaller of the two intervals and thus remain an open set.
However, context clues (following this same logic results in way more than 3 valid topologies), and a broader but very similar question: Ex.1.1 9 from topology without tears suggests that my logic is flawed.
Looking through the answer to the question linked above, the explanation for why this was not a topology is as follows:
"$\tau_3$ is not a topology since for some irrational $a$ we have
$$\bigcup_{n=1}^\infty(-r_n,r_n) = (-a,a)\notin \tau_3$$ where $\lim_{n\to\infty}r_n = a$."
I don't follow this explanation. Why would an infinite union of these rational open intervals become an irrational open interval? Why would it not just be some infinitely large rational open interval? I feel like this alludes to some key piece of information that I'm missing that would likely be very helpful for me throughout the rest of this book.
Best Answer
Consider the decimal expansion of an irrational number $a$: $$ a = a_0.a_1a_2a_3\cdots $$ Let $r_i$ be the expression above truncated to the $i$th decimal place. So, $r_1=a_0.a_1$, $r_2=a_0.a_1a_2$, etc. Each $r_i$ is rational. Checking definitions shows that $$ (-a,a) = \bigcup_{i=1}^\infty (-r_i,r_i). $$ But this cannot be open since $a$ is irrational.