In my textbook I find:
Definition of a topology
$\tau$ as a collection of subsets of $X$ is a topology on $X$ if:
- $\varnothing, X \in \tau$
- If $\mathscr{F}\subseteq \tau$ then $\bigcup_{U\in \mathscr{F}} U \in \tau$
- $U,V\in \tau$ then $U\cap V\in \tau$
While working through topology, I started to get confused with the concept of a topology. Two rather basic (even trivial?) questions arose?
First question
On wikipedia I read:
- Any intersection of finitely many elements of $\tau$ is an element of $\tau$
But isn't that a different definition from the one in my textbook?
The definition my textbook keeps the possibility of a infinite, but countable intersection? Which one is valid?
Second question
Why couldn't 2. be written as $U, V\in \tau$ then $U\cup V \in \tau$.
I guess it has to do with the countability of sets, but could some give an example of an uncountable union of sets? Would this mean something like $\tau_\text{discrete}$ on $\mathbb{R}$.
Best Answer
Q1. If the intersection of any two open sets is open, then a straightforward induction tells you that the intersection of any finite number of open sets is open. (And if any intersection of a finite number of open sets is open then taking $n = 2$ we get that the intersection of any two open sets is open). This shows that they are equivalent. Neither of these tells you anything about infinite intersections of open sets. (You say "The definition my textbook keeps the possibility of a infinite, but countable intersection", but it doesn't do any such thing - why did you think it did? The only thing it tells you is that the intersection of any two open sets is open.)
For an example, consider $U_n = (-1/n, 1/n) \subset \mathbb{R}$. This is a countable family of open sets whose intersection is $\{0\}$, which is not open.
Q2. Again, the union of any two open sets being open tells you nothing about infinite unions of open sets, including countable unions.