I didn't want my first question here to be that stupid, but this is something that bothers me very much as it seems simple and yet I get confused when reading and thinking over it. Not clearly understood basics always get me.
Here are some details. I have this problem:
Let $X$ be $\Bbb R$, and let $\Omega$ consist of the empty set and the complements of all finite subsets of $\Bbb R$. Is $\Omega$ a topological structure?
Before that I solved another one:
Let $X$ be $\Bbb R$, and let $\Omega$ consist of the empty set and all infinite subsets of $\Bbb R$. Is $\Omega$ a topological structure?
I proved it by assuming that $[a, b]$ is a finite subset without giving it much thought, so I looked at all of the five other types of intervals.
As I started expanding on this I said, OK, here's the common form of all finite subsets of $\Bbb R$: $[a, b]$, when it struck me: there should be an infinite number of points in this interval (e.g. in $[0, 1]$, the ray 1/n, n -> Infinity), so it must not be finite, which means that I should look only at explicitly defined intervals, e.g., $[a < b < c < d < \cdots]$ where the number of elements is finite. And here I'm lost.
I found this question: Closed Infinite intervals where the different types of infinite intervals were described in the answer and the form $[a, b]$ was not listed.
So common sense tells me the a closed interval is infinite, but I cannot find it written anywhere. (I may have also other faults in my thinking but I'd appreciate if you could help me with the main question.)