A clarification on the definitions of Topological space

Question

I have a question regarding the following statement from the definitions of Topological Space.

“ The intersection of a finite number of sets in T is also in T.”

Here, what is the role of “finite” here? I do understand the proof of the statement but have no idea how the intersection of infinite sets behave.

Edit: The statement seems to suggest that the intersection of infinite number of sets in T might not be in T. Is that even possible?

Also, the statement is from https://mathworld.wolfram.com/TopologicalSpace.html

Thanks

Answer 1

The sets in $T$ are like the "open" sets. So the axiom says that the intersection of finitely many open sets is open. You don't want to extend that to infinite intersections. For example in $\mathbb{R}$ each set $(-\frac{1}{n},\frac{1}{n})$ is open for all $n$. But the intersection of all of these is $\{0\}$, which is not open.