I am currently starting to study topology through the Topology without Tears book, and I came across the Example 1.1.5 and I am not sure if I understand why is the result like it is. The example says as it follows:
Let $\mathbb{N}$ be the set of all natural numbers and let $\tau_4$ consist of $\mathbb{N}, \emptyset$, and all finite substes of $\mathbb{N}$. Then $\tau_4$ is not a topology on $\mathbb{N}$ because the infinite union $\{2\} \cup \{3\} \cup ... \cup \{n\} \cup ... = \{2,3,...,n, ...\}$ is not in $\tau_4$.
I guess that the result comes from the fact that only finite subsets are considered, but I don't see why an infinite union of finite set wouldn't generate an infinite set.
To put an example of my thinking about this example. Since all the finite subsets in $\mathbb{N}$ is in $\tau_4$ and any combination $\{k, k+1\}$ is a finite subset of $\mathbb{N}$, given that $k \in \mathbb{N}$, and any combination of them is also a a subset of $\mathbb{N}$, why is it not in $\tau_4$?
P.S. I think I start to get it, is it because we are explicitly indicating that only the finite sets are part of $\tau_4$?