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> Let $\mathbb N$ be the set of all positive integers and $\mathcal T\_2$ consist of $\mathbb N$, $\emptyset$ and every set $\{ n, n+1, \dots \}$ for any positive integer $n$.
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> Prove that $\mathcal T\_2$ is a topology on $\mathbb N$.
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Clearly $\mathbb N$ and $\emptyset \in \mathcal T\_2$.

Let $A\_i, A\_j \in \mathcal T\_2$ and define $A = A\_i \cap A\_j$. We must show that $A \in \mathcal T\_2$. If $A\_i = A\_j$, then clearly $A \in \mathcal T\_2$. If $A\_i \neq A\_j$, but $A\_i \subset A\_j$, then $A = A\_i \in \mathcal T\_2$. Similarly, if $A\_j \subset A\_i$, then $A = A\_j \in \mathcal T\_2$. Also note that for any $A\_i \in \mathcal T\_2$, we have $A\_i \cap \mathbb N = A\_i \in \mathcal T\_2$ and $A\_i \cap \emptyset = \emptyset \in \mathcal T\_2$.

I now still need to show that $\mathcal T\_2$ is closed under (finite or infinite) union. I am not sure how to do this in this example?