Initial segment topology

Question

I want to prove that $(\mathbb{N},\tau)$ is a topological space, where $\tau=\{ \mathbb N,\emptyset\}\cup\{\{1,\dots,n\}:n\in \mathbb{N}\}$.

I need a hint to prove that an arbitrary union of elements of $\tau$ is in $\tau$.

Answer 1

Hint: either the union is infinite or it has a largest element.

Answer 2

Your hint is: consider the maximum number in the family over which you are taking the union.

Answer 3

Let $\{U_i: i\in I\}$ be an indexed family of elements of $\tau$. Then $\bigcup_{i\in I}{U_i}=\mathbb N$, or $\bigcup_{i\in I}{U_i}=\emptyset$, or $\bigcup_{i\in I}{U_i}=U_k$ for some $k\in I$ such that $\max{U_k}\ge \max{U_j}$ for all $j\neq k$. In all cases, the union is in $\tau$.

Is this correct?