# Connectedness and Compactness of $\mathbb{N}$ with basic sets of the form $\mathbb{N}_{\ge n}$?

We have $$X = \mathbb N$$. The topology is generated by the basic sets $$A_n = \{n,n+1,n+2,...\} , n\in \mathbb N$$. Is $$X$$ connected and compact? My guess is that if we take any two open sets, one of them has to be contained in the other and hence, $$X$$ is both connected and compact.

• Your conclusion is right. Jan 24 at 9:28
• Hint: If $S$ is any nonempty set of natural numbers and $N$ is the smallest element of $S$ then $\bigcup_{n\in S}A_n=A_N$. Jan 24 at 10:23

Formally, the closed sets are of the form $$\{1,2\dots,n\}$$. So the only clopen sets are empty and whole set itself, so connected.
Compactness follows from the fact that $$A_1=\Bbb N$$ is a member of every open cover of $$\Bbb N$$ (as the only open set that contains $$1$$).