True or False:Let $X$ be a topological space such that any two nonempty openset in $X$ intersect, then $X$ is compact.

Question

Let $X$ be a topological space such that any two nonempty open sets in $X$ intersect,then $X$ is compact. True /false ?

I don't know how to think about this problem.

Any hints/solution ?

Answer 1

The statement is false. Consider the topology over $\mathbb{N}$ generated by $\{\{1, k\}: k\in\mathbb{N}\}$.