I have another problem, but in this one I have no idea how to start.
Let be $(X,\leq)$ a POSET with a first element and gifted with the topology $\{ (a,\rightarrow) : a \in X \}$ (principal order filters). Show that $X$ is compact.
I really have no idea how to start, I tried using the definition of compact, but I don't know how.