Let $X=\{2,3,4,5,6,7,8, \ldots\}$ and for each $n \in X$ let

$$U_{n}=\{d \in X \mid d \text { divides } n\} .$$

Prove that the set of unions of the sets $U_{n}$ forms a topology on $X$.

Prove or disprove each of the following:

(i) $X$ is Hausdorff;

(ii) $X$ is compact.

If $Y$ and $Z$ are topological spaces, $Y$ is the union of closed subspaces $A$ and $B$, and $f: Y \rightarrow Z$ is a function such that both $\left.f\right|_{A}: A \rightarrow Z$ and $\left.f\right|_{B}: B \rightarrow Z$ are continuous, show that $f$ is continuous. Hence show that $X$ is path-connected.