Subsets with Trivial topology

I was thinking about the trivial topology, in which the only open sets are $X$ and $\emptyset$, where $X$ is the entire space we are working in.

My doubt is: are all the subset of $X$ compact and connected?I understand that they are all closed, since the only open sets are $X$ and $\emptyset$. But compact means also bounded, so if for example $X = \mathbb{R}$, how can be, for example, $(1, +\infty)$ bounded?

Also I'm having problems in understanding connectedness. Say we have $[1, 3]\cup [4, 6]$, how is this connected in the trivial topology?