It is a well known fact that in many contexts, every bounded monotone sequence converges.
I wonder what is the most general context where this happens. That is,
Let $X$ be a set equipped with a topology and with a partial order. Let $(x_\lambda)_{\lambda\in\Lambda}$ be a monotone net (i.e. $x_\lambda\le x_\mu$ if $\lambda\le\mu$), bounded in the sense that for all $\lambda$ we have that $x_\lambda\le y$ for some fixed $y\in X$. Under which additional conditions can we conclude that $(x_\lambda)$ converges?
One such condition is for example that $X$ is compact. Obviously this is not the most general case. Is there an established theory about these facts?