Compact subspaces of the Poset On page 172, James Munkres' textbook Topology(2ed), there is a theorem about compact subspaces of the real line:

Let $X$ be a simply-ordered set having the least upper bound property. In the order topology, each closed interval in $X$ is compact.

My question is whether there is a generalized theorem about a Poset(or a lattice, complete lattice, maybe). Is there some elegant way to define a topology on Poset?

Many topologies have been defined on partial orders and lattices of various types. One of the most important is the Scott topology. Let $\langle P,\preceq\rangle$ be a partial order. A set $A\subseteq P$ is an upper set if ${\uparrow\!\!x}\subseteq A$ whenever $x\in A$, where ${\uparrow\!\!x}=\{y\in P:x\preceq y\}$. A set $U\subseteq P$ is open in the Scott topology iff $U$ is an upper set with the following property:

if $D\subseteq P$ is a directed set in $P$, and $\bigvee D\in U$, then $D\cap U\ne\varnothing$. (In this case $U$ is said to be inaccessible by directed joins.)

The upper topology is the topology that has $\{P\,\setminus\!\downarrow\!\!x:x\in P\}$ as a subbase. The lower topology is generated by the subbase $\{P\setminus{\uparrow\!\!x}:x\in P\}$.
The Lawson topology is the join (coarsest common refinement) of the Scott and lower topologies.
A number of interval topologies have also been defined, the first ones by Frink and by Birkhoff; this paper deals with a number of such topologies.
These terms should at least give you a start for further search if you’re interested.
