I am studying the concept of covering spaces in algebraic topology, and I am interested in understanding how to construct the covering space of a lattice. A lattice is a partially ordered set in which every pair of elements has a least upper bound and a greatest lower bound. A covering space of a lattice is a continuous and surjective map $p : X → L$ from a topological space $X$ to a lattice $L$, such that for every element $l ∈ L$, there exists an open neighborhood $U$ of $l$ and a discrete space $F$, such that $p^{−1}(U)$ is the disjoint union of open subsets of $X$, each homeomorphic to $U × F$ via $p$.
I know that there are some general methods to construct covering spaces, such as using the fundamental group or the universal cover, but I am not sure how to apply them to lattices. I also wonder if there are some special properties or examples of covering spaces of lattices that are different from other types of spaces.
Can anyone help me with this question? Any references or hints are welcome. Thank you in advance.
