For some context I have some partial understanding of lattices and an intermediate understanding of topology.
I at some point in the past week started thinking about a funny way to view a topology on a space as constituted by a lattice of open sets. In the sense that a union of two sets corresponds to a join and a intersection to a meet. Then in some sense the axioms of a topology on a space can be translated to a set of axioms on our lattice For example since the space itself and the null set are both open then there exist maximal and minimal elements. Since arbitrary unions of opens are opens then we can take joins of arbitrary number of elements, but since only finite intersections of opens are (gaurenteed to be) open then we can only take meets for finite sets of elements. I see these ideas more or less reflected in the wikipedia page for the stone duality https://en.wikipedia.org/wiki/Stone_duality#The_adjunction_of_Top_and_Loc.
However most resources on this connection are fairly advanced often bringing in a good chunk of category theory. My background in category theory is very limited.
While I understand that a formal treatment of this duality inevitably will involve some construction of some pair of functors between topological spaces and lattices, I was wondering if there was a resource that might give a softer and more intuitive introduction to this area. Or is it simply time to bite the category theory bullet
Furthermore I am interested if there is any work in applying topological notions to the appropriate types of lattices. For example it seems that we can extend notions of homotopy equivalences between spaces to homotopy equivalences between latices. Similarly we can extend compactness quite easily. Furthermore one could also construct algebraic invariants like the homology or fundamental groups of a lattice.
This doesn't seem to hold for all the usual topological concepts since it seems by switching to lattice one loses the notion of points and thus cannot straightforwardly talk about Hausdorfness, limits etc. Thus it seems one would recover a slightly altered version of the usual topological notions for suitable lattices.
I can find bits and pieces of these ideas scattered across several papers and articles and blogs. Is there any centralized resource focused on this type of stuff?