Our authority is Garrett Birkhoff in his book, Lattice Theory, pp. 39-47.

"It is well known that the topology of the real continuum can be defined in terms of order; this can be generalized to partially ordered sets. The generalization to chains came first historically ...."

Birkhoff then defines chains, open intervals, closed intervals, neighborhoods, in order to connect with "Hausdorff topology":

"THEOREM 6. Any chain is a normal Hausdorff space under the intrinsic topology -- and the latter is invariant under automorphisms and dual automorphisms."

Birkhoff then devotes pages to developing the above thesis about topology and order.

Since geometry is a topology with a metric, and since Rank in posets is a metric, it follows that geometry can be subsumed under ordinology.