Uphill graph topologies and degree-constrained realizations of posets

The uphill topology of a simple undirected graph has open sets closed under paths of nondecreasing vertex degree. Its T0 plateau quotients are exactly the well-founded set-sized posets; the minimum degree-spectrum order type equals ordinal height. Every finite poset of height h has a polynomial-size bipartite realization using the optimal h degree values 3, …, h+2. The bound h+2 on maximum degree is uniformly sharp over height-h posets for h ≥ 2, independently of cover-graph branching. A countable well-founded poset has a countable locally finite realization exactly when each element has finite rank. Consequences include finite simplicial weak homotopy types with degree at most dimension plus three, and #P-complete open-set counting for bipartite graphs with degrees exactly three and four. Compactness, countable compactness and sequential compactness coincide in every subspace and are characterized by minimal plateaus. The four-point crown has a unique smallest bipartite realization up to isomorphism: seven vertices and eleven edges. Under the standard weak patch refinement, degree at most three forces a discrete plateau quotient; a degree-four tree contains a convergent-sequence quotient subspace. Another countable locally finite realization has a metrizable refined quotient without isolated points.