Finite Models of Topological Varieties of Two Dimensions by Means of Directed Graphs

This article develops a finite directed-graph framework for representing selected topological spaces through an associated abstract simplicial complex. We distinguish arbitrary reflexive relations from preorders: reflexivity alone does not define a finite topology, and antisymmetry and transitivity are not assumed unless explicitly stated. For a finite looped digraph D, we define K(D) by directed chains and verify that the resulting collection is closed under taking faces. Using this construction, we describe graph models of the circle, cylinder, torus, cone, and sphere, and check their simplex counts and topological invariants. We also formulate a dominated-vertex reduction criterion directly in terms of K(D); every permitted deletion preserves the homotopy type of its geometric realization, and the finite reduction algorithm terminates. We also give the standard spanning-tree presentation for π₁(|K(D)|). The displayed constructions are not claimed to be minimal unless a separate lower-bound argument is provided. The principal computations yield π₁(S¹) ≅ ℤ, π₁(S²) = 1, and π₁(S¹ × S¹) ≅ ℤ².