Example. A triangle boundary and a filled triangle [fdip-004N]

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Take the simple undirected triangle on vertices \(\{0,1,2\}\). Give its three edges any orientations, then forget them here; any parallel copies would also have to be merged before this simple undirected comparison. The one-dimensional realization retains the three edges without a face. In the nonempty-face convention of [maypishevar2025finitespaces, Definition 6.1.1, p. 61], its boundary complex and the filled complex have faces \[\begin {aligned} K_{\partial } &=\{\{0\},\{1\},\{2\},\{0,1\},\{1,2\},\{0,2\}\},\\ K_{\Delta }&=K_{\partial }\cup \{\{0,1,2\}\}. \end {aligned}\] The added face consists of three pairwise adjacent vertices: it is the filled clique triangle. This specifies these two complexes, without declaring a general clique-construction functor.

Choose three noncollinear points \(v_0,v_1,v_2\) in the plane. Realize each listed face by the convex hull of its vertices, as in [maypishevar2025finitespaces, Definitions 6.2.1 and 6.2.6, pp. 62-63]. Then \(|K_{\partial }|\) is the perimeter and \(|K_{\Delta }|\) is the closed triangular region. Traversing the three line segments identifies the perimeter with \([0,3]/(0\sim 3)\): the induced map is a continuous bijection from a compact space to a Hausdorff space, hence a homeomorphism. The map \(t\mapsto \exp (2\pi i t/3)\) gives the same identification with \(S^1\).

The filled triangle contracts to its barycenter \(b=(v_0+v_1+v_2)/3\) by \[H(x,t)=(1-t)x+tb,\qquad 0\leq t\leq 1.\] Convexity keeps \(H(x,t)\) in the triangle, and the formula is continuous, with \(H(x,0)=x\) and \(H(x,1)=b\). The two constructions have the same vertices and edges, but only the second supplies a two-dimensional face. Incidence alone therefore does not choose between keeping the perimeter and filling it. Each specified realization is nevertheless a well-defined construction.

These comparisons are elementary deductions from the cited realization definitions. Mathlib's AbstractSimplicialComplex uses nonempty faces and includes each vertex; its Convex.contractibleSpace supplies contractibility of a nonempty convex subset of a real topological vector space. These support the conventions and the contraction step, not the complete incidence-to-triangle comparison.