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In mathematics, a Δ-set S, often called a semi-simplicial set, is a combinatorial object that is useful in the construction and triangulation of topological spaces, and also in the computation of related algebraic invariants of such spaces. A Δ-set is somewhat more general than a simplicial complex, yet not quite as general as a simplicial set.
As an example, suppose we want to triangulate the 1-dimensional circle S 1 {\displaystyle S^{1}}. To do so with a simplicial complex, we need at least three vertices, and edges connecting them. But delta-sets allow for a simpler triangulation: thinking of S 1 {\displaystyle S^{1}} as the interval with the two endpoints identified, we can define a triangulation with a single vertex 0, and a single edge looping between 0 and 0.
Abstraction useful in the construction and triangulation of topological spaces