Abstraction useful in the construction and triangulation of topological spaces
In mathematics, a Δ-set, often called a Δ-complex or 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 sophisticated as a simplicial set. Simplicial sets have additional structure, so that every simplicial set is also a semi-simplicial set.
Three Delta-set structures on the circle, with the third also being a simplicial complex.
As an example, suppose we want to triangulate the 1-dimensional circle . 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 as the interval [0,1] with the two endpoints identified, we can define a triangulation with a single vertex 0, and a single edge looping between 0 and 0.
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}}...
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{\begin{aligned}A\,\Delta \,B&=B\,\Delta \,A,\\(A\,\Delta \,B)\,\Delta \,C&=A\,\Delta \,(B\,\Delta \,C).\end{aligned}}} The empty set is neutral, and every set is its...
{\displaystyle f} . The set of points at which a function between metric spaces is continuous is a G δ {\displaystyle G_{\delta }} set – this follows from...
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images in that simplicial set of the morphisms δ n , 0 , … , δ n , n : [ n − 1 ] → [ n ] {\displaystyle \delta ^{n,0},\dotsc ,\delta ^{n,n}\colon [n-1]\to...
(contains a dense open set), a comeagre set need not be a G δ {\displaystyle G_{\delta }} set (countable intersection of open sets), but contains a dense...
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