Simplicial complex

A simplicial complex is a collection of simplices in which every face of a simplex is included and any two simplices intersect in a shared face or not at all. It encodes a space through combinatorial data.

Simplex: A simplex is the convex hull of a finite set of affinely independent points. Simplices are the pieces from which the complex is assembled.
Simplicial chain complex: A sequence of abelian groups generated by oriented simplices, linked by boundary maps whose successive composition is zero. It converts the complex's simplices and faces into a sequence suitable for homology.
Persistent homology: A method in topological data analysis that tracks homology as a parameter changes. It studies how topological features persist across a nested sequence of complexes.
Cell complex: A cell complex is a space assembled from cells attached along their boundaries. Unlike a simplicial complex, its cells need not be simplices or meet in a single face.
Face (geometry): A face of a polytope or simplex is the intersection with a supporting hyperplane, including the polytope itself. Closure under faces is a defining requirement for every simplicial complex.
Boundary operator: A linear map that sends a chain to its oriented boundary. It maps each simplex to a signed sum of its codimension-one faces.
Vietoris–Rips complex: A simplicial complex formed from a metric space by making a finite set a simplex when all its pairwise distances meet a threshold. It turns point-cloud distances into a scale-dependent complex for persistent homology.
CW complex: A CW complex is a topological space built by attaching open cells in stages under closure-finiteness and weak-topology conditions. CW complexes allow more general cells while preserving a useful cellular homology theory.
Vertex: A vertex is a zero-dimensional simplex, represented by a single point. Vertices are the basic elements from which every simplex is formed.
Orientation (mathematics): An orientation is a consistent choice of direction or handedness for a geometric object or basis. Oriented simplices give boundary terms their signs.

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