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The dimension of a simplex is the number of vertices it contains minus one.

A topological space constructed from an abstract simplicial complex via a mapping that places vertices in a Euclidean space and linearly interpolates the simplices.

The set of all simplices in the simplicial complex that contain a given simplex as a face.

A collection of finite sets closed under the operation of taking subsets, where each set represents a simplex.

The set of simplices that do not intersect a given simplex but are connected to it by higher dimensional simplices in the complex.

The faces of a simplex are the simplices determined by a proper subset of its vertex set.

Groups associated with a topological space that are invariant under homotopy equivalence, useful for distinguishing non-homeomorphic spaces.

A homology theory for simplicial complexes, providing algebraic invariants (homology groups) that classify spaces in terms of holes of different dimensions.

An abstract simplicial complex constructed from an open cover of a topological space, where each set in the cover corresponds to a vertex, and simplices correspond to non-empty intersections.

The basic building blocks of a simplicial complex, consisting of vertices, edges, faces, etc. identified together in a geometric space.

A process that subdivides each simplex in a simplicial complex into smaller simplices in such a way that the smaller simplices fit together to form the original simplex.