N-skeleton in the context of Simplex

⭐ Core Definition: N-skeleton

In mathematics, particularly in algebraic topology, the $n$ -skeleton of a topological space $X$ presented as a simplicial complex (resp. CW complex) refers to the subspace $X n$ that is the union of the simplices of $X$ (resp. cells of $X$ ) of dimensions $m \leq n .$ In other words, given an inductive definition of a complex, the $n$ -skeleton is obtained by stopping at the $n$ -th step.

These subspaces increase with $n$ . The 0-skeleton is a discrete space, and the 1-skeleton a topological graph. The skeletons of a space are used in obstruction theory, to construct spectral sequences by means of filtrations, and generally to make inductive arguments. They are particularly important when $X$ has infinite dimension, in the sense that the $X n$ do not become constant as $n \to \infty.$

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N-skeleton in the context of Hypercube

In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3); the special case for n = 4 is known as a tesseract. It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in n dimensions is equal to ${\sqrt {n}}$ .

An n-dimensional hypercube is more commonly referred to as an n-cube or sometimes as an n-dimensional cube. The term measure polytope (originally from Elte, 1912) is also used, notably in the work of H. S. M. Coxeter who also labels the hypercubes the γn polytopes.

View the full Wikipedia page for Hypercube

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⭐ Core Definition: N-skeleton

N-skeleton in the context of Hypercube