This article is not about the topological skeleton

Topological skeleton

In shape analysis, skeleton of a shape is a thin version of that shape that is equidistant to its boundaries. The skeleton usually emphasizes geometrical and topological properties of the shape, such as its connectivity, topology, length, direction, and width...

 concept of computer graphics

Computer graphics

Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....

In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, particularly in algebraic topology

Algebraic topology

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...

, the n-skeleton of a topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

 X presented as a simplicial complex

Simplicial complex

In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...

 (resp. CW complex

CW complex

In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...

) refers to the subspace X_n that is the union of the simplices of X (resp. cells of X) of dimensions m ≤ 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

Discrete space

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...

, and the 1-skeleton a topological graph. The skeletons of a space are used in obstruction theory

Obstruction theory

In mathematics, obstruction theory is a name given to two different mathematical theories, both of which yield cohomological invariants.-In homotopy theory:...

, to construct spectral sequence

Spectral sequence

In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations...

s by means of filtration

Filtration

Filtration is commonly the mechanical or physical operation which is used for the separation of solids from fluids by interposing a medium through which only the fluid can pass...

s, and generally to make inductive argument

Mathematical induction

Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...

s. They are particularly important when X has infinite dimension, in the sense that the X_n do not become constant as n → ∞.

In geometry

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a k-skeleton of n-polytope

Polytope

In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...

P (functionally represented as skel_k(P)) consists of all i-polytope elements of dimension up to k.

For example:

skel₀(cube) = 8 vertices

skel₁(cube) = 8 vertices, 12 edges

skel₂(cube) = 8 vertices, 12 edges, 6 square faces