A significant number of topological spaces are (or can be modeled by) simplicial complexes—spaces that are "glued" out of simplexes, attaching them face to face. Given a subcomplex

of a complex

, one would often like to consider a neighborhood

of

in

that approximates

fairly well. It is not always possible to build such a

out of simplexes of

; For example, if

, the only simplicial neighborhood of

is the whole segment

, which fails to capture the disconnectedness of

.

However, there are such neighborhoods that are simplicial in a double barycentric subdivision

of

(i.e. each simplex is subdivided barycentrically twice). Given any subcomplex

of

, such a neighborhood is given by

in

, where

is defined as the union of all open simplexes with a vertex in

.

The Demonstration illustrates this concept with the 3-simplex as an example of

. Each controller toggles the display of stars (in the double subdivision) of centers of corresponding elements. For example, selecting "vertices" and "edges" gives a neighborhood of a 1-skeleton of

(i.e. of the collection of its 1-simplexes).

Barycentric subdivision is defined inductively.

** **Subdivision of a point is the point itself. To define a subdivision

of a simplex

, take the subdivision

of its boundary

and take

as the collection of simplexes with base the simplex in

and vertex the barycenter of

.