What does the zeroth Betti number count?

You interpret a complete graph as a simplex by keeping its vertices and edges, then filling in higher-dimensional cells such as 2-dimensional triangular faces, 3-dimensional solids, and 4-dimensional spaces. A complete graph on k vertices forms $a (k-1)$-simplex.

Conditions: The starting structure is a complete graph on k vertices.; Higher-dimensional cells are filled successively between existing lower-dimensional ones.

Betti numbers are a sequence of topological invariants that indicate how many holes of each dimension an object has. For example, the first Betti number counts one-dimensional circular holes, and the second counts two-dimensional holes.

Conditions: Applies to topological spaces, especially simplicial complexes.; Each entry in the sequence corresponds to a different hole dimension.

Individual simplices in a simplicial complex can be completely separate or stuck together by sharing a face, an edge, or a vertex. Crucially, the intersection must happen along an entire face, never just a partial face.

Conditions: The object is a simplicial complex.; Intersections occur between any two simplices within the complex.

The first Betti number counts the number of one-dimensional, or circular, holes. A torus has two such independent circular directions: one along the longitude and one along the meridian, resulting in a first Betti number of 2.

Conditions: The object is a torus.; The first Betti number counts 1-dimensional holes.

A k-simplex is a k-dimensional object formed by taking $k+1$ vertices and filling in all the space between them, which is known as the convex hull of those vertices.

Conditions: The dimension index k is a non-negative integer.; The construction involves $k+1$ vertices.