Abstract simplicial complexes have had quite a renaissance recently. Simplicial complexes were originally used to describe pre-existing topological spaces such as manifolds, as in the question. Now they are also the key tool in constructing topological spaces from "real life" situations. The nerve of a covering of a set is a simplicial complex - if the set is a topological space and the subspaces are contractible (plus some technical conditions) the nerve has the same homotopy type as the space. The http://en.wikipedia.org/wiki/Rips_complex Wikipedia article is a good introduction.

Abstract simplicial complexes have had quite a renaissance recently. Simplicial complexes were originally used to describe pre-existing topological spaces such as manifolds, as in the question. But now they are the key tool in constructing discrete models for topological spaces. The nerve of a covering of a set is a simplicial complex - if the set is a topological space and the subspaces are contractible (plus some technical conditions) the nerve has the same homotopy type as the space. The Wikipedia article http://en.wikipedia.org/wiki/Rips_complex is a good introduction.

Abstract simplicial complexes have had quite a renaissance recently. Simplicial complexes were originally used to describe pre-existing topological spaces such as manifolds, as in the question. But now they are the key tool in constructing discrete models for topological spaces. The nerve of a covering of a set is a simplicial complex - if the set is a topological space and the subspaces are contractible (plus some technical conditions) the nerve has the same homotopy type as the space. The http://en.wikipedia.org/wiki/Rips_complex Wikipedia article is a good introduction.

Abstract simplicial complexes have had quite a renaissance recently. Simplicial complexes were originally used to describe pre-existing topological spaces such as manifolds, as in the question. Now they are also the key tool in constructing topological spaces from "real life" situations. The nerve of a covering of a set is a simplicial complex - if the set is a topological space and the subspaces are contractible (plus some technical conditions) the nerve has the same homotopy type as the space. The http://en.wikipedia.org/wiki/Rips_complex Wikipedia article is a good introduction.