Singular simplex

Definition
A singular simplex in a topological space is a continuous map to it from a standard simplex (The standard $$n$$-simplex is homeomorphic to the $$n$$-dimensional closed unit disk). The map need not be injective and it certainly need not be a homeomorphism.

Note that the use of in (as in, singular simplex in a topological space) simply indicates a map to the space and does not connote injectivity. In fact, the adjective singular captures the fact that the map may be far from injective.

If the simplex has dimension $$n$$, we get what is called a singular $$n$$-simplex.

The set of singular simplices is denoted $$S_n(X)$$.