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 $S_{n}(X)$.

Particular cases

Value of $n$	Standard $n$-simplex is homeomorphic to ...	Description of the set of singular simplices
0	one-point space	The set of singular simplices is identified with the underlying set of $X$. For each point $x\in X$, the corresponding singular simplex is the continuous map sending the one-point space to $x$.
1	closed unit interval $[0,1]$	The set of singular simplices is identified with the set of paths in $X$.
2	closed filled-in equilateral triangle; closed unit disk in Euclidean plane	Fill this in later