Singular simplex: Difference between revisions

Revision as of 21:03, 9 January 2011

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. The use of in simply indicates a map to the space and does not connote injectivity.

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

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 ==Definition==
-A '''singular simplex''' in a topological space is a [[continuous map]] to it from a [[standard simplex]] (The standard <math>n</math>-simplex is homeomorphic to the <math>n</math>-dimensional [[closed unit disk]]). The map need not be injective and it certainly need not be a homeomorphism.
+A '''singular simplex''' ''in'' a topological space is a [[continuous map]] to it from a [[standard simplex]] (The standard <math>n</math>-simplex is homeomorphic to the <math>n</math>-dimensional [[closed unit disk]]). The map need not be injective and it certainly need not be a homeomorphism. The use of ''in'' simply indicates a map ''to'' the space and does not connote injectivity.
 If the simplex has dimension <math>n</math>, we get what is called a singular <math>n</math>-simplex.
+The set of singular simplices is denoted <math>S_n(X)</math>.
 ==Particular cases==