Singular simplex: Difference between revisions

Latest revision as of 21:04, 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.

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

@@ Line 1: / Line 1: @@
 ==Definition==
-A '''singular simplex''' in a topological space is a continuous map to it from a [[standard simplex]]. 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.
+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 <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==
+{| class="sortable" border="1"
+! Value of <math>n</math> !! Standard <math>n</math>-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 <math>X</math>. For each point <math>x \in X</math>, the corresponding singular simplex is the continuous map sending the one-point space to <math>x</math>.
+|-
+| 1 || [[closed unit interval]] <math>[0,1]</math> || The set of singular simplices is identified with the set of [[path]]s in <math>X</math>.
+|-
+| 2 || closed filled-in equilateral triangle; closed unit disk in Euclidean plane || {{fillin}}
+|}