Vipul: Created page with '==Definition== Suppose $X$ is a topological space and $n$ is a natural number. The '''configuration space of ordered points''' $F_n(X)$ ...'

2010-12-30T18:52:10Z

Created page with '==Definition== Suppose <math>X</math> is a topological space and <math>n</math> is a natural number. The '''configuration space of ordered points''' <math>F_n(X)</math> ...'

New page

==Definition==

Suppose <math>X</math> is a [[topological space]] and <math>n</math> is a [[natural number]]. The '''configuration space of ordered points''' <math>F_n(X)</math> is defined as follows:

* As a set, it is the set of ordered <math>n</math>-tuples of ''distinct'' points of <math>X</math>.
* The topology is given as follows. First, note that <math>F_n(X)</math> is the subset of <math>X^n</math> comprising all <math>n</math>-tuples where no two coordinates are equal, i.e., it is the complement of the ''fat diagonal''. We first give the [[defining ingredient::product topology]] on <math>X^n</math> and then give <math>F_n(X)</math> the [[defining ingredient::subspace topology]] arising from that.

Note that the term ''configuration space'' is typically used for the [[configuration space of unordered points]] <math>C_n(X)</math>, which is the [[quotient topology|quotient]] of this space under the equivalence relation induced by the action of the symmetric group <math>S_n</math> by coordinate permutation.

Configuration space of ordered points - Revision history

Vipul: Created page with '==Definition== Suppose X is a topological space and n is a natural number. The '''configuration space of ordered points''' F_n(X) ...'

Vipul: Created page with '==Definition== Suppose $X$ is a topological space and $n$ is a natural number. The '''configuration space of ordered points''' $F_n(X)$ ...'