Separating set

转自：http://en.wikipedia.org/wiki/Separating_set

Separating set

          In mathematics aset offunctionsS from a set D to a set C is called a separating set forD or said to separate the points of D if for any two distinct elementsx and y of D, there exists a function f in S so that f(x) ≠ f(y).

Examples

• The singleton set consisting of theidentity function onR separates the points of R.

• If X is a normal topological space, then Urysohn's lemma states that the set C(X) ofcontinuous functions onX with real (orcomplex) values separates points onX.