separating set

{{About|separating sets for functions|use in graph theory|connectivity (graph theory)}}

In mathematics, a set $S$ of functions with domain $D$ is called a {{em|separating set for $D$ }} and is said to {{em|separate the points of $D$ }} (or just {{em|to separate points}}) if for any two distinct elements $x$ and $y$ of $D,$ there exists a function $f \in S$ such that $f(x) \neq f(y).$ {{citation|last=Carothers|first=N. L.|title=Real Analysis|publisher=Cambridge University Press|year=2000|isbn=9781139643160|pages=201–204|url=https://books.google.com/books?id=eokhAwAAQBAJ&pg=PT201}}.

Separating sets can be used to formulate a version of the Stone–Weierstrass theorem for real-valued functions on a compact Hausdorff space $X,$ with the topology of uniform convergence. It states that any subalgebra of this space of functions is dense if and only if it separates points. This is the version of the theorem originally proved by Marshall H. Stone.

Examples

The singleton set consisting of the identity function on $\Reals$ separates the points of $\Reals.$
If $X$ is a T1 normal topological space, then Urysohn's lemma states that the set $C(X)$ of continuous functions on $X$ with real (or complex) values separates points on $X.$
If $X$ is a locally convex Hausdorff topological vector space over $\Reals$ or $\Complex,$ then the Hahn–Banach separation theorem implies that continuous linear functionals on $X$ separate points.

References

Category:Set theory

separating set

Examples

See also

References