Separating set

A points separating quantity in mathematics is a set of functions on a given space, making it easy to two points in this space differ according to their function values ​​with respect to these functions. The term is used in the general topology and functional analysis.

Definition

Be a lot. A lot of functions with domain ie points separating if for any two elements exists with a function such that.

Use

Be turn a lot and a lot of functions. Now, can the evaluation map

By define ( was in the process of the target amount ). This is injective, if points divisive.

If a topological space and the set of all -valued continuous functions, so is the conclusion of the image of the Stone - Čech compactification of. So is separating points ( ie is complete Hausdorff space ), so providing an identification of the crowd with a subset of the Stone - Čech compactification.

Be more generally any set of functions in topological spaces. The evaluation map is an embedding if and only if the initial topology with respect and contributes points is divisive. This initial topology is also called weak topology with respect, especially in the functional analysis, when a set of linear functionals on a vector space. If the target space of each function into a Hausdorff space, then the weak topology with respect to Hausdorff if and only if is a separating points. Is a set of linear functionals on a vector space, the separation points and thus the Hausdorf fig stem the weak topology can be characterized by the condition that

Applies. In particular, it follows from the Hahn- Banach theorem that the set of all continuous linear functionals on a locally convex Hausdorff space separating points and thus the weak topology is Hausdorff on such a space.

The theorem of Stone - Weierstrass provides that a subalgebra of the algebra of functions on a locally compact Hausdorff space if and only is dense in, if it is separating points and no point always maps to the.