Bing metrization theorem

The set of Bing - Nagata - Smirnov ( by RH Bing, J. Nagata and Smirnov JM ) is a set from the mathematical branch of topology which characterizes those topological spaces whose topology can be defined by a metric.

Problem

In a first abstraction of the investigated convergence in or it is noted that it is sufficient to have a notion of distance. This leads to the informal notion of metric space. In a further abstraction is only referring to open sets and thus comes to the topological space.

Not every topological space is metrizable: Not for every topological space, there exists a metric, so that the open sets that arise precisely from the topology defined by the distance metric concept of open balls. It is therefore a natural question of which topological spaces are metrizable, where one looks for conditions that are not about structures or properties argue that can not be the basis to define arbitrary topological spaces (such as metrics: even in the case metrizable spaces can not be the metric of the space to be defined). This is the so-called Metrisationsproblem that was open long and was dissolved by the set of Bing - Nagata - Smirnov.

Topological terms

The measures necessary for the characterization of metric spaces topological terms are summarized briefly here. Room classes with purely topological definitions are:

• Hausdorff space: A topological space is Hausdorff space if disjoint to two distinct points, open sets are with and.
• Regular room: A topological space is called regular if there is any closed set and every disjoint open sets with and.

The following terms are purely topological nature, that is, their definitions only use open sets:

• A family of subsets of a topological space is called discrete if there is any point in an open set with and for all but at most one exception.

• A family of subsets of a topological space is called locally finite if there is any point in an open set with and for all but at most finitely many exceptions.

• A family of subsets of a topological space is called - discrete if there is a countable set of discrete systems with. According means - locally finite if there are countably many locally finite systems.

• A family of subsets of a topological space is called a basis of the space if every open and every open set can be written as a union of sets from.

Wording of the sentence

The following theorem of Bing - Nagata - Smirnov solves the Metrisationsproblem:

For a topological space the following are equivalent:

• Is metrizable.
• Is a regular Hausdorff space with a - discrete base.
• Is a regular Hausdorff space with a locally - finite basis.

Comments

Historical Note

The Metrisierbarkeitssatz was found in the early 1950s regardless of Bing, Nagata and Smirnov, the version with the discrete - base comes from Bing, the version with the- locally finite base originates, also independently, by Nagata and Smirnov.

Already in the 1920s were of Urysohn special cases have been proved:

• A normal space with a countable base is homeomorphic to a subset of the Hilbert space and therefore metrizable.
• A compact Hausdorff space is metrizable if and only if it has a countable basis.

Spaces with a countable base

An important implication of the above set of Bing - Nagata - Smirnov:

For topological spaces with a countable basis of the following statements are equivalent:

The implications 1 2 3 4 are comparatively simple. Since a countable basis is, of course, discreet, follows 4 1 from the set of Bing - Nagata - Smirnov.

This set is also known as Metrisierbarkeitssatz of Urysohn.

Generalizations of metric spaces

The set of Bing - Nagata - Smirnov has led to generalizations of the metric space in which the conditions have been weakened to the properties of the base. A family of subsets of a topological space is called degree - preserving if the relationship exists for each part family, and the family is, degree - preserving if it is a countable union of termination -preserving families.

This is called a regular Hausdorff space which has a degree -preserving base, a room. Da- locally finite families are degree - preserving, shows the above set of Bing - Nagata - Smirnov that spaces are generalizations of metric spaces. Further weakening of this kind lead to another room classes.