Pseudometric space

The pseudo- metric, metric or even half- span is a mathematical notion of distance that attenuates specific concept of the metric. By a pseudo- metric, more often, by a system of pseudo- metrics on a set a uniform structure on this set is introduced in the mathematical branch topology. Conversely, each uniform structure can be induced by a system of tensioning. For uniform spaces that have a countable fundamental system, even applies: your uniform structure can be induced by a single span.

Definition and properties

Be an arbitrary set. A mapping is called pseudo- metric, metric or semi- span, must be satisfied for arbitrary elements, and of the following conditions:

The only difference to the definition of a metric: it lacks the Definitheitsbedingung. There may be elements that are different, but between which the margin is still 0:

• .

Is there such elements, then we can also say the span is a real pseudo- metric. Is there not, then the span is even a metric.

Follows from the conditions that there is no margin can be negative, as it is.

Some terms that are defined in metric spaces with a metric, can be literally define the same also with tension, for example, the bounded subsets of, limited to pictures, uniformly bounded families of mappings according to (see: boundedness ).

As an example, only the concept of uniform continuity executed: Be and quantities with the tensioning or. Then say a picture is uniformly continuous if, for every positive a positive, so that

Margins and uniform structures

Definition of a uniform structure by tensioning

Let X be a set with the span, d then the system F is of all relations of the form

A fundamental system of a uniform structure on X. This structure is called the span of d defined.

Is given on X is a family of margins, then that means the supremum defined by di -uniform structures, ie the coarsest uniform structure in which all di are uniformly continuous, defined by the family of uniform structure.

Definition of a span by a uniform structure

The following construction is a proof sketch for the statement made in the introduction: "Every uniform structure on X, which has a countable fundamental system, can be defined by a margin '. To this end let X now such a uniform space and a countable fundamental system.

Now the neighborhoods are first symmetrized and tailored, we will replace with symmetric neighborhoods with the properties, and (with the concatenation is here meant in the sense ratio ). The auxiliary function

Is symmetrical and disappears on the diagonal. To satisfy the triangle inequality, the shortest path must be found now. For this purpose let M be the set of all finite sequences of points of X with initial point x and endpoint y. The required margin is then

The range d is of course not uniquely determined by the uniform structure on X. The, as defined above by d structure agrees but then the same as the original uniform structure.

Examples and construction of spans

• Each metric is a margin of each example of a metric space (M, d) therefore provides an example of a margin.
• The zero span d ( x, y) = 0 generates the indiscrete topology on any set X, which thus proves to be a uniform structure.
• On the set of positive unit fractions are defined by the metric amount and by the discrete metric depending on a range given ( even a metric ). Tensioning both induce the same, namely the discrete topology, to B, so are topologically equivalent. However, they define different uniform structures on B.
• A finite number of spans to X can be added to a new range.
• Countable number of spans on X can span