Pseudometrics

The pseudometrics , also half-metric or span, is a mathematical concept of distance that weakens the more specific concept of metric . By a pseudo metric, more frequently, through a system of pseudo-metrics on a set is in the mathematical branch topology a uniform structure introduced on this set. The reverse applies: Every uniform structure can be induced by a system of spans. For uniform spaces that have a countable fundamental system, the following even applies: Their uniform structure can be induced by a single span.

Definition and characteristics

Be any set. A picture is pseudometric , half metric or margin if for any element , and of they meet the following conditions: ${\displaystyle X}$ ${\displaystyle d\colon X\times X\to \mathbb {R} }$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$ ${\displaystyle X}$

1. ${\displaystyle d(x,x)=0}$ ("The span between a point and the point itself is 0."),
2. ${\displaystyle d(x,y)=d(y,x)}$ (Symmetry: "The span between two points does not depend on the order.") And
3. ${\displaystyle d(x,y)\leq d(x,z)+d(z,y)}$( Triangle inequality : "The range is shortest on the direct route.").

It follows from the conditions that no range can be negative because it holds . ${\displaystyle d(x,y)={\tfrac {1}{2}}(d(x,y)+d(y,x))\geq {\tfrac {1}{2}}d(x,x)=0}$

The only difference to the definition of a metric is that the definiteness condition is missing: There can be elements that are different, but between which the range is still 0:

${\displaystyle x\neq y\land d(x,y)=0}$.

If there are such elements in , then one also says that the range is a real pseudometric . If it doesn't exist, then the range is actually a metric. ${\displaystyle X}$ ${\displaystyle d}$ ${\displaystyle d}$

Some terms that are defined in metric spaces with the help of a metric can also be defined literally with ranges, for example the restricted subsets of , restricted mappings according to , equally restricted families of mappings according to (see: Boundedness ). ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X}$

As an example, only the concept of uniform continuity is given here: Let and quantities with the ranges or . Then a mapping is called uniformly continuous if there is a positive for every positive such that ${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$ ${\displaystyle d_{1}}$ ${\displaystyle d_{2}}$ ${\displaystyle f\colon X_{1}\to X_{2}}$ ${\displaystyle \varepsilon }$ ${\displaystyle \delta }$

${\displaystyle \forall x,y\in X_{1}\colon d_{1}(x,y)<\delta \Rightarrow d_{2}(f(x),f(y))<\varepsilon }$ applies.

Tension and uniform structures

Definition of a uniform structure by tensioning

Be a lot with the range . Then the system forms all relations of form ${\displaystyle X}$ ${\displaystyle d}$ ${\displaystyle F}$

${\displaystyle d^{-1}([0;a[)=\{(x,y)\in X\times X\mid d(x,y)<a\}}$,

${\displaystyle F:=\{d^{-1}({[0;a[})\mid a\in \mathbb {R} _{+}\}}$

a fundamental system of a uniform structure . This structure is called defined by the span . ${\displaystyle X}$ ${\displaystyle d}$

If a family of ranges is given, then the supremum of the uniform structures defined by , i.e. the coarsest uniform structure in which all are uniformly continuous, is called the uniform structure defined by the family. ${\displaystyle X}$ ${\displaystyle (d_{i})_{i\in I}}$ ${\displaystyle d_{i}}$ ${\displaystyle d_{i}}$

Definition of a span through a uniform structure

The following construction is a proof sketch for the statement from the introduction: "Every uniform structure that has a countable fundamental system can be defined by a range". For this purpose, now be such a uniform space and a countable fundamental system. ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle (N_{k})_{k\in \mathbb {N} }}$

Now the neighborhoods are first symmetrized and tailored, we replace them with symmetrical neighborhoods with the properties and ( here we mean the concatenation in the relational sense). The auxiliary function ${\displaystyle (N_{k})_{k\in \mathbb {N} }}$ ${\displaystyle (S_{k})_{k\in \mathbb {N} }}$ ${\displaystyle S_{0}\subseteq N_{0}}$ ${\displaystyle S_{k+1}^{3}\subseteq S_{k}\cap N_{k}}$ ${\displaystyle S^{3}}$

${\displaystyle g(x,y):={\begin{cases}1,\;{\mbox{falls}}\;(x,y)\not \in S_{0},\\\inf\{2^{-k-1}|(x,y)\in S_{k}\}\;{\rm {sonst}}\end{cases}}}$

is symmetrical and disappears on the diagonal. In order to satisfy the triangle inequality, the shortest path must now be found. Let us be the set of all finite sequences of points with a starting point and an end point . The range you are looking for is then ${\displaystyle M}$ ${\displaystyle X}$ ${\displaystyle x}$ ${\displaystyle y}$

${\displaystyle d(x,y):=\inf \left\{\sum _{j=0}^{n-1}g(z_{j},z_{j+1})\mid (z_{j})_{j=0,\dotsc ,n}\in M\right\}}$.

The range is of course not clearly determined by the uniform structure . However , the structure defined by as described above then corresponds to the original uniform structure. ${\displaystyle d}$ ${\displaystyle X}$ ${\displaystyle d}$

Examples and construction of spans

• Every metric is a range, so every example of a metric space provides an example of a range. ${\displaystyle (M,d)}$
• The zero span creates the indiscreet topology on every set , which turns out to be a uniform structure. ${\displaystyle d(x,y)=0}$ ${\displaystyle X}$
• On the set of positive fractions , the amount metric and the discrete metric each have a range (even a metric). Both spans induce the same, namely the discrete topology, and are therefore topologically equivalent. However, they define different uniform structures . ${\displaystyle B:=\{{\tfrac {1}{z}}\mid z\in \mathbb {Z} _{+}\}}$ ${\displaystyle B}$ ${\displaystyle B}$
• Finite number of clamping on to a new range will be added. ${\displaystyle d_{i};\;1\leq i\leq n}$ ${\displaystyle X}$ ${\displaystyle d(x,y)=d_{1}(x,y)+d_{2}(x,y)+\dotsb +d_{n}(x,y)}$
• Countably many spans on can add up to the span ${\displaystyle d_{i};\;i\in \mathbb {N} }$ ${\displaystyle X}$

${\displaystyle d(x,y)=\sum \limits _{i=0}^{\infty }2^{-i}{\frac {d_{i}(x,y)}{1+d_{i}(x,y)}}}$

be put together.

literature

• Boto von Querenburg : Set theoretical topology. 3rd revised and expanded edition. Springer-Verlag, Berlin et al. 2001, ISBN 3-540-67790-9 ( Springer textbook ).