Prametric space

In topology, a prametric space generalizes the concept of a metric space by not requiring the conditions of symmetry, indiscernability and the triangle inequality. Prametric spaces occur naturally as maps between metric spaces.

Definition

A prametric space $(M,mathrm\{d\})$ is a set $M$ together with a function $mathrm\{d\}:M\; imes\; M\; omathbb\{R\}$ (called a prametric) which satisfies the following conditions:
#$,!mathrm\{d\}(x,y)ge0$ ("non-negativity");
#$,!mathrm\{d\}(x,x)=0;$

The definition of a prametric allows for the case of $,!mathrm\{d\}(x,y)=0$ even if $x\; e\; y$. A prametric is said to be separating if $,!mathrm\{d\}(x,y)=0$ implies that $x=y$, for all $x,yin\; M$ (this is the identity of indiscernibles).

A prametric is called symmetric if $,!\; mathrm\{d\}(x,y)=d(y,x)$ for all $x,yin\; M$.

A symmetric, separating prametric is called a semimetric, and the corresponding space is a semimetric space.

A prametric which obeys the triangle inequality is called a hemimetric; a separating hemimetric is a quasimetric; a symmetric hemimetric is a pseudometric.

Examples

Another example is that of the distance between subsets of a metric space. That is, given a metric space $(X,\; ho)$ and some collection of subsets $\{V\_i:\; V\_isubset\; X,,\; iin\; I\}$ indexed by a set $I$, one defines

:$d(i,j)=\; ho(V\_i,V\_j).$

This distance is a symmetric prametric on the index set $I$.----A second example is the non-symmetric prametric on the reals:

:
x-y| & mbox{for } xle y \ 1 & mbox{for } x > yend{cases}

The topology generated by this prametric (as described below) is that of the Sorgenfrey line.----The set $\{0,1\}$ with the prametric $d(0,1)=1$ and $d(1,0)=0$ generates the connected two-point topology for this set, which makes it a ml|Particular_point_topology|Sierpi.C5.84ski_space|Sierpinski space. Thus, Sierpinski space is prametrizable but not metrizable.

Topology

For a prametric, define the ball as

:

At the most basic level, the definition of an open set for a prametric is as one might expect: every point must be an inner point with respect to this ball. That is, a subset $Usubset\; M$ is defined to be open if and only if, for each point $pin\; U$, there exists an $r>0$ such that $B\_r(p)\; subset\; U$.

What is unusual is that any given ball need not be an open set. The set of balls will not typically be a base for the topology; to obtain a topology, one instead works with the collection of open sets, as defined above.

In general, the interior of a ball $B\_r(p)$ may fail to contain "p", and the interior may even be empty; this is in sharp contrast to what one expects for a metric space.

Another unusual aspect is that a point in a closed set may have a distance from the closed set that is greater than zero. That is, if $overline\{A\}subset\; M$ is a closed set, and $xinoverline\{A\}$, it may not be true that $d(x,overline\{A\})=0$. The converse does hold: if $d(x,overline\{A\})=0$, then $xinoverline\{A\}$. The set of points at distance zero from a set defines a kind of closure, a praclosure.

To be clear, in the above, a set $Csubset\; M$ is defined to be closed if and only if $d(p,C)>0$ for all .

Such topologies do have some nice properties: a topological space with a topology generated by a prametric is a sequential space.

A topological space is said to be a prametrizable topological space if the space can be given a prametric such that the prametric topology coincides with the given topology on the space. With the additional appropriate axioms, one may say that a space is semimetrizable, quasimetrizable, etc.

Axioms

The following table shows the various special cases, according to applicable axioms:

References

* A.V. Arkhangelskii, L.S.Pontryagin, "General Topology I", (1990) Springer-Verlag, Berlin. ISBN 3-540-18178-4