Premetric space

In mathematics, 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 $\mathrm {d} :M\times M\to \mathbb {R}$ (called a prametric) which satisfies the following conditions:

1. $\,\!\mathrm {d} (x,y)\geq 0$ (non-negativity);
2. $\,\!\mathrm {d} (x,x)=0;$

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

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

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

If $(X,\rho )$ is a metric space, and $f:X\to Y$ is a map, then

$d(y_{1},y_{2})=\rho (f^{-1}(y_{1}),f^{-1}(y_{2}))$

is a symmetric prametric.

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Another example is that of the distance between subsets of a metric space. That is, given a metric space $(X,\rho )$ and some collection of subsets $\{V_{i}:V_{i}\subset X\,,i\in I\}$ indexed by a set $I$, one defines

$d(i,j)=\rho (V_{i},V_{j})$

This distance is a symmetric prametric on the index set $I$.

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A third example is the non-symmetric prametric on the reals:

$d(x,y)={\begin{cases}|x-y|&{\mbox{for }}x\leq y\\1&{\mbox{for }}x>y\end{cases}}$

The topology generated by this prametric (as described below) is that of the Sorgenfrey line.

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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 {{#invoke: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

$B_{r}(p)=\{x\in M\mid d(x,p)<r\}.$

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 $U\subset M$ is defined to be open if and only if, for each point $p\in U$, 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 $x\in {\overline {A}}$, it may not be true that $d(x,{\overline {A}})=0$. The converse does hold: if $d(x,{\overline {A}})=0$, then $x\in {\overline {A}}$. 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 $C\subset M$ is defined to be closed if and only if $d(p,C)>0$ for all $p\in M\backslash C$.

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:

Triangle inequality	Distinguishability	Symmetry	Type
No	No	No	prametric space
No	No	Yes	symmetric prametric space
No	Yes	No	separating prametric space
No	Yes	Yes	semimetric space
Yes	No	No	hemimetric space
Yes	No	Yes	pseudometric space
Yes	Yes	No	quasimetric space
Yes	Yes	Yes	metric space

References

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