Prametric space
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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.
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[edit] Definition
A prametric space (M,d) is a set M together with a function (called a prametric) which satisfies the following conditions:
- (non-negativity);
The definition of a prametric allows for the case of even if . A prametric is said to be separating if implies that x = y, for all (this is the identity of indiscernibles).
A prametric is called symmetric if for all .
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.
[edit] Examples
If (X,ρ) is a metric space, and is a map, then
- d(y1,y2) = ρ(f − 1(y1),f − 1(y2))
is a symmetric prametric.
Another example is that of the distance between subsets of a metric space. That is, given a metric space (X,ρ) and some collection of subsets indexed by a set I, one defines
- d(i,j) = ρ(Vi,Vj).
This distance is a symmetric prametric on the index set I.
A third example is the non-symmetric prametric on the reals:
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 Sierpinski space . Thus, Sierpinski space is prametrizable but not metrizable.
[edit] 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 is defined to be open if and only if, for each point , there exists an r > 0 such that .
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 Br(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 is a closed set, and , it may not be true that . The converse does hold: if , then . 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 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.
[edit] 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 |
[edit] References
- A.V. Arkhangelskii, L.S.Pontryagin, General Topology I, (1990) Springer-Verlag, Berlin. ISBN 3-540-18178-4