Metric space

In Mathematical, a metric space is a whole in which a concept of distance between the elements of the unit is defined. It is a topological particular case of Espace.

The example corresponding more to our intuitive experiment of space is the Euclidean Espace with three dimensions. The Euclidean Métrique of this space defines the distance between two points like the length of the segment connecting them.

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Definitions

  • One calls distance on a Ensemble E \, , a d application: E \ times E \ rightarrow \ mathbb R_+ such as:
* \ forall X, there \ in E, D (X, there) =d (there, X) (symmetry);
*d (X, there) =0 \ Leftrightarrow x=y (separation);
*d (X, Z) \ Leq D (X, there) +d (there, Z) ( triangular inequality ).

  • One calls swell (open) centered in a \ in E and of ray r \ in \ mathbb R_+, the unit \ {X \ in E/d (X, a). It is often noted B (has, R) \, .

  • One calls swell closed centered in a \ in E and of ray r \ in \ mathbb R_+, the unit \ {X \ in E/d (X, a) \ Leq R \} \ subset E. It is often noted B_f (has, R) \, .
See also swell.

  • the distance provides E \, of a topology, by defining a left U \, like open when: \ forall U \ in U, \ exists \ varepsilon>0/B (U, \ varepsilon) \ subset U. Open is thus a part which has a certain “thickness” around its points. A topological space is known as métrisable if there exists a distance defining its topology; this distance is almost never single and one will take guard which the concepts of ball, limited (i.e included in a ball), of continuation of Cauchy, uniform continuity, etc are not topological but metric concepts, likely to vary according to the selected distance. In this topology, the point neighborhoods are all the subsets containing an open ball centered on this point. Usual topology on the line (of the real numbers), the plan, etc are definable examples of topologies using metric.

  • an interesting property of topological spaces métrisables is the property of separation. Indeed, if one chooses two distinct elements x \, and y \, of a metric space E \, , their distance d \, is nonnull, consequently open the B (X, D/2) \, and B (there, D/2) \, is disjoined and is vicinities of x \, and y \, .

Examples

  • commonplace distance (or discrete distance): on a nonempty unit, one decides that the distance between two distinct points is 1. With such a distance, one easily checks that topology is then the whole of the parts of E, i.e. for very part F of E is open.
  • topological spaces R and] 0,1 are homeomorphic, but provided with the usual distances, they are not isomorphous as metric spaces; for example ''' R ''' is complete but 0,1 is not it. * if one provides R+ with the distance D (X, there) =|ex- ey| , one finds usual topology on R+ but maintaining all the functions polynomials are uniformly continuous.
  • the metric discrete one, for which D (X, there) = 1 for all X different of there and D (X, X) = 0 is another simple example being able to be applied to any unit E .
  • the distance to the Échecs makes it possible to know the number of blows necessary to the play of failure to go with the king of a box x1, y1 with a box x2, y2 and is defined by D_ {Echec} = \ max \ left (\ left | \ left (x_2 - x_1 \ right) \ right | , \ left | \ left (y_2 - y_1 \ right) \ right | \ right)
  • the distance from Manhattan: in the plan \ mathbb R^2: D (has, b)=|x_b-x_a|+|y_b there _a|.c' is of course the distance induced by standard 1.

Equivalence of metric spaces

By comparing two metric spaces it is possible to distinguish various degrees from equivalence. To preserve minimum the topological structure induced by the metric one has, a Fonction continues between the two is necessary.

That is to say two metric spaces ( M 1, D 1) and ( M 2, D 2). M 1 and M 2 is called

  • topologically isomorphic (or homeomorphic ) if there exists a Homéomorphisme between them.

  • uniformly isomorphic if there exists a uniform Isomorphisme between them.

  • isométriquement isomorphic if there exists a bijective isometry between them. In this case two spaces are primarily identical. A Isométrie is a function F : M 1 → M 2 which preserves the distances: D 2 ( F ( X ), F ( there )) = D 1 ( X , there ) for all X , there in M 1. The isométries are inevitably injective.

  • similar if there exists a positive constant K > 0 and a bijective function F , called Similarité such as F : M 1 → M 2 and D 2 ( F ( X ), F ( there )) = K D 1 ( X , there ) for all X , there in M 1.

  • similar (of the second type) if there exists a bijective function F , called Similarité such as F : M 1 → M 2 and D 2 ( F ( X ), F ( there )) = D 2 ( F ( U ), F ( v )) if and only if D 1 ( X , there ) = D 1 ( U , v ) for all X , there , U , v in M 1.

In the case of an Euclidean space with the metric usual one, the two concepts of similarity are equivalent.

Traps

  • the bond between a closed ball and the adherence of the corresponding open ball is in general a simple inclusion:
\ overline B (has, R) \ subset B_f (has, R) (it is a traditional trap of topology, discussed in Adhérence (mathematics) and Boule (mathematics)).

See too

Zh-classical: 度量空間

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