Metric space aimed at its subspace

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[edit] Introduction

In mathematics, a metric space aimed at its subspace is a categorical construction that has a direct geometric meaning. It is also a useful step toward the construction of the metric envelope, or tight span, which are basic (injective) objects of the category of metric spaces.

Following (Holsztyński 1966)), a notion of a metric space  Y  aimed at its subspace  X   is defined.

Informally, imagine terrain Y, and its part X, such that wherever in Y you place a sharpshooter, and an apple at another place in Y, and then let the sharpshooter fire, the bullet will go through the apple and will always hit a point of X, or at least it will fly arbitrarily close to points of X – then we say that Y is aimed at X.

A priori, it may seem plausible that for a given X the superspaces Y that aim at X can be arbitrarily large or at least huge. We will see that this is not the case. Among the spaces, which aim at a subspace isometric to X there is a unique (up to isometry) universal one, Aim(X), which in a sense of canonical isometric embeddings contains any other space aimed at (an isometric image of) X. And in the special case of an arbitrary compact metric space X every bounded subspace of an arbitrary metric space Y aimed at X is totally bounded (i.e. its metric completion is compact).

[edit] Definitions

Let (Y,d) be a metric space. Let X be a subset of Y, so that (X,d | X2) (the set X with the metric from Y restricted to X) is a metric subspace of (Y,d). Then

Definition.  Space Y aims at X if and only if, for all points y,z of Y, and for every real ε > 0, there exists a point p of X such that

| d(p,y) − d(p,z) | > d(y,z) − ε.

Let Met(X) be the space of all real valued metric maps (non-contractive) of X. Define

\text{Aim}(X) := \{f \in \text{Met}(X) : f(p) + f(q) \ge d(p,q) \text{ for all } p,q\in X\}

Then

d(f,g) := \sup_{x\in X} |f(x)-g(x)| < \infty

for every f, g\in \text{Aim}(X) is a metric on Aim(X). Furthermore, \delta_X : x\mapsto d_x, where dx(p): = d(x,p), is an isometric embedding of X into Aim(X); this is essentially a generalisation of the Kuratowski-Wojdysławski embedding of bounded metric spaces X into C(X), where we here consider arbitrary metric spaces (bounded or unbounded). It is clear that the space Aim(X) is aimed at δX(X).

[edit] Properties

Let  i : X \rightarrow Y\,  be an isometric embedding. Then there exists a natural metric map  j : Y \rightarrow Aim(X)\,  such that  j \circ i = \delta_X:

(j(y))(x) := d(x,y)\,

for every  x\in X\, and y\in Y\,.

Theorem Above, space  Y  is aimed at subspace  X   if and only if the natural mapping  j : Y \rightarrow Aim(X)\,   is an isometric embedding.

Thus it follows that every space aimed at  X  can be isometrically mapped into  Aim(X),   with some additional (essential) categorical requirements satisfied.

Space  Aim(X)   is injective (hyperconvex in the sense of Aronszajn-Panitchpakdi) – given a metric space M,  which contains  Aim(X)   as its metric subspace, there is a canonical (and explicit) metric retraction of M  onto Aim(X)   (Holsztyński 1966)

[edit] References

  • Holsztyński, W. (1966), “On metric spaces aimed at their subspaces.”, Prace Mat. 10: 95-100, MR0196709