Metric space aimed at its subspace

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.

Informal introduction

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).

Definitions

Let [math]\displaystyle{ (Y, d) }[/math] be a metric space. Let [math]\displaystyle{ X }[/math] be a subset of [math]\displaystyle{ Y }[/math], so that [math]\displaystyle{ (X,d |_X) }[/math] (the set [math]\displaystyle{ X }[/math] with the metric from [math]\displaystyle{ Y }[/math] restricted to [math]\displaystyle{ X }[/math]) is a metric subspace of [math]\displaystyle{ (Y,d) }[/math]. Then

Definition.  Space [math]\displaystyle{ Y }[/math] aims at [math]\displaystyle{ X }[/math] if and only if, for all points [math]\displaystyle{ y, z }[/math] of [math]\displaystyle{ Y }[/math], and for every real [math]\displaystyle{ \epsilon \gt 0 }[/math], there exists a point [math]\displaystyle{ p }[/math] of [math]\displaystyle{ X }[/math] such that

[math]\displaystyle{ |d(p,y) - d(p,z)| \gt d(y,z) - \epsilon. }[/math]

Let [math]\displaystyle{ \text{Met}(X) }[/math] be the space of all real valued metric maps (non-contractive) of [math]\displaystyle{ X }[/math]. Define

[math]\displaystyle{ \text{Aim}(X) := \{f \in \operatorname{Met}(X) : f(p) + f(q) \ge d(p,q) \text{ for all } p,q\in X\}. }[/math]

Then

[math]\displaystyle{ d(f,g) := \sup_{x\in X} |f(x)-g(x)| \lt \infty }[/math]

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

Properties

Let [math]\displaystyle{ i\colon X \to Y }[/math] be an isometric embedding. Then there exists a natural metric map [math]\displaystyle{ j\colon Y \to \operatorname{Aim}(X) }[/math] such that [math]\displaystyle{ j \circ i = \delta_X }[/math]:

[math]\displaystyle{ (j(y))(x) := d(x,y)\, }[/math]

for every [math]\displaystyle{ x\in X\, }[/math] and [math]\displaystyle{ y\in Y\, }[/math].

Theorem The space Y above is aimed at subspace X if and only if the natural mapping [math]\displaystyle{ j\colon Y \to \operatorname{Aim}(X) }[/math] 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.

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

References

• Holsztyński, W. (1966), "On metric spaces aimed at their subspaces.", Prace Mat. 10: 95–100