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, 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 $$(Y, d)$$ be a metric space. Let $$X$$ be a subset of $$Y$$, so that $$(X,d |_X)$$ (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 $$\epsilon > 0$$, there exists a point $$p$$ of $$X$$ such that


 * $$|d(p,y) - d(p,z)| > d(y,z) - \epsilon.$$

Let $$\text{Met}(X)$$ be the space of all real valued metric maps (non-contractive) of $$X$$. Define


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

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 $$\text{Aim}(X)$$. Furthermore, $$\delta_X\colon x\mapsto d_x$$, where $$d_x(p) := d(x,p)\,$$, is an isometric embedding of $$X$$ into $$\operatorname{Aim}(X)$$; 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 $$\operatorname{Aim}(X)$$ is aimed at $$\delta_X(X)$$.

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


 * $$(j(y))(x) := d(x,y)\,$$

for every $$x\in X\,$$ and $$y\in Y\,$$.


 * Theorem The space Y above is aimed at subspace X if and only if the natural mapping $$j\colon Y \to \operatorname{Aim}(X)$$ 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.

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).