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.

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 | 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 $\epsilon > 0$, there exists a point p of X such that

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

Let 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 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 δ_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) (Holsztyński 1966).

References

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