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 $(Y,d)$ 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

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

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) (Holsztyński 1966).

References

• Holsztyński, W. (1966), "On metric spaces aimed at their subspaces.", Prace Mat., 10: 95–100, MR 0196709