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

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

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

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

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

Then

${\displaystyle d(f,g):=\sup _{x\in X}|f(x)-g(x)|<\infty }$

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

Properties

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

${\displaystyle (j(y))(x):=d(x,y)\,}$

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

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