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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 Template:Harv, 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 $(X,d{|}_X)$ (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 $ϵ>0$, there exists a point $p$ of $X$ such that

$|d(p,y)-d(p,z)|>d(y,z)-ϵ.$

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

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

Then

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

for every $f,g\in \text{Aim}(X)$ is a metric on $\text{Aim}(X)$. Furthermore, ${\delta }_X:x\mapsto d_x$, where $d_x(p):=d(x,p)$, is an isometric embedding of $X$ into $\mathrm{Aim}(X)$; 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 $\mathrm{Aim}(X)$ is aimed at ${\delta }_X(X)$.

Properties

Let $i:X\to Y$ be an isometric embedding. Then there exists a natural metric map $j:Y\to \mathrm{Aim}(X)$ 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 $j:Y\to \mathrm{Aim}(X)$ 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) Template:Harv.

References

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