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

:$\text(X) := \.$

Then

:$d(f,g) := \sup_ , f(x)-g(x), < \infty$

for every $f, g\in \text(X)$ is a metric on $\text(X)$. Furthermore, $\delta_X\colon x\mapsto d_x$, where $d_x(p) := d(x,p)\,$, is an isometric embedding of $X$ into $\operatorname(X)$; 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 $\operatorname(X)$ is aimed at $\delta_X(X)$.

Properties

Let $i\colon X \to Y$ be an isometric embedding. Then there exists a natural metric map $j\colon Y \to \operatorname(X)$ 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 $j\colon Y \to \operatorname(X)$ 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) .

References

*{{citation, mr=0196709, last= Holsztyński, first= W., title= On metric spaces aimed at their subspaces. , journal= Prace Mat., volume= 10, year= 1966, pages= 95–100

Metric geometry