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

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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

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Let   be a metric space. Let   be a subset of  , so that   (the set   with the metric from   restricted to  ) is a metric subspace of  . Then

Definition.  Space   aims at   if and only if, for all points   of  , and for every real  , there exists a point   of   such that

 

Let   be the space of all real valued metric maps (non-contractive) of  . Define

 

Then

 

for every   is a metric on  . Furthermore,  , where  , is an isometric embedding of   into  ; this is essentially a generalisation of the Kuratowski-Wojdysławski embedding of bounded metric spaces   into  , where we here consider arbitrary metric spaces (bounded or unbounded). It is clear that the space   is aimed at  .

Properties

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Let   be an isometric embedding. Then there exists a natural metric map   such that  :

 

for every   and  .

Theorem The space Y above is aimed at subspace X if and only if the natural mapping   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

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