# 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 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 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

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

**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) Template:Harv.

## References

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