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 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 universal one, Aim, which in a sense of canonical isometric embeddings contains any other space aimed at 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.

Definitions

Let be a metric space. Let be a subset of, so that 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 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. It is clear that the space is aimed at.

Properties

Let be an isometric embedding. Then there exists a natural metric map such that :
for every and.
Thus it follows that every space aimed at X can be isometrically mapped into Aim, with some additional categorical requirements satisfied.
The space Aim is injective – given a metric space M, which contains Aim as a metric subspace, there is a canonical metric retraction of M onto Aim.