Metric space aimed at its subspace
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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 (Holsztyński 1966), 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 (
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 :
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
References
- Holsztyński, W. (1966), "On metric spaces aimed at their subspaces.", Prace Mat., 10: 95–100, MR 0196709