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 (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 ) {\displaystyle (Y,d)} be a metric space. Let X {\displaystyle X} be a subset of Y {\displaystyle Y} , so that ( X , d | X ) {\displaystyle (X,d|_{X})} (the set X {\displaystyle X} with the metric from Y {\displaystyle Y} restricted to X {\displaystyle X} ) is a metric subspace of ( Y , d ) {\displaystyle (Y,d)} . Then Definition. Space Y {\displaystyle Y} aims at X {\displaystyle X} if and only if, for all points y , z {\displaystyle y,z} of Y {\displaystyle Y} , and for every real ϵ > 0 {\displaystyle \epsilon >0} , there exists a point p {\displaystyle p} of X {\displaystyle X} such that
| d ( p , y ) − d ( p , z ) | > d ( y , z ) − ϵ . {\displaystyle |d(p,y)-d(p,z)|>d(y,z)-\epsilon .}
Let Met ( X ) {\displaystyle {\text{Met}}(X)} be the space of all real valued metric maps (non-contractive) of X {\displaystyle X} . Define
Aim ( X ) := { f ∈ Met ( X ) : f ( p ) + f ( q ) ≥ d ( p , q ) for all p , q ∈ X } . {\displaystyle {\text{Aim}}(X):=\{f\in \operatorname {Met} (X):f(p)+f(q)\geq d(p,q){\text{ for all }}p,q\in X\}.}
Then
d ( f , g ) := sup x ∈ X | f ( x ) − g ( x ) | < ∞ {\displaystyle d(f,g):=\sup _{x\in X}|f(x)-g(x)|<\infty }
for every f , g ∈ Aim ( X ) {\displaystyle f,g\in {\text{Aim}}(X)} is a metric on Aim ( X ) {\displaystyle {\text{Aim}}(X)} . Furthermore, δ X : x ↦ d x {\displaystyle \delta _{X}\colon x\mapsto d_{x}} , where d x ( p ) := d ( x , p ) {\displaystyle d_{x}(p):=d(x,p)\,} , is an isometric embedding of X {\displaystyle X} into Aim ( X ) {\displaystyle \operatorname {Aim} (X)} ; this is essentially a generalisation of the Kuratowski-Wojdysławski embedding of bounded metric spaces X {\displaystyle X} into C ( X ) {\displaystyle C(X)} , where we here consider arbitrary metric spaces (bounded or unbounded). It is clear that the space Aim ( X ) {\displaystyle \operatorname {Aim} (X)} is aimed at δ X ( X ) {\displaystyle \delta _{X}(X)} .
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