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 , 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 formula_1 be a metric space. Let formula_2 be a subset of formula_3, so that formula_4 (the set formula_2 with the metric from formula_3 restricted to formula_2) is a metric subspace of formula_8. Then Definition.  Space formula_3 aims at formula_2 if and only if, for all points formula_11 of formula_3, and for every real formula_13 , there exists a point formula_14 of formula_2 such that Let formula_17 be the space of all real valued metric maps (non-contractive) of formula_2. Define Then for every formula_21 is a metric on formula_22. Furthermore, formula_23, where formula_24, is an isometric embedding of formula_2 into formula_26; this is essentially a generalisation of the Kuratowski-Wojdysławski embedding of bounded metric spaces formula_2 into formula_28, where we here consider arbitrary metric spaces (bounded or unbounded). It is clear that the space formula_26 is aimed at formula_30. Properties. Let formula_31 be an isometric embedding. Then there exists a natural metric map formula_32 such  that formula_33: for every formula_35 and formula_36. 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) .