Ultra-Quasi-Metrically Tight Extensions of Ultra-Quasi-Metric Spaces
The concept of the tight extension of a metric space was introduced and studied by Dress. It is known that Dress theory is equivalent to the theory of the injective hull of a metric space independently discussed by Isbell some years earlier. Dress showed in particular that for a metric space the tight extension is maximal among the tight extensions of . In a previous work with P. Haihambo and H.-P. Künzi, we constructed the tight extension of a -quasi-metric space. In this paper, we continue these investigations by presenting a similar construction in the category of -metric spaces and nonexpansive maps.
In  a concept of tight extension that is appropriate in the category of -quasi-metric spaces and nonexpansive maps was studied. In particular such an extension was constructed and it was shown that this extension is maximal among the tight extensions.
In this paper we will show how the studies of  can be modified in order to obtain a theory that is appropriate for -metric spaces. By UQP-metric space in the following, we mean -ultra-quasi-metric spaces. Even though our studies follow essentially [1, 2], we found it imperative to work out every detail of this theory in this paper.
We will show that every -metric space has a -tight extension which is maximal amongst the -tight extensions of . This agrees with the result we have for -quasi-metric spaces (check ).
We mention that the ultra-quasi-pseudometric spaces should not be confused with the quasi-ultrametrics as they are discussed in the theory of dissimilarities (check, e.g., ).
Definition 1 (compare [4, page 2]). Let be a set and let be a mapping into the set of nonnegative reals. Then is an ultra-quasi-pseudometric or, for short, a uqp-metric on on if for all , whenever .We remark here that the conjugate of where whenever is also -metric on .
If also satisfies the condition,for any , implies that ,then is called -metric on .
Notice that is ultrametric on .
Example 2 (compare [4, Example ]). Note that, for , the pair is -metric space, where is such that if and , and if and . We show the strong triangle inequality whenever since the other conditions are obvious. For , the result is trivial, since . Similarly the case that and is obvious, since and . In the remaining case that , we have by transitivity of that , and thus . It is obvious that satisfies the -condition.
Notice also that, for , we have if and if . The ultrametric is complete on since and are bicomplete on . Recall that a -metric space is said to be bicomplete if the ultrametric space is complete.
Furthermore is the only nonisolated point of . Indeed is a compact subspace of .
In some cases we will replace with and, in this case, we will speak of an extended -metric.
Lemma 3 (compare [5, Proposition ]). Let . Then the following are equivalent:(a),(b).
. To reach a contradiction, suppose that . Since , we have by part and the way was defined. Thus, we would get , a contradiction.
. Suppose that, on the contrary, would hold. Then and, hence, would hold which would imply in contradiction to our assumption .
The following corollaries are immediate. Their proofs rely on Lemma 3.
Corollary 4 (see ). Let be -metric space. Consider a map and let . Then the following are equivalent:(a),(b).
Definition 5. A map between two -metric spaces and is called nonexpansive if holds for any .
Corollary 6 (see ). Let be -metric space. Then(a) is a nonexpansive map if and only if holds for all ,(b) is a nonexpansive map if and only if holds for all .
Definition 7. A map between two -metric spaces and is said to be an isometry provided that whenever . Two -metric spaces and are said to be isometric provided that there exists a bijective isometry between them. Note that if is a -metric space, then is injective.
3. Ultra-Ample Function Pairs on -Metric Space
We will recall some results from the theory of hyperconvex hulls of -metric spaces due to .
Definition 8 (compare [4, Definition , page 4]). Let be -metric space. One will say that a function pair on where is ultra-ample if, for all , one has .
Let us denote by the set of all ultra-ample function pairs on -metric space . For each , definewhere the -metric (of course we can use -metric here since the function pairs take values in ) is as defined in Example 2. Then is an extended -metric on .
Lemma 9. Let be -metric space and let . Then one has that whenever is an ultra-ample function pair belonging to .
We say that a function pair is -minimal among the ultra-ample function pairs on if it is an ultra-ample function pair and if is an ultra-ample function pair on and for each and , which implies . We will also call -minimal ultra-ample function pair -extremal (ultra-ample) function pair. By we will denote the set of all -extremal function pairs on equipped with the restriction of to , which we will still denote by . Of course is actually -metric on (compare Corollary of ). We will call the ultra-quasi-metrically injective hull of .
Lemma 10. Let be -metric space and let . For all implies that and implies that .
Proof. See the proof of Lemma of .
As a corollary, we have the following.
Corollary 11. Let be -metric space. If is a minimal ultra-ample function pair on , then and whenever . Thus the maps and are contracting maps (check, e.g., Corollary 6).
Lemma 12. Let be a minimal ultra-ample function pair on a -ultra-quasi-metric space . Then whenever .
Proof. See the proof of Lemma of .
The following lemma and its proof are found in .
Lemma 13. If and are minimal ultra-ample function pairs on a -metric space , thenAs a consequence of Lemmas 12 and 13 we have the following corollary.
Corollary 14. Any minimal ultra-ample function pair on -metric space satisfies the following:whenever .
4. -Metric Tight Extensions
In this paper we will study -tight extensions as defined below in Definition 19. Moreover, by -tight extensions, we will mean tight extensions of -metric spaces.
Proposition 15 (compare [2, Section ]). Let be -metric space. Then consists of all functions pairs which are “minimal” in .
Proof. To prove this proposition, we prove that there is no with but . This is so since, on the one hand, and imply ThusUsing (6) and the condition that , we have that .
In the same manner we can show that so as to conclude that .
On the other hand, suppose that, for some and , we have that .
For each and set if and .
Similarly for each and set if and .
We show first that is ultra-ample. We will consider the following cases.
Case 1. If and , then the result holds since .
Case 2. If and , then , so thatCase 3. Consider and . In this case and , so that Case 4. In a manner similar to case 3, the result can be shown.
Thus is ultra-ample and also satisfies by the way it was constructed.
Thus by taking , we can conclude that, for any , .
Lemma 16 (compare [4, Lemma ]). Let be -metric space. Then, for any , we have that
Proposition 17 (compare [1, Proposition ]). Let be -metric space. There exists a retraction map , that is, a map that satisfies the following conditions: whenever . whenever . (In particular we have that whenever .)
Proof. We will prove Proposition 17 by the use of Zorn’s lemma.
Indeed, let be -metric space and let be the set of all maps from to satisfying conditions and in Proposition 17.
Order byfor all and . Then since the identity map belongs to .
We have to check now that is actually a partial order.
Reflexivity is obvious since every map is equal to itself.
Let now such that and . Consider and imply that . In a similar manner, we have that so that we can conclude that .
Also and imply that . This shows that is antisymmetric.
Suppose now that such that and : and imply that by transitivity of as a subset of with the usual ordering . Similarly, we can show that .
Also and imply that . Thus . This proves that is transitive. Therefore is a partially ordered set.
Next we show that every chain in has a lower bound.
Let be a chain and define bywhenever . Since , we have that .
Indeed observe that , for all . Thus and condition is satisfied.
To check condition , we check that .
Indeed Thus we have that condition (b) is satisfied and since is a map from to , we conclude that and is a lower bound of the chain by construction. We therefore appeal to Zorn’s lemma to conclude that has a minimal element, say , with respect to the partial order .
To complete the proof, we show that whenever .
For each , we have that and (where is as defined in the proof of Proposition 15). Thus by minimality of , we have . It therefore follows that, for each , whenever . Thus by the way elements in are defined, we conclude that whenever .
Proposition 18 (compare [1, Proposition ]). Let be -metric space and let be a subspace of . Then there exists an isometric embedding such that whenever .
Proof. Fix and let be a retraction satisfying the conditions of Proposition 17. Also let be such that , where whenever , and whenever . The coordinate of pair is defined similarly.
We will consider the following cases to prove that belongs to .
Case 1. Consider that and .
Case 2. One has that and .
Then Case 3. Consider that and .
Case 4. Consider that and .
Define map . Then whenever since . Thus , and is minimal on .
Moreover for any , we have The last equality follows from the definition of and . Hence we have that which implies that and hence is an isometric map.
Definition 19. Let be a subspace of -metric space . Then is called -tight extension of if for any -metric on that satisfies and agrees with on , we have that .
Remark 20. For any -metric -tight extension of , any -metric extension of and any nonexpansive map satisfying whenever must necessarily be an isometric map.
Indeed if that is not the case, then the -metric defined by would contradict the fact that the span of is ultra-ample.
It was shown in  that the map from -metric space to its ultra-quasi-metrically injective hull defined by whenever is an isometric embedding. We will proceed now with the help of Lemma 16 to show that is -tight extension of .
Proposition 21. Let be -metric space and let be as defined above. Then is -tight extension of .
Proof. Let be -metric on such that and whenever . By Lemma 16 and the fact that , for any , we have Thus .
Proposition 22. Let be -metric -tight extension of . Then the restriction map defined by whenever is a bijective isometric map .
Proof. Let be a retraction map that satisfies the conditions of Proposition 17 and let denote the composition of the retraction map with the restriction map. Then one sees immediately that is nonexpansive. Thus by Lemma 3, must be an isometry, because is -tight extension of (this is so since is -tight extension of and is -tight extension of ).
Choose an isometric embedding such that for every (compare Proposition 18). We therefore haveThis implies that is onto. The fact that is injective is clear since is -metric space (see, e.g., the last line of Definition 7). Thus is bijective. In this case, the inverse of has to be the inverse of and hence for any , we have , which is the mapthat maps onto , without it being composed of . Hence for any -metric -tight extension of , the mapis a bijective isometry between and .
Theorem 23. Let be a subspace of the -metric space . Then the following are equivalent:(a) is -metric -tight extension of .(b) whenever .(c), , is minimal on whenever and the map defined by is an isometric embedding.
. Let be -metric -tight extension of . By Proposition 22, the restriction map is a bijective isometry between and . Thus the extension satisfies condition , since satisfies it by Lemma 16.
. Let and . Then we have that . Thus by condition we have that . AlsoHence for we have by condition that Similarly we have that whenever and so that by condition we get . It therefore follows that, for each and , Thus for we see by that Thus we conclude that .
As we have above, for any we have thatObserve that if we substitute for in (24) and for in (25) we obtain the following equations:whenever and andwhenever and . We have therefore that restriction is minimal on whenever (compare Lemma 12).
. Let be -metric on such that whenever and whenever . Then by part and the fact that is minimal whenever , we have By substitutinginto formulawe obtain whenever . The last inequality holds by the light of the inequalityand the fact that and . Thus we have that whenever and hence follows.
Remark 24. We see from Theorem 23 that there is only one isometric embedding satisfying whenever , since for such an embedding we havetherefore . Similarly, one can show that whenever .
Thus we see that the -tight extension of can be understood as a subspace of extension of . Hence is maximal among the -metric -tight extensions of .
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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