Some characterizations of ultrametrically injective spaces
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Some characterizations of ultrametrically injective spaces Collins Amburo Agyingi1,2 Received: 12 August 2020 / Accepted: 19 September 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract In this article, we present a proof (in two different ways) of the fact that every hyperconvex ultrametric space is ultrametrically injective. The first proof will be constructive in nature and the second proof will appeal to Zorn’s Lemma. We show also that the ultrametrically injective hull TX , of an ultrametric space (X , m), as constructed by Bayod et al. and K(X ) (the space of all ultra-Katˇetov functions on an ultrametric space X ) are both ultrametrically injective. Furthermore, it is shown that ultrametrically injective spaces do not contain proper essential extensions. In the end, we give a number of characterizations of the ultrametrically injective hull TX of an ultrametric space (X , m). Keywords Ultrametrically injective · Hyperconvex · Retraction · Ultra–Katˇetov functions · Ultrametrically injective hull · Essential extensions Mathematics Subject Classification 54E15 · 54E35 · 54E55 · 54E50
1 Introduction Hyperconvex metric spaces were introduced by Aronszajn and Panitchpakdi in [4] (see for instance [6]). It is, nevertheless, necessary to point out that earlier investigations had been done on that subject by Aronszajn in his Ph.D. thesis [5] and that thesis was never published. In Sect. 2 of [11], Isbell introduced the notion of a hyperconvex hull and since then many studies have been carried out and published on that subject (consult for example [10,13,16]). It is well known that a metric space is hyperconvex if and only if it is injective in the category of metric spaces and nonexpansive maps. Let us recall that a metric space Y is said to be injective
This paper is dedicated to the late Professor Hans-Peter Künzi of the University of Cape Town, South Africa. C.A. Agyingi was supported in part by grant number 115223 from the National Research Foundation of South Africa.
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Collins Amburo Agyingi [email protected]
1
Department of Mathematical Sciences, University of South Africa, UNISA, 003, P.O. Box 392, Pretoria, South Africa
2
African Center for Advanced Studies (ACAS), P.O. Box 4477, Yaounde, Cameroon
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C.A. Agyingi
if for any metric space X and any subspace A of X , any non-expansive map f : A → Y has a non-expansive extension g : X → Y . Furthermore, we know that every metric space can be isometrically embedded in a hyperconvex metric space, its so-called hyperconvex hull (check for instance [11]). Similar studies about hyperconvexity in the setting of ultrametric spaces have been carried out by Bayod and Martínez-Maurica in [7]. They used the term ultrametrically injective hull to mean the hyperconvex hull of an ultrametric space and they denoted it by TX , where X is the particular ultrametric space in question. In their paper, they actually gave a construction of TX . They characterized TX as a space consisting of ultra-extrema
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