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Some characterizations of ultrametrically injective spaces

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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 \(T_X\), of an ultrametric space (X, m), as constructed by Bayod et al. and \({\mathcal {K}}(X)\) (the space of all ultra-Katětov 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 \(T_X\) of an ultrametric space (X, m).

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Notes

  1. If there is no confusion on the ultrametric m, we shall denote simply by \(P_X\).

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of South Africa, UNISA, 003, P.O. Box 392, Pretoria, South Africa

    Collins Amburo Agyingi

  2. African Center for Advanced Studies (ACAS), P.O. Box 4477, Yaounde, Cameroon

    Collins Amburo Agyingi

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  1. Collins Amburo Agyingi
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Correspondence to Collins Amburo Agyingi.

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This paper is dedicated to the late Professor Hans-Peter Künzi of the University of Cape Town, South Africa.

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C.A. Agyingi was supported in part by grant number 115223 from the National Research Foundation of South Africa.

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Agyingi, C.A. Some characterizations of ultrametrically injective spaces. Afr. Mat. 32, 517–529 (2021). https://doi.org/10.1007/s13370-020-00841-x

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  • DOI: https://doi.org/10.1007/s13370-020-00841-x

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