Based on the theory of ball spaces introduced by Kuhlmann and Kuhlmann we introduce and study Caristi-Kirk and Oettli-Théra ball spaces. We show that if the underlying metric space is complete, then these have a very strong property: every ball contains a singleton ball. This fact provides quick proofs for several results which are equivalent to the Caristi-Kirk Fixed Point Theorem, namely Ekeland's Variational Principles, the Oettli-Théra Theorem, Takahashi's Theorem and the Flower Petal Theorem.2010 Mathematics Subject Classification. Primary 54H25; Secondary 47H09, 47H10.

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We study spherical completeness of ball spaces and its stability under expansions. We give some criteria for ball spaces that guarantee that spherical completeness is preserved when the ball space is closed under unions of chains. This applies in particular to the spaces of closed ultrametric balls in ultrametric spaces with linearly ordered value sets, or more generally, with countable narrow value sets. Finally, we show that in general, chain union closures of ultrametric spaces with partially ordered value sets do not preserve spherical completeness.

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