Penot's compactness property in ultrametric spaces with an application (Q2028655)

In this paper, the compactness property over ultrametric spaces is investigated. The following is the main result of this exposition. Let \((M,d)\) be a metric space. By an admissible subset we mean any intersection of closed balls. The class of all these is denoted as \(\mathcal{A}(M)\). \textbf{Theorem.} Let \((M,d)\) be an ultrametric space. The following are equivalent (1) \(\mathcal{A}(M)\) is compact (in the sense of Penot): for each family \(\{B_i=B(a_i,r_i)\mid i\in I\}\) of closed balls, \(\cap\{B_i\mid i\in I\}\ne\emptyset\) provided \(\cap\{B_j\mid j\in J\}\ne \emptyset\), for each finite subset \(J\) of \(I\) (2) \((M,d)\) is spherically complete. An application of these to the fixed point theory over such structures is then given. Finally, some other aspects occasioned by these developments are also being discussed.