In this work, we investigate the compactness property in the sense of Penot in ultrametric spaces. Then, we show that spherical completeness is exactly the Penot’s compactness property introduced for convexity structures. The spherical completeness property misled some mathematicians to it to hyperconvexity in metric spaces. As an application, we discuss some fixed point results in spherically complete ultrametric spaces.

中文翻译：

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超空间中Penot的紧致性及其应用

在这项工作中，我们研究了超度量空间中Penot意义上的紧致性。然后，我们证明球形完整性恰好是为凸度结构引入的Penot紧致性。球面完整性属性使某些数学家误以为它具有度量空间中的超凸性。作为应用，我们讨论了球形完整超度量空间中的一些不动点结果。