The theory of X-normed spaces over non-Archimedean valued fields with valuations of higher rank was introduced by H. Ochsenius and W. H. Schikhof in [9] and further developed in [10–12, 16, 17] and [13]. In order to obtain results like the Open Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem, H. Ochsenius and W. H. Schikhof used 1st countability conditions in the value group of the based field. In this article the author develops a new tool to work with transfinite induction simplifying the techniques employed in X-normed spaces, thus accomplishing a Generalized Baire Category Theorem that allows the proof of an Open Mapping Theorem for X-normed spaces without restrictions on the value group of the based field. Additionally, some contributions to the theory of X-normed spaces are presented regarding quotient spaces.

中文翻译：

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X-范数空间的广义开映射定理

H. Ochsenius 和 WH Schikhof 在 [9] 中引入了非阿基米德值域上的 X 范数空间理论，并在 [10-12, 16, 17] 和 [13] 中进一步发展。为了得到开映射定理、闭图定理和一致有界定理等结果，H. Ochsenius 和 WH Schikhof 在基域的值组中使用了第一可数条件。在本文中，作者开发了一种新工具来处理超限归纳，简化 X 范数空间中采用的技术，从而实现广义贝尔范畴定理，该定理允许证明 X 范数空间的开放映射定理，而对值没有限制基于字段的组。此外，还介绍了有关商空间的对 X 范数空间理论的一些贡献。