A unified completion of ⊤-filter spaces

This paper develops a unified approach to the completion of ⊤-filter spaces. First, a novel equivalence relation on a ⊤-filter structure is introduced, which serves as the foundation for defining ⊤-pre-Cauchy spaces. This equivalence relation establishes the connection between ⊤-filter spaces and ⊤-convergence spaces, and provides a rigorous basis for defining completeness in ⊤-filter spaces. On this basis, an equivalence-embedding completion of a ⊤-filter space is constructed, together with corresponding extension theorems. Subsequently, the completion method is applied to both ⊤-pre-Cauchy and ⊤-Cauchy spaces. Moreover, alternative completion methods for these spaces are introduced, and a detailed comparison of their interrelations is carried out. In particular, the finest T1 equivalence-embedding completion is characterized in the setting of ⊤-pre-Cauchy spaces.