⊤-ultrafilters and their applications in ⊤-convergence spaces

Ultrafilters serve as an important tool for studying compactness and Choquet convergence structures in classical convergence spaces. In the framework of ⊤-convergence spaces, we provide three characterizations of ⊤-ultrafilters and consider their applications from three aspects. Firstly, we use ⊤-ultrafilters to study the ⊤-compactness of a ⊤-convergence space, including the Tychonoff theorem and the relationships between the compactness of a classical convergence space and its induced ⊤-convergence space. Secondly, we use ⊤-ultrafilters to construct the one-point T₂ ⊤-compactification of a ⊤-convergence space and present the necessary and sufficient conditions for one-point T₂ ⊤-compactification to be the smallest. Finally, we employ ⊤-ultrafilters to define Choquet ⊤-convergence spaces and investigate their function spaces as well as their relationships with other types of ⊤-convergence spaces.