The largest T₂ ⊤-compactification of ⊤-convergence spaces

The primary objective of this paper is to study the properties of ⊤-closed sets and the largest T₂ ⊤-compactification of ⊤-convergence spaces. Firstly, we study some properties of ⊤-ultrafilters and introduce the concept of a ⊤-closed set and the concept of a ⊤-compact set, examining the relationship between them. Secondly, we present the notion of essentially ⊤-compact ⊤-convergence spaces and explore the necessary and sufficient conditions for a ⊤-convergence space to have the largest T₂ ⊤-compactification. Finally, we construct the Richardson ⊤-compactification of a ⊤-convergence space and identify the necessary and sufficient conditions for the Richardson ⊤-compactification to be the largest T₂ ⊤-compactification within the framework of Kent ⊤-convergence spaces.