New class of spaces links topology and completion methods
On U_c-admitting spaces and U_c-completions of T_0 spaces
Logic in Computer Science
Summary
This paper introduces a new kind of mathematical space called U_c-admitting spaces and shows how they relate to other known spaces in topology. The authors prove that these spaces fit into a larger framework called H-sober spaces, which helps understand how they behave. They also identify when a space and its related Smyth power space share this property. Finally, they describe a way to complete any given space into a U_c-admitting one, providing a method for extending spaces in this class.
What this means in practice
- •For topological data analysts: Provide new structure and completion techniques to better analyze and represent T_0 spaces in data modeling.
- •For domain theory practitioners: Use U_c-admitting space properties to guide construction and completion of domains with desired topological features.
A theory result. No direct application yet.
Authors
Kangming Lu, Qingguo Li, Shengwen Wang, Rongqi Xiao
Abstract
In this paper, we introduce a new class of spaces, termed U_c-admitting spaces, and investigate their relationships with U_s-admitting spaces and well-filtered spaces. Adopting the perspective of H-sobriety, we show that every U_c-admitting space is H-sober, so that U_c-admitting spaces arise as a special case of H-sober spaces. Consequently, the category U_c-Sob of U_c-admitting spaces and continuous maps is complete and forms a reflective subcategory of Top_0. In addition, we prove that a T_0 space X is U_c-admitting if and only if its Smyth power space is U_c-admitting. Finally, we present the standard U_c-completion of an arbitrary T_0 space.