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New class of spaces links topology and completion methods

On U_c-admitting spaces and U_c-completions of T_0 spaces

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.

Thu 22 OctLogic in Computer Science

The gist

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.

Open → 2610.30967v1