Computable Bases

TL;DR

The paper addresses extending admissibility of representations beyond second-countable ${\mathsf{T}}_0$ spaces by introducing computable presubbases, prebases, and bases, and proving a Presubbase Theorem that generalizes the Kreitz–Weihrauch result to topologies generated by compact intersections ${\bigcap_{y\in K} B_y}$. It develops a dual framework of computable bases and a Galois connection between representations and presubbases, establishing that presubbase representations are admissible for the corresponding compact-intersection topology and that prebases yield representations admissible for the generated topology, with a rich set of closure properties. The work shows that the category of computable Kolmogorov spaces is cartesian closed and that hyperspace and function-space topologies arise as sequentializations of standard topologies (e.g., compact-open, Vietoris, Fell), enabling robust, computation-friendly analysis of topological spaces in computable analysis. Overall, the results provide a unified, constructive approach to describing and manipulating topologies via subbase-based descriptions, with a natural pathway to implementations and further theoretical development across generalized bases and represented spaces.

Abstract

In computable analysis typically topological spaces with countable bases are considered. The Theorem of Kreitz-Weihrauch implies that the subbase representation of a second-countable $T_0$ space is admissible with respect to the topology that the subbase generates. We consider generalizations of this setting to bases that are representable, but not necessarily countable. We introduce the notions of a computable presubbase and a computable prebase. We prove a generalization of the Theorem of Kreitz-Weihrauch for the presubbase representation that shows that any such representation is admissible with respect to the topology generated by compact intersections of the presubbase elements. For computable prebases we obtain representations that are admissible with respect to the topology that they generate. These concepts provide a natural way to investigate many topological spaces that have been studied in computable analysis. The benefit of this approach is that topologies can be described by their usual subbases and standard constructions for such subbases can be applied. Finally we discuss a Galois connection between presubbases and representations of $T_0$ spaces that indicates that presubbases and representations offer particular views on the same mathematical structure from different perspectives.

Figures (2)

• Figure 1: Topologies associated to a presubbase $B$ of a represented space.
• Figure 2: Function and hyperspace topologies and their subbases.

Theorems & Definitions (50)

• Definition 1: Admissibility
• Theorem 2: Schröder 2002
• Proposition 3: Admissibility
• Theorem 4: Kreitz-Weihrauch 1985
• Definition 5: Presubbase
• Definition 6: Presubbase representation
• Theorem 7: Presubbase theorem
• Proposition 8: Scott convergence
• Corollary 9: Schröder 2002
• proof : Proof of Theorem \ref{['thm:presubbase']}