Continuous Domains for Function Spaces Using Spectral Compactification

TL;DR

The paper addresses computing function spaces over non-core-compact domains by introducing spectral compactification to embed the original space into a core-compact ambient space $\hat{\mathbb{X}}$. A Galois connection between $[\mathbb{X} \to \mathbb{D}]$ and $[\hat{\mathbb{X}} \to \mathbb{D}]$ provides a bridge for computable analysis, with a left adjoint $^*$ and a right adjoint $_*$ linking the spaces. When $\mathbb{X}$ is not core-compact, a bc-domain structure is recovered via step-function bases on the spectral compactification, and the approach is shown equivalent to an abstract-bases construction, ensuring a robust, implementable domain-theoretic framework. The framework is applicable to temporal discretization in IVPs and potentially to PDEs, offering a computable semantics for non-core-compact topologies through spectral compactification and Stone duality. Overall, the work unifies topological compactification, domain theory, and computable analysis to enable rigorous computation on a broader class of function spaces.

Abstract

We introduce a continuous domain for function spaces over topological spaces which are not core-compact. Notable examples of such topological spaces include the real line with the upper limit topology, which is used in solution of initial value problems with temporal discretization, and various infinite dimensional Banach spaces which are ubiquitous in functional analysis and solution of partial differential equations. If a topological space $\mathbb{X}$ is not core-compact and $\mathbb{D}$ is a non-singleton bounded-complete domain, the function space $[\mathbb{X} \to \mathbb{D}]$ is not a continuous domain. To construct a continuous domain, we consider a spectral compactification $\mathbb{Y}$ of $\mathbb{X}$ and relate $[\mathbb{X} \to \mathbb{D}]$ with the continuous domain $[\mathbb{Y} \to \mathbb{D}]$ via a Galois connection. This allows us to perform computations in the native structure $[\mathbb{X} \to \mathbb{D}]$ while computable analysis is performed in the continuous domain $[\mathbb{Y} \to \mathbb{D}]$, with the left and right adjoints used for moving between the two function spaces.

Figures (1)

• Figure 1: The correct semi-continuity of the bounds of the successive approximations cannot be guaranteed, even when the process starts from an initial enclosure with continuous bounds (in green). The curve in magenta depicts the true solution, and the black (respectively, blue) piecewise affine curves represent the upper (respectively, lower) bounds of the enclosures obtained from applying $\Phi$ once (solid) and twice (dashed).

Theorems & Definitions (79)

• definition 1: Domain
• theorem 1
• proof
• definition 2
• proposition 1
• proof
• lemma 1
• proof
• definition 3: Quasi-embedding, Embedding
• proposition 2