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.

翻译：本文为拓扑空间上的函数空间引入了一种连续域，这些拓扑空间并非核心紧的。此类拓扑空间的显著例子包括具有上极限拓扑的实数线（用于含时间离散化的初值问题求解），以及泛函分析和偏微分方程求解中普遍存在的各种无限维巴拿赫空间。若拓扑空间 $\mathbb{X}$ 非核心紧且 $\mathbb{D}$ 是非单点的有界完备域，则函数空间 $[\mathbb{X} \to \mathbb{D}]$ 不是连续域。为构造连续域，我们考虑 $\mathbb{X}$ 的一个谱紧化 $\mathbb{Y}$，并通过伽罗瓦连接将 $[\mathbb{X} \to \mathbb{D}]$ 与连续域 $[\mathbb{Y} \to \mathbb{D}]$ 关联起来。这使得我们能够在原生结构 $[\mathbb{X} \to \mathbb{D}]$ 中进行计算，而可计算分析则在连续域 $[\mathbb{Y} \to \mathbb{D}]$ 中执行，左右伴随函子用于在两个函数空间之间进行转换。