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：本文针对非核紧拓扑空间上的函数空间引入了一种连续域。此类拓扑空间的典型例子包括具有上极限拓扑的实数线（用于带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 $\mathbb{X}$ 非核紧且 $\mathbb{D}$ 为非单元素有界完备域，则函数空间 $[\mathbb{X} \to \mathbb{D}]$ 不构成连续域。为构造连续域，我们考虑 $\mathbb{X}$ 的谱紧化 $\mathbb{Y}$，并通过伽罗瓦连接将 $[\mathbb{X} \to \mathbb{D}]$ 与连续域 $[\mathbb{Y} \to \mathbb{D}]$ 建立关联。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，左右伴随函子用于两个函数空间之间的转换。