In this paper, we establish the link between continuous lattices and closure spaces. By generalizing the notion of algebraic closure space to continuous closure space, we show that continuous lattices can be represented by continuous closure spaces, just as algebraic lattices can be represented by algebraic closure spaces. We also introduce the notion of approximable mappings between continuous closure spaces, which produces the category equivalent to that of continuous lattices with Scott-continuous functions. These results demonstrate the capability of closure spaces in representing continuous domains.

中文翻译：

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基于闭合空间的连续格子的表示

在本文中，我们建立了连续格子和闭合空间之间的联系。通过将代数闭空间的概念推广到连续闭空间，我们表明连续格可以用连续闭空间表示，就像代数格可以用代数闭空间表示一样。我们还介绍了连续闭包空间之间的近似映射的概念，它产生的范畴等同于具有 Scott 连续函数的连续格的范畴。这些结果证明了封闭空间在表示连续域方面的能力。