Abstract The function space plays an indispensable role in investigating the Cartesian closedness of categories of domains. This paper focuses on the quasicontinuity and topological properties of the function spaces associated with quasicontinuous domains. For any quasicontinuous domain P with property M ⁎ and bounded complete algebraic domain X, it is shown that: (i) the function space [ X → P ] is a quasicontinuous domain; especially, a counterexample is given to illustrate that property M ⁎ is necessary for the quasicontinuity of the function space; (ii) the Isbell topology and Scott topology are equal in the function space [ X → P ] .

中文翻译：

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Isbell 和 Scott 拓扑在准连续域的函数空间上的巧合

摘要 函数空间在研究域范畴的笛卡尔封闭性中起着不可或缺的作用。本文重点研究与准连续域相关的函数空间的准连续性和拓扑性质。对于任何具有性质 M ⁎ 和有界完全代数域 X 的拟连续域 P，证明： (i) 函数空间 [ X → P ] 是一个拟连续域；特别地，给出了一个反例来说明性质 M ⁎ 对于函数空间的拟连续性是必要的；(ii) Isbell 拓扑和 Scott 拓扑在函数空间 [ X → P ] 中相等。