The Cartesian closedness of c-spaces

Xiaolin Xie; Hui Kou; Xiaolin Xie; Hui Kou

doi:10.3934/math.2022891

AIMS Mathematics

2022, Volume 7, Issue 9: 16315-16327. doi: 10.3934/math.2022891

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Research article

The Cartesian closedness of c-spaces

Xiaolin Xie ,
Hui Kou ^,

Department of Mathematics, Sichuan University, Chengdu, 610064, China

Received: 16 April 2021 Revised: 22 June 2022 Accepted: 29 June 2022 Published: 04 July 2022
MSC : 06B35, 54A20, 54B30, 54H10

Directed space was defined by Hui Kou in 2014 ^[21], which is equivalent to $ T_0 $ monotone determined space. Its main purpose is to build an extended framework for domain theory. In this paper, we study the category of c-spaces which is a subcategory of directed spaces. The main results are: (1) we will describe c-spaces using a new definition, which give us the convenience to construct new classes of spaces; (2) we give some conditions such that categorical products and topological products agree in $ {\bf Dtop} $; (3) the category of c-spaces is not Cartesian closed; (4) we define a new class of spaces, namely, FS-spaces, which forms a Cartesian closed category.
- dcpo,
- domain,
- directed spaces,
- continuous spaces,
- monotone determined spaces,
- c-spaces
Citation: Xiaolin Xie, Hui Kou. The Cartesian closedness of c-spaces[J]. AIMS Mathematics, 2022, 7(9): 16315-16327. doi: 10.3934/math.2022891

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Abstract

Directed space was defined by Hui Kou in 2014 ^[21], which is equivalent to $ T_0 $ monotone determined space. Its main purpose is to build an extended framework for domain theory. In this paper, we study the category of c-spaces which is a subcategory of directed spaces. The main results are: (1) we will describe c-spaces using a new definition, which give us the convenience to construct new classes of spaces; (2) we give some conditions such that categorical products and topological products agree in $ {\bf Dtop} $; (3) the category of c-spaces is not Cartesian closed; (4) we define a new class of spaces, namely, FS-spaces, which forms a Cartesian closed category.

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