AI 中文总结
该研究建立了基本零维空间及其基本连续映射的范畴，作为闭包空间等多个范畴的共同推广，得到了原像刻画、Joyal-Tierney型定理，并利用对偶原理关联开闭映射结果，提供了连续性与开性的统一框架。
AI 中文摘要
遵循Erné、Picado和Pultr一篇论文中的建议，我们建立了基本零维空间及其基本连续映射的范畴。该范畴的对象是具有分配闭包系统的闭包空间，该闭包系统包含补元的特定交基，这使得该范畴成为闭包空间、零维拓扑空间、Heyting半格以及locale（框架）范畴的共同推广。我们研究了基本零维空间之间映射的像、原像、连续性、开性与闭性。在主要结果中，我们得到了基本连续映射原像的有用刻画（类似于localic原像映射作为余框架同态的刻画），以及基本开映射的Joyal-Tierney型定理。基本零维空间范畴的一个新颖之处在于其对偶原理，该原理除其他作用外，还能从基本开映射的结果中直接推导出基本闭映射的结果。
英文摘要
Following a suggestion in a paper by Erné, Picado and Pultr, we establish the category of basic zero-dimensional spaces and their basic continuous maps. The objects are closure spaces with a distributive closure system containing a specified meet-base of complemented members that make the category a common generalization of those of closure spaces, zero-dimensional topological spaces, Heyting semilattices, and locales (frames). We treat images, preimages, continuity, openness and closedness of maps between basic zero-dimensional spaces. Among our main results, we have a useful description of preimages for basic continuous maps (analogous to the one for localic preimage maps as coframe homomorphisms), and a Joyal-Tierney type theorem for basic open maps. A novel aspect of the category of basic zero-dimensional spaces is a duality principle that, among other things, enables results for basic closed maps to be obtained for free from those for basic open maps.
Comments26 pages
