内部邻里空间
Internal Neighbourhood Spaces
摘要：
本文的主要目的是给出具有有限余积的完备范畴中的邻域算子的一个刻画和一个适当的分解系统，使得容许子余积的半格构成分配完全格.证明了邻域、Kuratowski内算子和伪框架集之间的等价性。此外，内部邻域的类别是拓扑的。描述了邻域范畴的正则满态，并探讨了保证正则满态遗传的条件。证明了拓扑空间的内邻域范畴是双拓扑空间范畴，而在locale范畴中，每个locale都具有一个自然的内拓扑.
英文摘要：
The main aim of this paper is to provide a description of neighbourhood operators in finitely complete categories with finite coproducts and a proper factorisation system such that the semilattice of admissible subobjects make a distributive complete lattice. The equivalence between neighbourhoods, Kuratowski interior operators and pseudo-frame sets is proved. Furthermore the categories of internal neighbourhoods is shown to be topological. Regular epimorphisms of categories of neighbourhoods are described and conditions ensuring hereditary regular epimorphisms are probed. It is shown the category of internal neighbourhoods of topological spaces is the category of bitopological spaces, while in the category of locales every locale comes equipped with a natural internal topology.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
