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Internal neighbourhood structures

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Abstract

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 is a complete lattice. The equivalence between neighbourhood systems, Kuratowski interior operations and pseudo-frame sets is proved. Furthermore the categories of internal neighbourhood spaces is shown to be topological. Regular epimorphisms of categories of internal neighbourhood spaces are described and conditions ensuring hereditary regular epimorphisms are probed. It is shown the category of internal neighbourhood spaces of topological spaces is the category of bitopological spaces, while in the category of locales every locale comes equipped with a natural internal topology.

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Acknowledgements

I am gratefully indebted: (1) To M. Korostenski-Davies for painstakingly going through the draft version of this document and suggesting several changes. (2) To Z. Janelidze for his stimulating ideas during our talks on several occasions. (3) To A. Razafindrakato, D. Holgate and their students for sharing their experiences. (4) To G. Janelidze for his valuable help, support and guidance at several stages. (5) Last, and most importantly, to the very positive remarks, criticisms and corrections offered by the referee, which has given the present shape of the paper. The errors that still remain are wholly my misconceptions which needs to be purged with time and forbearance.

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  1. Department of Mathematical Sciences, University of South Africa, UNISA Science Campus, corner of Christiaan de Wet & Pioneer Avenue, Florida, Johannesburg, 1709, Gauteng, South Africa

    Partha Pratim Ghosh

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  1. Partha Pratim Ghosh
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Correspondence to Partha Pratim Ghosh.

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Communicated by J. Adámek.

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The research was funded from the Topology Research Chair at the University of South Africa and European Union Horizon 2020 MCSA Irses Project 731143, both of which are thankfully acknowledged.

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Ghosh, P.P. Internal neighbourhood structures. Algebra Univers. 81, 12 (2020). https://doi.org/10.1007/s00012-020-0640-2

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