Internal Neighbourhood Structures II: Closure and closed morphisms

Document Type : Research Paper


Department of Mathematical Sciences, University of South Africa, Unisa Science Campus, corner of Christian de Wet \& Pioneer Avenue, Florida 1709, Johannesburg, Gauteng, South Afric, National Institute for Theoretical and Computational Sciences (NITheCS), South Africa.


  Internal preneighbourhood spaces inside any finitely complete category with finite coproducts and proper factorisation structure were first introduced in \cite{2020}. This paper proposes a closure operation on internal preneighbourhood spaces and investigates closed morphisms and its close allies. Consequently it introduces  analogues of several well-known classes of topological spaces for preneighbourhood spaces. Some preliminary properties of these spaces are established in this paper. The results of this paper exhibit that preneighbourhood systems are more general than closure operators and conveniently allows identifying properties of classes of morphisms independent of \emph{continuity} of morphisms with respect to induced closure operators.


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