caa a22 4500 386359652 CHVBK 20180307111911.0 cr unu---uuuuu 161130e198902 xx s 000 0 eng 10.1017/S0004972700027969 doi S0004972700027969 pii (NATIONALLICENCE)cambridge-10.1017/S0004972700027969 Prokop Frank P. Department of Mathematics, The University of Wollongong, Wollongong, N.S.W., Australia Neighbourhood lattices - a poset approach to topological spaces [Elektronische Daten] [Frank P. Prokop] In this paper neighbourhood lattices are developed as a generalisation of topological spaces in order to examine to what extent the concepts of "openness”, "closedness”, and "continuity” defined in topological spaces depend on the lattice structure of P(X), the power set of X. A general pre-neighbourhood system, which satisfies the poset analogues of the neighbourhood system of points in a topological space, is defined on an -semi-lattice, and is used to define open elements. Neighbourhood systems, which satisfy the poset analogues of the neighbourhood system of sets in a topological space, are introduced and it is shown that it is the conditionally complete atomistic structure of P(X) which determines the extension of pre-neighbourhoods of points to the neighbourhoods of sets. The duals of pre-neighbourhood systems are used to generate closed elements in an arbitrary lattice, independently of closure operators or complementation. These dual systems then form the backdrop for a brief discussion of the relationship between preneighbourhood systems, topological closure operators, algebraic closure operators, and Čech closure operators. Continuity is defined for functions between neighbourhood lattices, and it is proved that a function f: X → Y between topological spaces is continuous if and only if corresponding direct image function between the neighbourhood lattices P(X) and P(Y) is continuous in the neighbourhood sense. Further, it is shown that the algebraic character of continuity, that is, the non-convergence aspects, depends only on the properites of pre-neighbourhood systems. This observation leads to a discussion of the continuity properties of residuated mappings. Finally, the topological properties of normality and regularity are characterised in terms of the continuity properties of the closure operator on a topological space. Copyright © Australian Mathematical Society 1989 Bulletin of the Australian Mathematical Society Cambridge University Press 39/1(1989-02), 31-48 0004-9727 39:1<31 1989 39 BAZ https://doi.org/10.1017/S0004972700027969 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1017/S0004972700027969 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Prokop Frank P. Department of Mathematics, The University of Wollongong, Wollongong, N.S.W., Australia NATIONALLICENCE 773 0- Bulletin of the Australian Mathematical Society Cambridge University Press 39/1(1989-02), 31-48 0004-9727 39:1<31 1989 39 BAZ CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge