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Convergence in neighbourhood lattices

Published online by Cambridge University Press:  17 April 2009

Frank P. Prokop
Frank P. Prokop
Affiliation:
Department of MathematicsThe University of WollongongWollongong, N.S.W.Australia
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Abstract

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A consideration of the separation properties of pre-neighbourhood lattices, leads to the definition and characterisation of T1-neighbourhood lattices in terms of the properties of the neighbourhood mapping, independently of points. It is then shown that if net convergence is defined in neighbourhood lattices as a consequence of replacing ‘point’ by ‘set’ in topological convergence, then the limits of convergent nets are unique. The relationship between continuity and convergence is established with the proof of the statement that a residuated function between conditionally complete T1-neighbourhood lattices is continuous if and only if it preserves the limit of convergent nets. If P(X) denotes the power set of X, then the observation that a filter in a topological space (X, T) is a net in P(X) leads to a discussion of the net convergence of a filter as a special case of net convergence. Particular attention is paid to maximal filters, Fréchet filters and to the filter generated by the limit elements of a net. Further, if the ‘filter’ convergence of a filter F in a topological space (X, T) is given by , if η(x) ⊆ F, then the relationship between ‘filter’ convergence and the net convergence of a filter in P(X) is established. Finally, it is proved that, in the neighbourhood system ‘lifted’ from a topological space to P(P(X)), the continuous image of a filter which converges to a singleton set is a convergent filter with the appropriate image set as the limit.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 40 , Issue 1 , August 1989 , pp. 129 - 145
Copyright
Copyright © Australian Mathematical Society 1989

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