Closure, interior and neighbourhood in a category

Year 2018, Volume: 47 Issue: 6, 1512 - 1520, 12.12.2018

David Holgate Josef Slapal

https://izlik.org/JA66XG78PW

Abstract

The natural correspondences in topology between closure, interior and neighbourhood no longer hold in an abstract categorical setting where subobject lattices are not necessarily Boolean algebras.  We analyse three canonical correspondences between closure, interior and neighbourhood operators in a category endowed with a subobject structure. While these correspondences coincide in general topology, the analysis highlights subtle differences which distinguish different approaches taken in the literature.

Keywords

Categorical closure operator , interior operator , neighbourhood operator

References

• Adamek, J., Herrlich, H., Strecker, G.E. Abstract and concrete categories, Whiley, 1990.
• Castellini, G. Categorical closure operators, Mathematics: Theory & Applications Birkhauser Boston Inc., Boston, MA, 2003.
• Castellini, G. Interior operators in a category: idempotency and heredity, Topology Appl. 158 (17), 2332--2339, 2011.
• Castellini, G. Some remarks on interior operators and the functional property, Quaest. Math. 39 (2), 275--287, 2016.
• Castellini, G., Holgate, D. Closure operator constructions depending on one parameter, Quaest. Math. 26 (3), 289--305, 2003.
• Castellini, G., Koslowski, J., Strecker, G.E. An approach to the dual of regular closure operators, Cahiers Topologie Géom. Différentielle Catégoriques 35, 109--128, 1994.
• Clementino, M. M., Giuli, E., Tholen, W. A functional approach to general topology, in: Categorical foundations 97, Encyclopedia Math. Appl., 103--163, Cambridge Univ. Press, Cambridge, 2004.
• Clementino, M. M., Gutierres, G. On regular and homological closure operators, Cah. Topol. Géom. Différ. Catég. 51 (2), 127--142, 2010.
• Dikranjan, D., Giuli, E. Closure operators I, Topology Appl. 27 (2), 129--143, 1987.
• Dikranjan, D., Giuli, E., Tholen, W. Closure operators II, in: Categorical topology and its relation to analysis, algebra and combinatorics, Prague, 1988, 297--335, World Sci. Publishing, Teaneck, NJ, 1989.
• Dikranjan, D., Tholen, W. Categorical structure of closure operators, Kluwer Academic Publishers Group, Dordrecht, 1995.
• Giuli, E., Slapal, J. Raster convergence with respect to a closure operator, Cah. Topol. Géom. Différ. Catég. 46 (4), 275--300, 2005.
• Giuli, E., Slapal, J. Neighborhoods with respect to a categorical closure operator, Acta Math. Hungar. 124 (1-2), 1--14, 2009.
• Giuli, E., Tholen, W. Openness with respect to a closure operator, Appl. Categ. Structures 8 (3), 487--502, 2000.
• Holgate, D., Slapal, J. Categorical neighborhood operators, Topology Appl. 158 (17), 2356--2365, 2011.
• Razafindrakoto, A., Holgate, D. Interior and neighbourhood, Topology Appl. 168, 144--152, 2014.
• Slapal, J., Net spaces in categorical topology, Ann. N. Y. Acad. Sci. 806, 393--412, 1996.
• Slapal, J., Neighborhoods and convergence with respect to a closure operator, Math. Slovaca 61 (5), 717--732, 2011.
• Slapal, J., Compactness and convergence with respect to a neighborhood operator, Collect. Math. 63 (2), 123--137, 2012.
• Vorster, S. J. R., Interior operators in general categories, Quaest. Math. 23 (4), 405--416, 2000.