Subbases for N-ary Convexities

Evgeny Sharikov

doi:10.68381/jca16020

Abstract

We introduce a notion of N-connectedness of a topological space with respect to a convexity on this space. For a given collection

\mathcal{H}

of subsets of a set X we introduce different convexities and topologies on both

\mathcal{H}

and X. Then we prove that the convexity

\mathcal{G}

generated by

\mathcal{H}

is of arity N whenever

\mathcal{H}

is connected and X is N-connected.

Author information

Contact details are reproduced from the original publication and may be historical.

Evgeny Sharikov

Centre for Informatics and Applied Optimization, School of Information Technology and Mathematical Sciences, University of Ballarat, Victoria, Australia

e.sharikov@mail.ru

Keywords

Axiomatic convexity
convexities of arity N
subbases for convexities and topologies

Mathematics Subject Classification

52A01

References

A. M. Rubinov: Abstract Convexity and Global Optimization, Kluwer, Dordrecht (2000).https://doi.org/10.1007/978-1-4757-3200-9 Google Scholar
E. V. Sharikov: About subdifferential calculus for abstract convex functions, J. Nonlinear Convex Anal. 8(2) (2007) 257–275.Google Scholar
E. V. Sharikov: Separation properties via connectedness of topological convexity spaces, Pac. J. Optim., submitted.Google Scholar
V. P. Soltan: Introduction to the Axiomatic Theory of Convexity, Shtiintsa, Kishinev (1984), (in Russian).Google Scholar
M. van de Vel: Theory of Convex Structures, North-Holland, Amsterdam (1993).Google Scholar

Suggested citation

Published by Heldermann Verlag, 2009. Rights now held by Banach Press LLC.

[1] A. M. Rubinov: Abstract Convexity and Global Optimization, Kluwer, Dordrecht (2000).https://doi.org/10.1007/978-1-4757-3200-9 Google Scholar

[2] E. V. Sharikov: About subdifferential calculus for abstract convex functions, J. Nonlinear Convex Anal. 8(2) (2007) 257–275.Google Scholar

[3] E. V. Sharikov: Separation properties via connectedness of topological convexity spaces, Pac. J. Optim., submitted.Google Scholar

[4] V. P. Soltan: Introduction to the Axiomatic Theory of Convexity, Shtiintsa, Kishinev (1984), (in Russian).Google Scholar

[5] M. van de Vel: Theory of Convex Structures, North-Holland, Amsterdam (1993).Google Scholar