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Metrizability of Finite Dimensional Spaces with a Binary Convexity

Published online by Cambridge University Press:  20 November 2018

M. Van de Vel
M. Van de Vel*
Affiliation:
Vrije Universiteit Amsterdam, Amsterdam, The Netherlands
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A convex structure consists of a set X, together with a collection of subsets of X, which is closed under intersection and under updirected union. The members of are called convex sets, and is a convexity on X. Fox A a subset of X, h (A) denotes the (convex) hull of A. If A is finite, then h(A) is called a polytope, is called a binary convexity if each finite collection of pairwise intersecting convex sets has a nonempty intersection. See [8], [21] for general references.

If X is also equipped with a topology, then the corresponding weak topology is the one generated by the convex closed sets. It is usually assumed that at least all polytopes are closed. is called normal provided that for each two disjoint convex closed sets C, D there exist convex closed sets C′, D′, with

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 1 , 01 February 1986 , pp. 1 - 18
Copyright
Copyright © Canadian Mathematical Society 1986

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