Topological convexity spaces

Victor Bryant
1974 Proceedings of the Edinburgh Mathematical Society  
Preliminaries We shall start by recalling the definition and some basic properties of a convexity space; a topological convexity space (tcs) will then be a convexity space together with an admissible topology, and will be a generalisation of a topological vector space (tvs). After showing that the usual tvs results connecting the linear and topological properties extend to this new setting we then prove a form of the Krein-Milman theorem in a tcs. Let J b e a non-empty set, let a, b, ... be
more » ... ents of X, and let A, B, ... be subsets of X. We do not distinguish between a point of X and the singleton subset which it defines. Thus in X the notation e is redundant and is replaced by <=. Also we write AxB for A meets B or AnB / 0. A join on A" is a mapping •: Xx X->2 X , i.e. it associates with each ordered pair (a, b) of elements of A' a subset a-b (or simply ab) of X. Given a join and a, bcX we define a/b = {c<= X: acbc). The operations • and / can be extended to subsets of X in the obvious way, for example AB = (J ab. In particular for a, b, c<=. X, a{bc) is a subset of X. Note also that A xBC if and only if A/BxC. Definition. A pair (X, •) is a convexity space if • is a join on X satisfying (i) ab ¥= 0, alb # 0 ; (ii) aa = a = aja; (iii) a(bc) = (ab)c; (iv) a/bxc/d=>adxbc; for all a, b, c, dc X. The most familiar example of a convexity space is a real vector space with join defined by ab = {Xa + (l -X)b: O<A<1}, and other examples are given in (2). If (X, •) and (Y, •) are convexity spaces and a join is defined on Xx Y by (a, b).(c, d) = a.cxb.d (a, ccX; b, d<= Y), then it is easily verified that this product space is a convexity space. Although the above axioms are algebraic in nature they have a strong geometric motivation based on the vector space example. The properties of joins are discussed in more detail in (2), (5) and some consequences of these axioms are given in (1), (3). We have included here only those properties necessary for our subsequent study of topologies on (X, •). One consequence of the axiom aa = a = a/a is that a<=-ab if and only
doi:10.1017/s0013091500010245 fatcat:wzgpo3fzdbeo7ngvncyg7uzsfu