Cardinality of some convex sets and of their sets of extreme points

Zbigniew Lipecki¹

¹ Institute of Mathematics Polish Academy of Sciences Wrocław Branch Kopernika 18 51-617 Wrocław, Poland

Colloquium Mathematicum, Tome 123 (2011) no. 1, pp. 133-147

Résumé

We show that the cardinality ${\mathfrak n}$ of a compact convex set $W$ in a topological linear space $X$ satisfies the condition that ${\mathfrak n}^{\aleph_0} = {\mathfrak n}$. We also establish some relations between the cardinality of $W$ and that of $\mathop{\rm extr}\nolimits{W}$ provided $X$ is locally convex. Moreover, we deal with the cardinality of the convex set $E(\mu)$ of all quasi-measure extensions of a quasi-measure $\mu$, defined on an algebra of sets, to a larger algebra of sets, and relate it to the cardinality of $\mathop{\rm extr}\nolimits E(\mu)$.

Détail

DOI : 10.4064/cm123-1-10

Keywords: cardinality mathfrak compact convex set topological linear space satisfies condition mathfrak aleph mathfrak establish relations between cardinality mathop extr nolimits provided locally convex moreover cardinality convex set quasi measure extensions quasi measure defined algebra sets larger algebra sets relate cardinality mathop extr nolimits