Remarks on a problem by Efimov
by
Mirna Dzamonja
University of East Anglia, UK
Coauthors: Grzegorz Plebanek

A well known problem of Efimov is if every infinite compact topological space has a convergent sequence or maps onto [0, 1]^w₁. No model is known in which the dichotomy is true, while counterexamples are known in various diamond-like models. Connecting this with another well known (solved) question about compact spaces, Haydon's problem, in such models it is also true that having a nonseparable Radon measure does not guarantee that the space maps onto [0, 1]^w₁ (as it does for example in models where MA + not CH is true). Therefore it makes sense to ask if a weaker dichotomy might be true, does every infinite compact space which does not have a convergent sequence must support a nonseparable Radon measure. We shall discuss the this problem and present some partial results.

Date received: March 6, 2005