General topology under the axiom of determinacy: The beauty of topology without choice.

摘要

I investigate the structure of topological spaces in L( R ), the canonical model of the Axiom of Determinacy (under the appropriate large cardinal hypotheses) in particular spaces with a well-ordered point set of size ℵ1 . I define notions of effective Hausdorffness, regularity, normality and first-countability and construct examples of normal spaces which are not effectively normal or Hausdorff. I show that there are no sequential Dowker spaces of cardinality ℵ1 which are effectively Hausdorff. I prove that every metric space of cardinality ℵ1 is a countable union of discrete subspaces. I also prove that every Lindelof space with a well-ordered dense subset of cardinality ℵ1 with no points of countable character is the continuous image of the space beta w 1 of ultrafilters on w1 hence it is well-orderable with cardinality ≤ ℵw .