The Existence of a Discontinuous Homomorphism Requires a Strong Axiom of Choice

• Main Author: Andersen, Michael Steven
• Format: Others
• Published: BYU ScholarsArchive 2014
• Subjects: inscrutable; inscrutability; scrutable; scrutability; axiom of choice; discontinuous; locally trivial; non-locally trivial; kernel invariance; shelah; pawlikowski; countable choice; choice; arbitrary choice; dependent choice; discontinuity; Mathematics
• Online Access: https://scholarsarchive.byu.edu/etd/4299; https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=5298&context=etd

Conner and Spencer used ultrafilters to construct homomorphisms between fundamental groups that could not be induced by continuous functions between the underlying spaces. We use methods from Shelah and Pawlikowski to prove that Conner and Spencer could not have constructed these homomorphisms with a weak version of the Axiom of Choice. This led us to define and examine a class of pathological objects that cannot be constructed without a strong version of the Axiom of Choice, which we call the class of inscrutable objects. Objects that do not need a strong version of the Axiom of Choice are scrutable. We show that the scrutable homomorphisms from the fundamental group of a Peano continuum are exactly the homomorphisms induced by a continuous function.We suspect that any proposed theorem whose proof does not use a strong Axiom of Choice cannot have an inscrutable counterexample.