Invariant Ideal Axiom

(joint work with A. Shibakov) We present an axiom relatively consistent with the usual axioms of set theory with profound impact on convergence structure of topological groups. In particular, the axiom implies that all countable sequential groups are definable (have Borel topologies), all separable Fréchet groups are metrizable and all sequential groups are either metrizable or have sequential order omega_1.