Maximal Equivariant Compactifications

The paper’s main theorem (Theorem 4.8) establishes A βG B iff for all neighborhoods V of the identity, VA β VB, and in normal spaces this is equivalent to V cl(A) ∩ V cl(B) ≠ ∅; the proof proceeds via the general proximity/uniformity framework (Theorem 4.1 and the maximal compatible uniformity) and is coherent and correct . The candidate solution proves the same equivalence by an averaging argument using Haar measure to produce RUCG separators and a uniform-continuity contradiction in the converse direction. It is essentially correct once one makes a minor, standard correction to the change-of-variables step (use left translations and F(x)=∫φ(g)f(g^{-1}·x)dμ(g)); it also properly uses local compactness of G and the normal-space reduction. Notably, the paper’s abstract states an existential quantifier (∃V) while the body and proof correctly use a universal quantifier (∀V); this appears to be a minor typographical slip in the abstract that should be fixed .