The paper studies the topological semigroups that admit the adjunction of a non-isolated absorbing element and the structure and permanence properties of the class AA of topological semigroups admitting this type of adjunctions. No precompact topological group can belong to the class AA. More generally, a subsemigroup X of a compact Hausdorff semigroup K belongs to AA iff X misses the Sushkevich kernel of the closure cl(X) in K. Every non-torsion abelian group belongs to AA when equipped with the discrete topology. The interest in the class AA stems from the question of quasi-uniformizability of semigroups (every topological AA-group, after the adjunction of a non-isolated absorbing element, gives rise to a non-quasi-uniformizable semigroup
