The Wallace Problem and Continuity of the Inverse in Pseudocompact Groups

After Bourbaki, a paratopological group is a group G with a topology such that the multiplication map m:G x G - G is continuous. For an infinite cardinal a, a paratopological group G is a-totally bounded if, for each neighbourhood V of the identity element, there is a subset A of G with |A| < α such that G = AV, For a paratopological group (G, τ), τ* is the topology consisting of the inverses of τ-open sets. Then (G, τ) is a topological group iff any of the following hold for all subsets A of G: (i) Cl_rA = Cl_t,A; (ii) ClA c ClA; (iii) ClA a Cl_r,A.