The concept of a [semi]
topological quasigroup is defined and the notions of induced groupoids and isotopes
are extended to the topological case. Necessary and sufficient conditions are found in
order for a continuously induced isotope of a semitopological quasigroup to be a
semitopological quasigroup. Given an injection i of a topological space (A,𝒜) into a
set S acted on by a group, G, a topology 𝒯A on S is introduced in a natural fashion
under which i is continuous. When S = Q is itself a semitopological quasigroup and
G is generated by the left or right translations of Q the continuity or openness
of i can be checked by comparing the topology 𝒯A− with that of Q. In
particular this method is applied in §3 to the study of topologically invariant
subloops.
