Ah! I missed that part. This has been bothering me all day, so thank you for the correction. I guess this provides yet another perspective on how/why a model complete theory may fail to eliminate quantifiers: for model complete theories, the above statement only holds whenever A also models T. For theories with qe, the above statement holds for all such substructures A.

I understand that continuity is not important because of its inverse-image-ness, but because it corresponds to the geometric notion. But WHY does the inverse-image-ness correspond to this geometric notion? It seems that there are seemingly-unrelated different geometric notions that ALSO corresponds to inverse-image-ness (such as measurable functions). I suppose the question is, why the prevalence of inverse-image-ness?