In the first part of our work we show a condition for a semi-homeomorphism in the sense of Crossley and Hildebrand (s.h.C.H) to be a semi-homeomorphism in the sense of Biswas (s.h.B). Certain relevant examples are provided. Next, we define strong semi-homeomorphisms Via “nice” restrictions of semi-homeomorphisms (“global condition”) and we show that the new class of functions actually coincides with semi-homeomorphisms. Then, in the third part we introduce local semi-homeomorphisms (l.s.h.C.H.) via a corresponding “local condition” for restrictions. A few results pertaining to the preservation of some topological properties under this new class of functions are examined.