The Characterization of the Continuity of Topologies

The following propositions are true: (1) Let S, T be non empty relational structures and f be a map from S into T . Suppose f is one-to-one and onto. Then f ·f−1 = idT and f −1 ·f = idS and f−1 is one-to-one and onto. (2) Let X, Y be non empty sets, Z be a non empty relational structure, S be a non empty relational substructure of Z [: X, Y :], T be a non empty relational substructure of (Z ) , and f be a map from S into T . If f is currying, one-to-one, and onto, then f−1 is uncurrying. (3) Let X, Y be non empty sets, Z be a non empty relational structure, S be a non empty relational substructure of Z [: X, Y :], T be a non empty relational substructure of (Z ) , and f be a map from T into S. If f is uncurrying, one-to-one, and onto, then f−1 is currying.