The Characterization of the Continuity of Topologies1

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 = id T and f −1 · f = id S 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 Y) X , 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 Y) X , and f be a map from T into S. If f is uncurrying, one-to-one, and onto, then f −1 is currying. (4) Let X, Y be non empty sets, Z be a non empty poset, S be a non empty full relational substructure of Z [: X,Y :] , T be a non empty full relational substructure of (Z Y) X , and f be a map from S into T. If f is currying, one-to-one, and onto, then f is isomorphic. (5) Let X, Y be non empty sets, Z be a non empty poset, T be a non empty full relational substructure of Z [: X,Y :] , S be a non empty full relational substructure of (Z Y) X , and f be a map from S into T. If f is uncurrying, one-to-one, and onto, then f is isomorphic. (i) the relational structure of S 1 = the relational structure of S 2 , and (ii) the relational structure of T 1 = the relational structure of T 2. Let f be a map from S 1 into T 1. Suppose f is isomorphic. Let g be a map from S 2 into T 2. If g = f , then g is isomorphic.