let T1, T2 be empty TopSpace; :: thesis: T1,T2 are_homeomorphic
reconsider E = {} as Function of T1,T2 by FUNCT_2:3, RELAT_1:60;
A1: ( [#] T2 = {} iff [#] T1 = {} ) ;
for P being Subset of T1 st P is open holds
(E " ) " P is open ;
then A2: E " is continuous by A1, TOPS_2:55;
for P being Subset of T2 st P is open holds
E " P is open ;
then E is continuous by A1, TOPS_2:55;
then E is being_homeomorphism by A1, A2, RELAT_1:60, TOPS_2:def 5;
hence T1,T2 are_homeomorphic by T_0TOPSP:def 1; :: thesis: verum