reserve A, B for non empty set,
  A1, A2, A3 for non empty Subset of A;
reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X0, X1, X2, Y1, Y2 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace;
reserve f for Function of X,Y;
reserve X,Y,Z for non empty TopSpace;
reserve f for Function of X,Y,
  g for Function of Y,Z;
reserve X, Y for non empty TopSpace,
  X0 for non empty SubSpace of X;
reserve f for Function of X,Y;
reserve f for Function of X,Y,
  X0 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace,
  X0, X1 for non empty SubSpace of X;
reserve f for Function of X,Y,
  g for Function of X0,Y;
reserve X0, X1, X2 for non empty SubSpace of X;
reserve f for Function of X,Y,
  g for Function of X0,Y;
reserve X for non empty TopSpace,
  H, G for Subset of X;
reserve A for Subset of X;
reserve X0 for non empty SubSpace of X;

theorem Th102:
  the carrier of (X modified_with_respect_to X0) = the carrier of
  X & for A being Subset of X st A = the carrier of X0 holds the topology of (X
  modified_with_respect_to X0) = A-extension_of_the_topology_of X
proof
  set Y = X modified_with_respect_to X0;
  reconsider A = the carrier of X0 as Subset of X by TSEP_1:1;
A1: Y = X modified_with_respect_to A by Def10;
  hence the carrier of (X modified_with_respect_to X0) = the carrier of X;
  thus thesis by A1;
end;
