reserve T,T1,T2 for TopSpace,
  A,B for Subset of T,
  F for Subset of T|A,
  G,G1, G2 for Subset-Family of T,
  U,W for open Subset of T|A,
  p for Point of T|A,
  n for Nat,
  I for Integer;
reserve Af for finite-ind Subset of T,
  Tf for finite-ind TopSpace;

theorem Th26:
  for A1 be Subset of T1,A2 be Subset of T2 st A1,A2
  are_homeomorphic holds A1 is finite-ind iff A2 is finite-ind
proof
  let A1 be Subset of T1,A2 be Subset of T2;
  assume A1,A2 are_homeomorphic;
  then
A1: T1|A1,T2|A2 are_homeomorphic by METRIZTS:def 1;
  hereby
    assume A1 is finite-ind;
    then T2|A2 is finite-ind by A1,Lm9;
    hence A2 is finite-ind by Th18;
  end;
  assume A2 is finite-ind;
  then T1|A1 is finite-ind by A1,Lm9;
  hence thesis by Th18;
end;
