reserve i,j,k,n,m for Nat;

theorem Th8:
  for A, B being Subset of TOP-REAL 2 st A is open & B
  is_a_component_of A holds B is open
proof
  let A, B be Subset of TOP-REAL 2 such that
A1: A is open and
A2: B is_a_component_of A;
A3: B c= A by A2,SPRECT_1:5;
A4: the TopStruct of TOP-REAL 2 = TopSpaceMetr Euclid 2 by EUCLID:def 8;
  then reconsider C = B, D = A as Subset of TopSpaceMetr Euclid 2;
A5: D is open by A1,A4,PRE_TOPC:30;
  for p being Point of Euclid 2 st p in C ex r being Real st r>0 &
  Ball(p,r) c= C
  proof
    let p be Point of Euclid 2;
    assume
A6: p in C;
    then consider r being Real such that
A7: r > 0 and
A8: Ball(p,r) c= D by A3,A5,TOPMETR:15;
    reconsider r as Real;
    take r;
    thus r>0 by A7;
    reconsider E = Ball(p,r) as Subset of TOP-REAL 2 by A4,TOPMETR:12;
A9: p in E by A7,GOBOARD6:1;
    then
A10: E is connected by Th7;
    B meets E by A6,A9,XBOOLE_0:3;
    hence thesis by A2,A8,A10,GOBOARD9:4;
  end;
  then C is open by TOPMETR:15;
  hence B is open by A4,PRE_TOPC:30;
end;
