reserve x,X,X2,Y,Y2 for set;
reserve GX for non empty TopSpace;
reserve A2,B2 for Subset of GX;
reserve B for Subset of GX;

theorem
  for B being Subset of GX, p be Point of GX st p in B holds
  Component_of(p,B) is connected
proof
  let B be Subset of GX, p be Point of GX;
  assume
A1: p in B;
  then reconsider B9 = B as non empty Subset of GX;
  Component_of Down(p,B9) is connected & Component_of(p,B)=Component_of
  Down(p,B) by A1,Th27;
  hence thesis by CONNSP_1:23;
end;
