
theorem Th16:
  for X being non empty TopSpace, Y being compact non empty
  TopSpace, x being Point of X, Z being Subset of [:Y, X:] st Z = [:[#]Y, {x}:]
  holds Z is compact
proof
  let X be non empty TopSpace, Y be compact non empty TopSpace, x be Point of
  X, Z be Subset of [:Y, X:];
  reconsider V = {x} as non empty Subset of X;
  Y, [: Y, X | V :] are_homeomorphic by Lm3;
  then
A1: [:Y, X | V:] is compact by Th14;
  assume
A2: Z = [:[#]Y, {x}:];
  then the TopStruct of [:Y, X | V:] = the TopStruct of ([:Y, X:] | Z) by Lm4;
  hence thesis by A2,A1,COMPTS_1:3;
end;
