 reserve n,m for Nat,
         p,q for Point of TOP-REAL n, r for Real;

theorem Th1: :: MFOLD_1:1
  for T being non empty TopSpace holds T , T | [#]T are_homeomorphic
proof
  let X be non empty TopSpace;
  set f = id X;
  A1: dom f = [#]X;
  A2: [#](X | ([#]X))= [#]X by PRE_TOPC:def 5;
  A3: rng f = [#](X | ([#]X)) by PRE_TOPC:def 5;
  reconsider XX=X|([#]X) as non empty TopSpace;
  reconsider f as Function of X, XX by A2;
ZZ:  for P being Subset of X st P is closed holds (f")"P is closed
  proof
    let P be Subset of X;
    assume P is closed; then
    A4: ([#]X) \ P in the topology of X by PRE_TOPC:def 2,def 3;
    A5: for x being object holds x in (f")"P iff x in P
    proof
      let x be object;
      hereby
        assume A6: x in (f")"P;
        x in f.:P by A6,A3,TOPS_2:54; then
        consider y be object such that
        A7: [y,x] in f & y in P by RELAT_1:def 13;
        thus x in P by A7,RELAT_1:def 10;
      end;
      assume A8: x in P; then
      [x,x] in id X by RELAT_1:def 10; then
      x in f.:P by A8,RELAT_1:def 13;
      hence x in (f")"P by A3,TOPS_2:54;
    end;
S1: ([#]X) \ P = ([#] (X | ([#]X))) \ (f")"P by A2,A5,TARSKI:2;
    ([#]X) \ P = ([#]X /\ [#]X) \ P
    .= (([#]X) \ P) /\ [#]X by XBOOLE_1:49; then
    ([#]X) \ P in the topology of (X | ([#]X)) by PRE_TOPC:def 4,
      A2,A4; then
    ([#] (X | ([#]X))) \ (f")"P is open by PRE_TOPC:def 2,S1;
    hence (f")"P is closed by PRE_TOPC:def 3;
  end;
S0: f is continuous by JGRAPH_1:45;
  f" is continuous by PRE_TOPC:def 6,ZZ; then
  f is being_homeomorphism by A1,A3,TOPS_2:def 5,S0;
  hence thesis by T_0TOPSP:def 1;
end;
