reserve X for set,
        A for Subset of X,
        R,S for Relation of X;
reserve QUS for Quasi-UniformSpace;
reserve SUS for Semi-UniformSpace;
reserve T for TopSpace;

theorem Th32:
  for T being non empty TopSpace holds
  Family_open_set(FMT_induced_by(Pervin_quasi_uniformity T)) =
  the topology of T
  proof
    let T be non empty TopSpace;
A1: Family_open_set(FMT_induced_by(Pervin_quasi_uniformity T)) c=
      the topology of T
    proof
      let x be object;
      assume x in Family_open_set(FMT_induced_by(Pervin_quasi_uniformity T));
      then consider O be open Subset of
        FMT_induced_by(Pervin_quasi_uniformity T) such that
A2:     x = O;
      reconsider O1 = O as Subset of T;
      for x be set holds x in O1 iff ex B be Subset of T st
        B is open & B c= O1 & x in B
      proof
        let x be set;
        hereby
          assume
A3:       x in O1;
          then reconsider t = x as Element of
            FMT_induced_by(Pervin_quasi_uniformity T);
          consider y being Element of the carrier of
            Pervin_quasi_uniformity T such that
A4:       t = y and
A5:       U_FMT t = Neighborhood y by Th31;
          O in Neighborhood y by A5,FINTOPO7:def 1,A3;
          then consider V be Element of the entourages of
            Pervin_quasi_uniformity T such that
A6:       O = Neighborhood(V,y);
          consider b be Element of
            FinMeetCl(subbasis_Pervin_quasi_uniformity(T)) such that
A7:         b c= V by CARDFIL2:def 8;
          FinMeetCl(subbasis_Pervin_quasi_uniformity(T)) c=
            <.FinMeetCl(subbasis_Pervin_quasi_uniformity(T)).]
            by CARDFIL2:18; then
A16:      b in the entourages of Pervin_quasi_uniformity T;
A17:      [y,y] in id the carrier of Pervin_quasi_uniformity T
            by RELAT_1:def 10;
          Pervin_quasi_uniformity T is axiom_U1; then
A18:      id the carrier of Pervin_quasi_uniformity T c= b by A16;
          reconsider B = {z where z is Element of the carrier of
                           Pervin_quasi_uniformity T:
          [y,z] in b} as set;
          B c= the carrier of Pervin_quasi_uniformity T
          proof
            let t be object;
            assume t in B;
            then ex z be Element of T st t = z & [y,z] in b;
            hence thesis;
          end;
          then reconsider B as Subset of T;
          now
            take B;
            thus B is open by Th30,PRE_TOPC:def 2;
            B c= O
            proof
              let t be object;
              assume t in B;
              then ex z be Element of T st t = z & [y,z] in b;
              hence thesis by A7,A6;
            end;
            hence B c= O1;
            thus x in B by A4,A18,A17;
          end;
          hence ex B be Subset of T st B is open & B c= O1 &
            x in B;
        end;
        assume ex B be Subset of T st B is open & B c= O1 &
          x in B;
        hence x in O1;
      end;
      hence thesis by A2,PRE_TOPC:def 2,TOPS_1:25;
    end;
    the topology of T c=
      Family_open_set(FMT_induced_by(Pervin_quasi_uniformity T))
    proof
      let x be object;
      assume
A20:  x in the topology of T;
      then reconsider y = x as Subset of T;
      reconsider z = y as Subset of Pervin_quasi_uniformity T;
      for u be Element of FMT_induced_by(Pervin_quasi_uniformity T) st
        u in z holds z in U_FMT u
      proof
        let u be Element of FMT_induced_by(Pervin_quasi_uniformity T);
        assume
A21:    u in z;
        consider w being Element of the carrier of
          Pervin_quasi_uniformity T such that
A22:    u = w and
A23:    U_FMT u = Neighborhood w by Th31;
        reconsider W = block_Pervin_quasi_uniformity(y) as Subset of
          [:the carrier of Pervin_quasi_uniformity T,
            the carrier of Pervin_quasi_uniformity T:];
A24:    W in subbasis_Pervin_quasi_uniformity(T) by A20;
A25:    subbasis_Pervin_quasi_uniformity(T) c=
          FinMeetCl(subbasis_Pervin_quasi_uniformity(T)) by CANTOR_1:4;
        FinMeetCl(subbasis_Pervin_quasi_uniformity(T)) c=
          <.FinMeetCl(subbasis_Pervin_quasi_uniformity(T)).] by CARDFIL2:18;
        then reconsider W as Element of the entourages of
          Pervin_quasi_uniformity T by A25,A24;
        {ww where ww is Element of T: [w,ww] in
          block_Pervin_quasi_uniformity(y)} = y
        proof
          thus {ww where ww is Element of T: [w,ww] in
            block_Pervin_quasi_uniformity(y)} c= y
          proof
            let a be object;
            assume a in {ww where ww is Element of T:
              [w,ww] in block_Pervin_quasi_uniformity(y)};
            then consider ww be Element of T such that
A27:        a = ww and
A28:        [w,ww] in [:(the carrier of T) \ y,the carrier of T:] \/
              [:the carrier of T,y:];
            [w,ww] in [:(the carrier of T) \ y,the carrier of T:] or
              [w,ww] in [:the carrier of T,y:] by A28,XBOOLE_0:def 3;
            then (w in (the carrier of T) \ y & ww in the carrier of T) or
              (w in the carrier of T & ww in y) by ZFMISC_1:87;
            hence thesis by A27,A21,A22,XBOOLE_0:def 5;
          end;
          let a be object;
          assume
A29:      a in y;
          then reconsider b = a as Element of T;
          [w,b] in [:the carrier of T,y:] by A29,ZFMISC_1:87;
          then [w,b] in block_Pervin_quasi_uniformity(y) by XBOOLE_0:def 3;
          hence thesis;
        end;
        then z = Neighborhood(W,w);
        hence thesis by A23;
      end;
      then z is open Subset of FMT_induced_by(Pervin_quasi_uniformity T)
        by FINTOPO7:def 1;
      hence thesis;
    end;
    hence thesis by A1;
  end;
