reserve T   for TopSpace,
        A,B for Subset of T;
reserve NT,NTX,NTY for NTopSpace,
        A,B        for Subset of NT,
        O          for open Subset of NT,
        a          for Point of NT,
        XA         for Subset of NTX,
        YB         for Subset of NTY,
        x          for Point of NTX,
        y          for Point of NTY,
        f          for Function of NTX,NTY,
        fc         for continuous Function of NTX,NTY;
reserve NT for T_2 NTopSpace;
reserve T   for non empty TopSpace,
        A,B for Subset of T,
        F   for closed Subset of T,
        O   for open Subset of T;
reserve T   for non empty strict TopSpace,
        A,B for Subset of T,
        x   for Point of T;
reserve A for Subset of FMT_R^1,
        x for Point of FMT_R^1,
        y for Point of RealSpace,
        z for Point of TopSpaceMetr RealSpace,
        r for Real;

theorem Th58:
  r > 0 implies ].x - r , x + r.[ is a_neighborhood of x
  proof
    assume
A1: r > 0;
    set S  = ]. x - r, x + r.[,
        BA = {].a,b.[ where a,b is Real: a < b};
    now
      x < x + r & x - r < x by A1,XREAL_1:29,44;
      then x - r < x + r by XXREAL_0:2;
      hence S in BA;
      thus BA c= Family_open_set(FMT_R^1) by Th57,FINTOPO7:def 14;
    end;
    then S in Family_open_set(FMT_R^1);
    then S in the set of all O where O is open Subset of FMT_R^1
      by FINTOPO7:def 11;
    then ex O be open Subset of FMT_R^1 st S = O;
    then S in U_FMT x by A1,TOPREAL6:15,FINTOPO7:def 1;
    hence thesis by FINTOPO7:def 5;
  end;
