reserve k, k1, n, n1, m for Nat;
reserve X, y for set;
reserve p for Real;
reserve r for Real;
reserve a, a1, a2, b, b1, b2, x, x0, z, z0 for Complex;
reserve s1, s3, seq, seq1 for Complex_Sequence;
reserve Y for Subset of COMPLEX;
reserve f, f1, f2 for PartFunc of COMPLEX,COMPLEX;
reserve Nseq for increasing sequence of NAT;
reserve h for 0-convergent non-zero Complex_Sequence;
reserve c for constant Complex_Sequence;
reserve R, R1, R2 for C_RestFunc;
reserve L, L1, L2 for C_LinearFunc;

theorem Th11:
  for X being Subset of COMPLEX holds ((for z0 be Complex st z0 in
  X holds ex N be Neighbourhood of z0 st N c= X) implies X is open)
proof
  let X be Subset of COMPLEX;
  assume that
A1: for z0 be Complex st z0 in X holds ex N be Neighbourhood of z0 st N
  c= X and
A2: not X is open;
  not X` is closed by A2;
  then consider s1 be sequence of COMPLEX such that
A3: rng s1 c= X` and
A4: s1 is convergent and
A5: not lim s1 in X`;
  lim s1 in COMPLEX by XCMPLX_0:def 2;
  then lim s1 in X by A5,SUBSET_1:29;
  then consider N be Neighbourhood of (lim s1) such that
A6: N c= X by A1;
  consider g be Real such that
A7: 0 < g and
A8: {y where y is Complex : |.y-(lim s1).| < g} c= N by Def5;
  consider n such that
A9: for m be Nat st n <= m
  holds |.(s1.m)-(lim s1).| < g by A4,A7,
COMSEQ_2:def 6;
  n in NAT by ORDINAL1:def 12;
  then n in dom s1 by FUNCT_2:def 1;
  then
A10: s1.n in rng s1 by FUNCT_1:def 3;
  |.s1.n-(lim s1).| < g by A9;
  then s1.n in {y where y is Complex : |.y-(lim s1).| < g};
  then s1.n in N by A8;
  hence contradiction by A3,A6,A10,XBOOLE_0:def 5;
end;
