
theorem Th17:
  for T being non empty TopSpace for A being Subset of T holds
  A is open iff
  for x being Point of T holds x in A implies ex B being Nbhd of x,T st B c= A
proof
  let T be non empty TopSpace;
  let A be Subset of T;
  thus A is open implies for x being Point of T st x in A ex B being Nbhd of x
  ,T st B c= A
  proof
    assume
A1: A is open;
    let x be Point of T;
    assume x in A;
    then ex B being Subset of T st B is a_neighborhood of x & B c= A by A1,
CONNSP_2:7;
    hence thesis;
  end;
  assume
A2: for x being Point of T holds x in A implies ex B being Nbhd of x,T
  st B c= A;
  for x being Point of T st x in A ex B being Subset of T st B is
  a_neighborhood of x & B c= A
  proof
    let x be Point of T;
    assume x in A;
    then ex B being Nbhd of x,T st B c= A by A2;
    hence thesis;
  end;
  hence thesis by CONNSP_2:7;
end;
