reserve X for non empty TopSpace;
reserve x for Point of X;
reserve U1 for Subset of X;

theorem Th15:
  X is locally_connected implies
  for S being Subset of X st S is a_component holds S is open
proof
  assume
A1: X is locally_connected;
  let S be Subset of X such that
A2: S is a_component;
  now
    let p be object;
    assume
A3: p in S;
    then reconsider x=p as Point of X;
A4: [#] X is a_neighborhood of x by Th3;
    X is_locally_connected_in x & S is_a_component_of [#] X by A1,A2,Th11;
    then S is a_neighborhood of x by A3,A4,Th13;
    hence p in Int S by Def1;
  end;
  then Int(S) c= S & S c= Int S by TOPS_1:16;
  then Int S = S by XBOOLE_0:def 10;
  hence thesis;
end;
