
theorem
  for X being Hausdorff non empty TopSpace, E being non empty Subset of
  X st X|E is dense locally-compact holds X|E is open
proof
  let X be Hausdorff non empty TopSpace, E be non empty Subset of X such that
A1: X|E is dense locally-compact;
A2: [#](X|E) = E by PRE_TOPC:8;
  Int E = E
  proof
    thus Int E c= E by TOPS_1:16;
    let a be object;
    assume
A3: a in E;
    then reconsider x = a as Point of X;
    reconsider xx = x as Point of X|E by A3,PRE_TOPC:8;
    set UE = {G where G is Subset of X : G is open & G c= E};
    consider K being a_neighborhood of xx such that
A4: K is compact by A1;
    reconsider KK = K as Subset of X by A2,XBOOLE_1:1;
A5: K c= [#](X|E);
    Int K in the topology of X|E by PRE_TOPC:def 2;
    then consider G being Subset of X such that
A6: G in the topology of X and
A7: Int K = G /\ E by A2,PRE_TOPC:def 4;
A8: G is open by A6;
    for P being Subset of X|E st P=KK holds P is compact by A4;
    then KK is compact by A5,COMPTS_1:2;
    then
A9: Cl(G /\ E) c= KK by A7,TOPS_1:5,16;
    E is dense by A1,A2,TEX_3:def 1;
    then
A10: Cl(G /\ E) = Cl G by A8,TOPS_1:46;
    G c= Cl G by PRE_TOPC:18;
    then G c= K by A10,A9;
    then G c= E by A2,XBOOLE_1:1;
    then
A11: G in UE by A8;
A12: xx in Int K by CONNSP_2:def 1;
    Int K c= G by A7,XBOOLE_1:17;
    then a in union UE by A12,A11,TARSKI:def 4;
    hence thesis by TEX_4:3;
  end;
  hence thesis by A2,TSEP_1:16;
end;
