
theorem
  for T being non empty TopSpace st T is locally-compact & T is T_2
  for x,y being Element of InclPoset the topology of T st x << y
  ex Z being Subset of T st Z = x & Cl Z c= y & Cl Z is compact
proof
  let T be non empty TopSpace such that
A1: T is locally-compact and
A2: T is T_2;
  set L = InclPoset the topology of T;
A3: L = RelStr(#the topology of T, RelIncl the topology of T#)
  by YELLOW_1:def 1;
  let x,y be Element of L;
  assume x << y;
  then consider Z being Subset of T such that
A4: x c= Z and
A5: Z c= y and
A6: Z is compact by A1,Th39;
  x in the topology of T by A3;
  then reconsider X = x as Subset of T;
  take X;
  thus X = x;
  Cl X c= Z by A2,A4,A6,TOPS_1:5;
  hence thesis by A5,A6,COMPTS_1:9;
end;
