
theorem
  for T being non empty TopSpace st T is regular holds T is
locally-compact iff for x being Point of T ex Y being Subset of T st x in Int Y
  & Y is compact
proof
  let T be non empty TopSpace such that
A1: T is regular;
  hereby
    assume
A2: T is locally-compact;
    let x be Point of T;
    ex Y being Subset of T st x in Int Y & Y c= [#]T & Y is compact by A2;
    hence ex Y being Subset of T st x in Int Y & Y is compact;
  end;
  assume
A3: for x being Point of T ex Y being Subset of T st x in Int Y & Y is compact;
  let x be Point of T, X be Subset of T;
  assume x in X & X is open;
  then
A4: x in Int X by TOPS_1:23;
  consider Y being Subset of T such that
A5: x in Int Y and
A6: Y is compact by A3;
  x in Int X /\ Int Y by A5,A4,XBOOLE_0:def 4;
  then x in Int(X /\ Y) by TOPS_1:17;
  then consider Q being Subset of T such that
A7: Q is closed and
A8: Q c= X /\ Y and
A9: x in Int Q by A1,Th27;
  take Q;
  thus x in Int Q by A9;
  X /\ Y c= X by XBOOLE_1:17;
  hence Q c= X by A8;
  X /\ Y c= Y by XBOOLE_1:17;
  hence thesis by A6,A7,A8,COMPTS_1:9,XBOOLE_1:1;
end;
