
theorem Th36:
  for T being non empty TopSpace
  for x being Element of InclPoset the topology of T
  for X being Subset of T st x = X holds x is compact iff X is compact
proof
  let T be non empty TopSpace;
  let x be Element of InclPoset the topology of T, X be Subset of T such that
A1: x = X;
  hereby
    assume x is compact;
    then
A2: x << x;
    thus X is compact
    proof
      let F be Subset-Family of T such that
A3:   X c= union F and
A4:   F is open;
      consider G being finite Subset of F such that
A5:   x c= union G by A1,A2,A3,A4,Th34;
      reconsider G as Subset-Family of T by XBOOLE_1:1;
      take G;
      thus G c= F & X c= union G & G is finite by A1,A5;
    end;
  end;
  assume
A6: for F being Subset-Family of T st F is Cover of X & F is open
  ex G being Subset-Family of T st G c= F & G is Cover of X & G is finite;
  now
    let F be Subset-Family of T;
    assume that
A7: F is open and
A8: x c= union F;
    F is Cover of X by A1,A8,SETFAM_1:def 11;
    then consider G being Subset-Family of T such that
A9: G c= F and
A10: G is Cover of X and
A11: G is finite by A6,A7;
    x c= union G by A1,A10,SETFAM_1:def 11;
    hence ex G being finite Subset of F st x c= union G by A9,A11;
  end;
  hence x << x by Th35;
end;
