reserve T for TopSpace;
reserve T for non empty TopSpace;
reserve F for Subset-Family of T;
reserve T for non empty TopSpace;

theorem Th78:
  for F being Subset-Family of T holds F c= Open_Domains_of T iff
  F is open-domains-family
proof
  let F be Subset-Family of T;
  thus F c= Open_Domains_of T implies F is open-domains-family
  proof
    assume
A1: F c= Open_Domains_of T;
    now
      let A be Subset of T;
      assume A in F;
      then A in Open_Domains_of T by A1;
      then A in {P where P is Subset of T : P is open_condensed} by
TDLAT_1:def 9;
      then ex Q being Subset of T st Q = A & Q is open_condensed;
      hence A is open_condensed;
    end;
    hence thesis;
  end;
  thus F is open-domains-family implies F c= Open_Domains_of T
  proof
    assume
A2: F is open-domains-family;
    for X being object holds X in F implies X in Open_Domains_of T
    proof
      let X be object;
      assume
A3:   X in F;
      then reconsider X0 = X as Subset of T;
      X0 is open_condensed by A2,A3;
      then X0 in {P where P is Subset of T : P is open_condensed};
      hence thesis by TDLAT_1:def 9;
    end;
    hence thesis;
  end;
end;
