
theorem
  for T being 1-element TopSpace holds
  {the carrier of T} is Basis of T &
  {} is prebasis of T & {{}} is prebasis of T
proof
  let T be 1-element TopSpace;
  set BB = {the carrier of T};
A1: the carrier of T c= the carrier of T;
A2: the topology of T = bool the carrier of T by Th9;
  reconsider BB as Subset-Family of T by A1,ZFMISC_1:31;
  set x = the Element of T;
A3: the topology of T = {{}, {x}} by Th9;
A4: the carrier of T = {x} by Th9;
A5: {} c= bool the carrier of T;
A6: {} c= BB;
A7: {} c= the carrier of T;
  reconsider C = {} as Subset-Family of T by XBOOLE_1:2;
  the topology of T c= UniCl BB
  proof
    let a be object;
    assume a in the topology of T;
    then a = {x} & union {{x}} = {x} & BB c= BB & BB c= bool the carrier of
    T or a = {} by A3,TARSKI:def 2,ZFMISC_1:25;
    hence thesis by A4,A5,A6,A7,CANTOR_1:def 1,ZFMISC_1:2;
  end;
  hence
A8: {the carrier of T} is Basis of T by A2,CANTOR_1:def 2,TOPS_2:64;
A9: {} c= the topology of T;
  BB c= FinMeetCl C
  proof
    let x be object;
    assume x in BB;
    then x = the carrier of T by TARSKI:def 1;
    then Intersect C = x by SETFAM_1:def 9;
    hence thesis by CANTOR_1:def 3;
  end;
  hence {} is prebasis of T by A8,A9,CANTOR_1:def 4,TOPS_2:64;
  {} c= the carrier of T;
  then reconsider D = {{}} as Subset-Family of T by ZFMISC_1:31;
A10: D c= the topology of T by A3,ZFMISC_1:7;
  BB c= FinMeetCl D
  proof
    let x be object;
    assume x in BB;
    then
A11: x = the carrier of T by TARSKI:def 1;
A12: Intersect C = the carrier of T by SETFAM_1:def 9;
    C c= D;
    hence thesis by A11,A12,CANTOR_1:def 3;
  end;
  hence thesis by A8,A10,CANTOR_1:def 4,TOPS_2:64;
end;
