
theorem Th9:
  for T being 1-element TopSpace holds
  the topology of T = bool the carrier of T & for x being Point of T holds
  the carrier of T = {x} & the topology of T = {{},{x}}
proof
  let T be 1-element TopSpace;
  thus the topology of T c= bool the carrier of T;
  consider x being Point of T such that
A1: the carrier of T = {x} by TEX_1:def 1;
A2: {} in the topology of T by PRE_TOPC:1;
A3: the carrier of T in the topology of T by PRE_TOPC:def 1;
A4: bool {x} = {{},{x}} by ZFMISC_1:24;
  hence bool the carrier of T c= the topology of T by A1,A2,A3,ZFMISC_1:32;
  let a be Point of T;
  a = x by STRUCT_0:def 10;
  hence the carrier of T = {a} & the topology of T c= {{},{a}} &
  {{},{a}} c= the topology of T by A1,A2,A3,A4,ZFMISC_1:32;
end;
