
theorem Th45:
  for T being non empty TopSpace, V being open Subset of T holds
  chi(V, the carrier of T) is continuous Function of T, Sierpinski_Space
proof
  let T be non empty TopSpace, V be open Subset of T;
  reconsider c = chi(V, the carrier of T) as Function of T, Sierpinski_Space
  by WAYBEL18:def 9;
A1: c"{0,1} = [#]T by FUNCT_2:40;
  c = chi(c"{1}, the carrier of T) by FUNCT_3:40;
  then
A2: V = c"{1} by FUNCT_3:38;
A3: c"{} = {}T;
A4: now
    let W be Subset of Sierpinski_Space;
    assume W is open;
    then W in the topology of Sierpinski_Space by PRE_TOPC:def 2;
    then W in {{}, {1}, {0,1}} by WAYBEL18:def 9;
    hence c"W is open by A2,A3,A1,ENUMSET1:def 1;
  end;
  [#]Sierpinski_Space <> {};
  hence thesis by A4,TOPS_2:43;
end;
