reserve T,T1,T2,S for non empty TopSpace;
reserve p,q for Point of TOP-REAL 2;

theorem
  for X being non empty TopSpace, a being Real holds ex g being
  Function of X,R^1 st (for p being Point of X holds g.p=a) & g is continuous
proof
  let X be non empty TopSpace,a be Real;
  reconsider a1 = a as Element of R^1 by TOPMETR:17,XREAL_0:def 1;
  set g1 = (the carrier of X) --> a1;
  reconsider g0=g1 as Function of X,R^1;
  for p being Point of X,V being Subset of R^1 st g0.p in V & V is open
  holds ex W being Subset of X st p in W & W is open & g0.:W c= V
  proof
    set f1=g0;
    let p be Point of X,V be Subset of R^1;
    assume that
A1: g0.p in V and
    V is open;
    set G1=V;
    f1.: [#]X c= G1
    proof
      let y be object;
      assume y in f1.: [#]X;
      then ex x being object
       st x in dom f1 & x in [#]X & y=f1.x by FUNCT_1:def 6;
      then y=a by FUNCOP_1:7;
      hence thesis by A1,FUNCOP_1:7;
    end;
    hence thesis;
  end;
  then ( for p being Point of X holds g1.p=a)& g0 is continuous by Th10,
FUNCOP_1:7;
  hence thesis;
end;
