 reserve
  S for non empty TopSpace,
  T for LinearTopSpace,
  X for non empty Subset of the carrier of S;
 reserve
    S,T for RealNormSpace,
    X for non empty Subset of the carrier of S;

theorem Th28:
for X being NormedLinearTopSpace,
    V being Subset of X
holds
V is closed iff
for s1 being sequence of X st rng s1 c= V & s1 is convergent holds
   lim s1 in V
proof
let X be NormedLinearTopSpace;
let V be Subset of X;
consider RNS be RealNormSpace such that
A1:  RNS = the NORMSTR of X
  & the topology of X = the topology of TopSpaceNorm RNS by Def7;
reconsider V0 = V as Subset of TopSpaceNorm RNS by A1;
reconsider V1 = V as Subset of RNS by A1;
hereby assume V is closed; then
([#] X) \ V is open; then
([#] (TopSpaceNorm RNS)) \ V0 is open by A1; then
V0 is closed; then
A2: V1 is closed by NORMSP_2:15;
thus for s1 being sequence of X st rng s1 c= V & s1 is convergent holds
   lim s1 in V
proof
  let s1 be sequence of X;
  assume A3: rng s1 c= V & s1 is convergent;
  reconsider s2=s1 as sequence of RNS by A1;
  s2 is convergent & lim s1=lim s2 by A1,A3,Th26;
  hence lim s1 in V by A2,A3;
end;
end;
assume
A4: for s1 being sequence of X st rng s1 c= V & s1 is convergent holds
   lim s1 in V;
for s1 being sequence of RNS st rng s1 c= V1 & s1 is convergent holds
   lim s1 in V1
proof
  let s2 be sequence of RNS;
  assume A5: rng s2 c= V1 & s2 is convergent;
  reconsider s1=s2 as sequence of X by A1;
  A6: s1 is convergent by A5,A1,Th27; then
  lim s1=lim s2 by A1,Th26;
   hence lim s2 in V1 by A4,A6,A5;
end; then
V1 is closed; then
V0 is closed by NORMSP_2:15; then
([#] (TopSpaceNorm RNS)) \ V0 is open; then
([#] X) \ V is open by A1;
hence V is closed;
end;
