reserve X for RealNormSpace;

theorem Th8:
  for X be RealNormSpace, x be Point of X,
      r be Real holds {y where
  y is Point of X:||.x-y.|| < r} is open Subset of TopSpaceNorm X
proof
  let X be RealNormSpace, x be Point of X, r be Real;
  reconsider z=x as Element of MetricSpaceNorm X;
  ( ex t be Point of X st t=x & Ball(z,r) = {y where y is Point of X: ||.t
-y.|| < r})& Ball(z,r) in Family_open_set(MetricSpaceNorm X) by Th2,PCOMPS_1:29
  ;
  hence thesis by PRE_TOPC:def 2;
end;
