The notion of topological free entropy dimension of $n-$tuples of elements in
a unital C$^*$ algebra was introduced by Voiculescu. In the paper, we compute
topological free entropy dimension of one self-adjoint element and topological
orbit dimension of one self-adjoint element in a unital C$^*$ algebra.
Moreover, we calculate the values of topological free entropy dimensions of
families of generators of some unital C$^*$ algebras (for example: irrational
rotation C$^*$ algebras or minimal tensor product of two reduced C$^*$ algebras
of free groups).
