Let A = C[z]¹, then Inv(C[z]) = C-{0}, hence the elements p_n(z) = 1 + [z/n] (n Î N) aren't invertible. But lim_n p_n(z) = 1 Î Inv(C[z]). Therefore A - Inv(A) isn't closed.

Footnotes:

¹The set C[z] of all polynomials in an indeterminate z with complex coefficients under usual operations on polynomials and with the norm ||p|| = sup_{|l| £ 1}|p(l)| is a normed algebra.