A normed algebra with non-open group of invertibles (and so the algebra is not Banach).

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Let A = C[z]^{1}, then Inv(C[z]) = C − {0}, hence the elements p_{n}(z) = 1 + \frac{z}{n}(n ∈ N) aren’t invertible. But lim_{n} p_{n}(z) = 1 ∈ Inv(C[z]). Therefore A − Inv(A) isn’t closed.

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_{|λ|≤1} |p(λ)| is a normed algebra.