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Part (c) makes $G(A)$ relatively open in $G_B(A)$. It is also relatively closed: if a sequence in $G(A)$ converges to $a\in G_B(A)$, part (e) forbids $a\notin G(A)$. Sequences detect closure here because the topology comes from a norm.

Intersecting this clopen subset with any connected component $C$ of $G_B(A)$ gives a clopen subset of the connected space $C$. It is therefore either empty or all of $C$. Thus $G(A)$ is the union of those connected components of $G_B(A)$ that it meets. This is connected-component permanence of subalgebra invertibility; it does not require connected components to be path connected.