Question about the definition of strings that GNFA's accept In Sipser's "Introduction to the Theory of Computation" it states "A GNFA accepts a string $w$ in $Σ^{*}$ if $w = w_1 w_2 · · · w_k$ , where each $w_i$ is in $Σ^{*}$". 

However, wouldn't a more precise definition be "A GNFA accepts a string $w$ in $Σ^{*}$ if $w = w_1 w_2 · · · w_k$ , where each $w_i$ is in $Σ$" ?
If a GNFA $A$ reconizes $\Sigma^{*}$, then $\epsilon \in \Sigma^{*}$ must be accepted. However $\epsilon \not\in \Sigma $.