Finite words, contiguous factors, avoidance and proper-prefix states

Let $\mathrm{\Sigma }$ be a finite set, called an alphabet. A word of length $n$ over $\mathrm{\Sigma }$ is a function from the natural number $n=\{0,\dots ,n-1\}$ (The natural numbers $\mathbb{N}$ (von Neumann)) to $\mathrm{\Sigma }$, written $a_0{a}_1\cdots a_{n-1}$. The unique word of length zero is the empty word $\epsilon$. Concatenation of words $u$ and $v$ is denoted by $uv$.

A word $u$ is a contiguous factor of $w=a_0\cdots a_{n-1}$ if $u=a_i\cdots a_{i+r-1}$ for some $0\le i\le i+r\le n$. It is a prefix if $i=0$, a suffix if $i+r=n$, and a proper prefix if it is a prefix other than the whole word.

For a set $B$ of words, a word avoids $B$ if none of its contiguous factors belongs to $B$. Its set of proper-prefix states is

$S_B:=\{u:u\text{ avoids }B\text{ and }u\text{ is a proper prefix of some }b\in B\}\mathrm{.}$

If $B$ is nonempty and every word in $B$ is nonempty, then $\epsilon \in S_B$. If $B$ is finite, then $S_B$ is finite.