Let $\mathcal{A}$ be a finite alphabet of letters and $\Sigma$ either the semi-infinite space or the doubly infinite space of sequences whose elements are drawn from $\mathcal{A}$. Define the natural topology on $\Sigma$. If $W$ is a set of words, denote by $\Sigma_{W}$ the subspace of $\Sigma$ consisting of those sequences none of whose subsequences is in $W$. Prove that $\Sigma_{W}$ is a closed subspace of $\Sigma$; and state and prove a necessary and sufficient condition for a closed subspace of $\Sigma$ to have the form $\Sigma_{W}$ for some $W$.

$$\begin{aligned} &\text { If } \mathcal{A}=\{0,1\} \text { and } \quad W=\{000,111,010,101\} \end{aligned}$$

what is the space $\Sigma_{W}$ ?