I am reffering to this site: http://www.scholarpedia.org/article/Topological_entropy
Definitionj of topological Entropy by Adler, Kohnheim
For an open cover $\mathcal{U}$ of $X$, let $N(\mathcal{U})$ denote the smallest cardinality of a subcover of $\mathcal{U}$. If $\mathcal{U}$ and $\mathcal{V}$ are open covers of $X$, then $$ \mathcal{U}\vee\mathcal{V}=\left\{U\cap V: U\in\mathcal{U},V\in\mathcal{V}\right\} $$ is called their common refinement.
Let $$ \mathcal{U}^n=\mathcal{U}\vee T^{-1}\mathcal{U}\vee T^{-2}\mathcal{U}\vee\cdots\vee T^{-(n-1)}\mathcal{U}, $$ where $T^{-k}\mathcal{U}=\left\{T^{-k}U: U\in\mathcal{U}\right\}$.
Define the entropy of $T$ relative to $\mathcal{U}$ as $$ h(\mathcal{U},T)=\lim_{n\to\infty}\log\frac{1}{n}N(\mathcal{U}^n). $$ The topological entropy of $T$ is defined as $$ h(T)=\sup_{\mathcal{U}}h(\mathcal{U},T), $$ where the supremum ranges over all open covers $\mathcal{U}$ of $X$.
In order to understand this definitions, I looked at the left sided shift, i.e. let $(X,T)$ be a subshift, i.e., T being the left shift map on a closed left shift invariant collection $X$ of sequences of symbols belonging to a finite alphabet.
It is said that the topological entropy of $T$ is given by $$ h(t)=\lim_{n\to\infty}\frac{1}{n}\log \#\mathcal{B}_n, $$ where $\mathcal{B}_n$ denotes the collection of all words of length $n$ appearing anywhere in the sequences $x\in X$.
Could you please explain me how I get from the definition of topological entropy (as given above) that $$ h(t)=\sup_{\mathcal{U}}h(\mathcal{U},T)=\lim_{n\to\infty}\frac{1}{n}\log \#\mathcal{B}_n? $$