Reading from Walkden's lecture notes on Ergodic Theory (https://personalpages.manchester.ac.uk/staff/charles.walkden/magic/lecture03.pdf), in the first two paragraphs of page 6 (i.e. §3.4.3 Bernoulli and Markov measures) it is claimed the following three:
1- Cylinders are open subsets of $\Sigma_k$. 2- Cylinders are closed subsets of $\Sigma_k$. 3- "1" and "2" above reflects the fact that $\Sigma_k$ is totally disconnected.
I want to prove those three claims:
1- Consider some a cylinder $U=[i_0, ..., i_n]_m$ and a point x \in U, say $(..., i_0, ..., i_n, x_0, x_1, ...)$ (Here I consider $\Sigma_k^+$ the full one-sided k-shift. The set of all points $(i_0, ..., i_n, y, x_1, ...)$ for all possible $y$ is an open set containing $x$ since it is a "neighborhood" of $x$ and its intersection with complement of $U$ in $\Sigma_k$ is empty. So $U$ is open and so is $\Sigma_k$. Is my argument correct?
2- Consider the same $U$ and $x$ in Item "1" above. The sequence of points $(..., i_0, ..., i_n, y_0, y_1, ...)$ such that if $y_0=x_0$ then we go on until we arrive at $y_k \ne x_k$ then we decrease or increase $y_k$ to be equal to $x_k$ and each step is a sequence in the sequences that the limit approaches $x$. So $\Sigma_k$ is closed. Is my argument correct?
3- According to the Wiki the singletons are the only connected proper subsets in a totally disconnected space. How is this related to every set being clopen?