Errata for "Topology inspired problems for cellular automata and a counterexample in topology"
In the proof of Theorem 1, compactness is proved incorrectly: \( (U_x(a))_{x \in \Sigma^{\mathbb{Z}}, a \in \Sigma} \) obviously has a finite subcover, indeed \( (U_x(a))_{a \in \Sigma} \) is one, for any \( x \in \Sigma^{\mathbb{Z}} \). For a correct proof, for each \( a \in \Sigma \) pick a dense set \( X_a \subset \Sigma^{\mathbb{Z}} \) such that \( X_a \cap X_b = \emptyset \) for \( a \neq b \). Then \( (U_x(a))_{a \in \Sigma, x \in X_a} \) is a cover (since cellular automata are continous) but clearly has no finite subcover.
Last update: 14.6.2023