Non-closedness of transitive cellular automata in the uniform Bernoulli space
Let denote the space of cellular automata equipped with the uniform Bernoulli topology, and call a cellular automaton transitive if it has the transitivity property under consideration. Non-closedness claim. Transitive cellular automata are not closed in the uniform Bernoulli space. This is stated as an open problem in the surrounding discussion of topology-inspired questions for cellular automata; the supplied text gives no resolution or further context for the claim.
References
Primary source
Ville Salo and Ilkka Törmä, “Topology Inspired Problems for Cellular Automata, and a Counterexample in Topology”, arXiv:1208.2783 (2012).
Progress summary
The conjecture remains open: no public proof or counterexample was found.
A 2012 paper records the assertion as Conjecture 1: transitive cellular automata are not closed in the uniform Bernoulli space. The cited work treats it as unresolved, and the scan found no later result addressing this exact conjecture.
Current status (as of August 2026): The non-closedness assertion remains an open conjecture; no proof, counterexample, verification, or credible objection was found in the retrieved sources.
Sources
Solutions 0
No solutions have been posted yet.