# Is there a name for infinite words containing every finite words?

Apparently, the closest thing I've found would be normal number http://mathworld.wolfram.com/NormalNumber.html

But requiring that every finite words occurs is weaker than this property. So I'm wondering if there are any study on this topic.

My original goal is to find a criterion for a Büchi automaton not to recognize some infinite word like this. There's been a post here Proof that the $\omega$-language consisting of all words containing every finite word as a factor is not rational/regular , but there were no references to such a class of words.

Maybe all this is obvious to specialists, but I couldn't find anything with our universal friend google :)

Thank you!

• I've used the term "universal" for this. – Joel David Hamkins May 18 '14 at 10:09
• In dynamical systems, this property is called transitive. – Anthony Quas May 19 '14 at 3:04