Omega language

An ω-language is a set of infinite-length sequences of symbols.

Let Σ be a set of symbols (not necessarily finite). Following the standard definition from formal language theory, Σ is the set of all finite words over Σ. Every finite word has a length, which is, obviously, a natural number. Given a word w of length n, w can be viewed as a function from the set {0,1,...,n-1} → Σ, with the value at i giving the symbol at position i. The infinite words, or ω-words, can likewise be viewed as functions from to Σ. The set of all infinite words over Σ is denoted Σ. The set of all finite and infinite words over Σ is sometimes written Σ.