Prove or disprove : A finite set of infinite words is $ω$-regular.
I am planning to prove this statement by constructing a NBA for every infinite word and then using the closure of $\omega$-regular languages under union.
But I am stuck in trying to show that every infinite word has a NBA or is captured by a LTL formula. Can someone help with regard to this?
Or else should I try to disprove this? Could anyone give some pointers?
Hint. Every non-empty $\omega$-regular set contains an ultimately periodic word.