  The study of various decision problems for logic fragments has a long history
in computer science. This paper is on the membership problem for a fragment of
first-order logic over infinite words; the membership problem asks for a given
language whether it is definable in some fixed fragment. The alphabetic
topology was introduced as part of an effective characterization of the
fragment $\Sigma_2$ over infinite words. Here, $\Sigma_2$ consists of the
first-order formulas with two blocks of quantifiers, starting with an
existential quantifier. Its Boolean closure is $\mathbb{B}\Sigma_2$. Our first
main result is an effective characterization of the Boolean closure of the
alphabetic topology, that is, given an $\omega$-regular language $L$, it is
decidable whether $L$ is a Boolean combination of open sets in the alphabetic
topology. This is then used for transferring Place and Zeitoun's recent
decidability result for $\mathbb{B}\Sigma_2$ from finite to infinite words.
