  In this note we introduce the notion of $t$-analytic sets. Using this
concept, we construct a class of closed prime ideals in Banach function
algebras and discuss some problems related to Alling's conjecture in
$H^\infty$. A description of all closed $t$-analytic sets for the disk-algebra
is given. Moreover, we show that some of the assertions in Daoui et al. (Proc.
Am. Math. Soc. 131:3211-3220, 2003) concerning the $O$-analyticity and
$S$-regularity of certain Banach function algebras are not correct. We also
determine the largest set on which a Douglas algebra is pointwise regular.
