  The Central Sets Theorem was introduced by H. Furstenberg and then afterwards
several mathematicians have provided various versions and extensions of this
theorem. All of these theorems deal with central sets, and its origin from the
algebra of Stone-Cech compactification of arbitrary semigroup, say $\beta S$.
It can be proved that every closed subsemigroup of $\beta S$ is generated by a
filter. We will show that, under some restrictions, one can derive the Central
Sets Theorem for any closed subsemigroup of $\beta S$ . We will derive this
theorem using the corresponding filter and its algebra. Later we will also deal
with how the notions of largeness along filters are preserved under some well
behaved homomorphisms and give some consequences.
