Central Sets Theorem near zero

In this paper, we introduce notions of J-set near zero and C-set near zero for a dense subsemigroup of ((0,+∞),+) and state the Central Sets Theorem near zero. Among the other results for a dense subsemigroup S⊆((0,+∞),+), we give some sufficient and equivalent algebraic conditions on a subset A⊂S to be a J-set near zero and to be a C-set near zero.