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