Summary: In this paper, we shall prove that acyclically monotonically \(T_2\) is a properly hereditary property, and every metric space is an acyclically monotonically \(T_2\) space. Moreover, we shall give a characterization of acyclically monotonically \(T_2\). Also, some open problems and conjectures concerning acyclically monotonically \(T_2\) spaces and their relation with strongly monotonically \(T_2\) space will be given.