In this paper, we introduce and investigate new types of Alexandroff spaces using well known types of posets; under the conditions that the corresponding poset is complete lattice, distributive lattice and Boolean Algebra. We present some results about these types. Some characterizations of spaces, spaces, space and spaces are obtained and it mainly shown that a space is space if and only if it is homeomorphic to a product of copies of Sierpinski space if and only if every join-irreducible element is an atom
