On Monotone Determined Spaces

In this paper, we investigate some basic properties, especially categorical properties, of monotone determined spaces. For a topology τ, we construct a monotone determined topology md(τ). The main results are: (1) for a space (X,τ), then md(τ) is the weakest monotone determined topology on X containing τ; (2) the category Topmd of monotone determined spaces with continuous maps is fully co-reflexive in the category Top of all topology spaces with continuous maps; (3) the category Topmd is cartesian closed.