Generalized Lowen functors

Abstract

According to their value ranges, L-topological spaces form different categories. Clearly, the investigation on their relationships is certainly important and necessary. Lowen was one of the first authors who had studied the relation between the category of I-topological spaces and that of topological spaces. He introduced two well-known functors: ω and ι. Later, these functors, namely Lowen functors, were extended by different authors for various kinds of lattices studying the relation between L-TOP and TOP. In this paper, we introduce two functors ω_f and ι_f between L₁-TOP and L₂-TOP for each Scott continuous mapping $\text{f:L}{}_2\text{→}\text{L}{}_1$, where L₁ and L₂ are arbitrary two completely distributive lattices. Some topological properties related to these functors are revealed, e.g., for L_i-topological space $\text{(X}{}_{\mathrm{i}}\text{,Δ}{}_{\mathrm{i}}\text{)}\text{(i}\text{=}\text{1,2)}$, both ω_f(X₁,Δ₁) and ι_f(X₂,Δ₂) are Lowen spaces; in the case that f is the identity mapping on a linearly ordered complete lattice, for L-topological space (X,Δ), ω_f(Δ) is the finest Lowen topology on X contained in Δ and ι_f(Δ) is the coarsest Lowen topology on X containing Δ, etc.