On H-sober spaces and H-sobrifications of T0 spaces

Abstract

In this paper, we provide a uniform approach to d-spaces, sober spaces and well-filtered spaces, and develop a general framework for dealing with all these spaces. The concepts of irreducible subset systems (R-subset systems for short), H-sober spaces and super H-sober spaces for a general R-subset system H are introduced. It is proved that the product space of a family of T0 spaces is H-sober iff each factor space is H-sober, and if H has a natural property (called property M), then the super H-sobriety is a special type of H-sobriety, and hence the product space of a family of T0 spaces is super H-sober iff each factor space is super H-sober. Let Top0 be the category of all T0 spaces with continuous mappings. For a T0 space X and an H-sober space Y, we show that the function space Top0(X,Y) equipped with the topology of pointwise convergence is H-sober. Furthermore, if H has property M and Y is a super H-sober space, then the function space Top0(X,Y) equipped with the topology of pointwise convergence is super H-sober. One immediate corollary is that for a T0 space X and a well-filtered space Y, the function space Top0(X,Y) equipped with the topology of pointwise convergence is well-filtered. For an R-subset system H having property M, the Smyth power space of an H-sober space is not H-sober in general. But for the super H-sobriety, we prove that a T0 space X is super H-sober iff its Smyth power space PS(X) is super H-sober. A direct construction of the H-sobrifications and super H-sobrifications of T0 spaces is given. So the category of all H-sober spaces is reflective in Top0, and the category of all super H-sober spaces is also reflective in Top0 if H has property M.