On H-sober spaces and H-sobrifications of $T_0$ spaces

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. For a subset system H, the theory of H-sober spaces and super H-sober spaces is established, and a direct construction of the H-sobrifcations and super H-sobrifications of $T_0$ spaces is given. Therefore, the category of all H-sober spaces is reflective in $\mathbf{Top}_0$, and the category of all super H-sober spaces is also reflective in $\mathbf{Top}_0$ if H has a natural property (called property M). It is shown that the H-sobrification preserves finite products of $T_0$ spaces, and the super H-sobrification preserves finite products of $T_0$ spaces if H has property M.

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