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.

中文翻译：

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关于 T0 空间的 H-清醒空间和 H-清醒

在本文中，我们为 $d$-空间、清醒空间和良好过滤空间提供了统一的方法，并开发了处理所有这些空间的通用框架。对于子集系统H，建立了H-清醒空间和超H-清醒空间的理论，并给出了$T_0$空间的H-清醒和超H-清醒的直接构造。因此，所有 H-清醒空间的范畴在 $\mathbf{Top}_0$ 中是反射的，如果 H 具有自然性质，则所有超 H-清醒空间的范畴也在 $\mathbf{Top}_0$ 中反射（称为属性 M）。结果表明，H-sobrification保留$T_0$空间的有限乘积，如果H具有性质M，超H-sobrification保留$T_0$空间的有限乘积。