A direct approach to $K$-reflections of $T_0$ spaces

In this paper, we provide a direct approach to $\mathbf{K}$-reflections of $T_0$ spaces. For a full subcategory $\mathbf{K}$ of the category of all $T_0$ spaces and a $T_0$ space $X$, let $\mathbf{K}(X)=\{A\subseteq X : A$ is closed and for any continuous mapping $f : X\longrightarrow Y$ to a $\mathbf{K}$-space $Y$, there exists a unique $y_A\in Y$ such that $\overline{f(A)}=\overline{\{y_A\}}\}$ and $P_H(\mathbf{K}(X))$ the space of $\mathbf{K}(X)$ endowed with the lower Vietoris topology. It is proved that if $P_H(\mathbf{K}(X))$ is a $\mathbf{K}$-space, then the pair $\langle X^k=P_H(\mathbf{K}(X)), \eta_X\rangle$, where $\eta_X :X\longrightarrow X^k$, $x\mapsto\overline{\{x\}}$, is the $\mathbf{K}$-reflection of $X$. We call $\mathbf{K}$ an adequate category if for any $T_0$ space $X$, $P_H(\mathbf{K}(X))$ is a $\mathbf{K}$-space. Therefore, if $\mathbf{K}$ is adequate, then $\mathbf{K}$ is reflective in $\mathbf{Top}_0$. It is shown that the category of all sober spaces, that of all $d$-spaces, that of all well-filtered spaces and the Keimel and Lawson's category are all adequate, and hence are all reflective in $\mathbf{Top}_0$. Some major properties of $\mathbf{K}$-spaces and $\mathbf{K}$-reflections of $T_0$ spaces are investigated. In particular, it is proved that if $\mathbf{K}$ is adequate, then the $\mathbf{K}$-reflection preserves finite products of $T_0$ spaces. Our study also leads to a number of problems, whose answering will deepen our understanding of the related spaces and their categorical structures.