2019/11/25 by Xiaoquan Xu, Xu, Xiaoquan · 1 citation
Mathematics · Medicine · #Algebraic structures and combinatorial models #FOS: Mathematics #General Topology (math.GN) #Homotopy and Cohomology in Algebraic Topology #Intracranial Aneurysms: Treatment and Complications
paper · pdf · doi:10.48550/arxiv.1911.11618
openalex publication_date 2019/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we provide a direct approach to K-reflections of T0 spaces. For a full subcategory K of the category of all T0 spaces and a T0 space X, let K(X)=\A⊆ X : A is closed and for any continuous mapping f : X\longrightarrow Y to a K-space Y, there exists a unique yA∈ Y such that f(A)=\yA\\ and PH(K(X)) the space of K(X) endowed with the lower Vietoris topology. It is proved that if PH(K(X)) is a K-space, then the pair ⟨ Xk=PH(K(X)), ηX⟩, where ηX :X\longrightarrow Xk, x↦\x\, is the K-reflection of X. We call K an adequate category if for any T0 space X, PH(K(X)) is a K-space. Therefore, if K is adequate, then K is reflective in Top0. 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 Top0. Some major properties of K-spaces and K-reflections of T0 spaces are investigated. In particular, it is proved that if K is adequate, then the K-reflection preserves finite products of T0 spaces. Our study also leads to a number of problems, whose answering will deepen our understanding of the related spaces and their categorical structures.