K-reflections of product spaces

Abstract Let Top d , Top w and Sob be the category of all d-spaces, that of all well-filtered spaces and that of all sober spaces respectively. For a full subcategory K of Top d containing Sob, it is proved that the product of an arbitrary family of K-determined sets is a K-determined set and if K is adequate, then the K-reflection preserves arbitrary products of T 0 spaces. In particular, the Keimel-Lawson reflection, well-filtered reflection and d-reflection all preserve arbitrary products of T 0 spaces, and DCPO-completion preserves the product of a family { P i : i ∈ I } of posets if the Scott topology of the product ∏ i ∈ I P i is the product of the Scott topologies of the factors.

[1]  Marcel Erné.  Categories of Locally Hypercompact Spaces and Quasicontinuous Posets , 2018, Appl. Categorical Struct..

[2]  Xiaoquan Xu,et al.  On well-filtered reflections of T0 spaces , 2019, Topology and its Applications.

[3]  Yuri Leonidovich Ershov.  On d-Spaces , 1999, Theor. Comput. Sci..

[4]  Klaus Keimel,et al.  Quasicontinuous Domains and the Smyth Powerdomain , 2013, MFPS.

[5]  Marcel Erné,et al.  The strength of prime separation, sobriety, and compactness theorems , 2018, Topology and its Applications.

[6]  Rudolf-E. Hoffmann,et al.  On the sobrification remainder $^{s}X-X$. , 1979 .

[7]  Xiaoyong Xi,et al.  Well-filtered spaces and their dcpo models , 2017, Math. Struct. Comput. Sci..

[8]  Dongsheng Zhao,et al.  Dcpo-completion of posets , 2010, Theor. Comput. Sci..

[9]  Xiaoquan Xu.  On H-sober spaces and H-sobrifications of T0 spaces , 2020, 2003.07737.

[10]  Jean Goubault-Larrecq,et al.  Non-Hausdorff topology and domain theory , 2013 .

[12]  Xiaoquan Xu,et al.  On T0 spaces determined by well-filtered spaces , 2019, Topology and its Applications.

[13]  Rudolf-E. Hoffmann,et al.  Sobrification of partially ordered sets , 1979 .

[14]  Jimmie D. Lawson,et al.  The spectral theory of distributive continuous lattices , 1978 .

[15]  K. Hofmann,et al.  Continuous Lattices and Domains , 2003 .

[16]  Qingguo Li,et al.  A direct characterization of the monotone convergence space completion , 2017 .

[17]  Xiaoquan Xu,et al.  On topological Rudin's lemma, well-filtered spaces and sober spaces , 2019, 1911.11617.

[18]  Reinhold Heckmann,et al.  An Upper Power Domain Construction in Terms of Strongly Compact Sets , 1991, MFPS.

[19]  Xiaoquan Xu.  A direct approach to K-reflections of T0 spaces , 2019, 1911.11618.

[20]  Xiaoquan Xu,et al.  First countability, ω-well-filtered spaces and reflections , 2020 .

[21]  Oswald Wyler,et al.  Dedekind complete posets and scott topologies , 1981 .

[22]  J. Lawson,et al.  On Well-Filtered Spaces and Ordered Sets , 2017 .

[23]  Xiaoquan Xu,et al.  A complete Heyting algebra whose Scott space is non-sober , 2019, Fundamenta Mathematicae.

[24]  Klaus Keimel,et al.  D-completions and the d-topology , 2009, Ann. Pure Appl. Log..