The Answer to a Problem Posed by Zhao and Ho

Bounded sobriety and k-bounded sobriety of Q-cotopological spaces

“…In the concluding remarks of [32], Zhao and Ho asked whether KB(X) (the set of all closed irreducible sets of a T 0 space X whose suprema exist) is the canonical k-bounded sobrification of X in the sense of Keimel and Lawson with respect to the map x −→ cl({x}). Zhao, Lu and Wang (see [30]) constructed a counterexample to illustrate that (KB(X), cl) is not the canonical k-bounded sobrification of a T 0 space X in the sense of Keimel and Lawson. Although we possess no answer on the above question, we know that (KB(X), ι) is not universal with respect to I, where I : KBSobQ-CTop−→ SQ-CTop is the inclusion functor.…”

Some open problems on well-filtered spaces and sober spaces

“…Recently, Zhao Bin, Lu Jing and Wang Kaiyun [57] gave a negative answer to Zhao and Ho's problem. Furthermore, it is proved in [37] that, unlike Sob and BSob, the category KBSob of all k-bounded sober spaces with continuous mappings is not reflective in Top 0 .…”

Non-reflective categories of some kinds of weakly sober spaces

“…In domain theory and non-Hausdorff topology, the sober spaces, well-filtered spaces and d-spaces form three of the most important classes (see [2][3][4][5][6][7][8][9][10][11][12][13][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]). Let Top 0 be the category of all T 0 spaces with continuous mappings and Sob the full subcategory of Top 0 containing all sober spaces.…”

“…In recent years, a number of classes of strongly sober spaces, strong d-spaces, weakly sober spaces, weakly well-filtered spaces and weakly d-spaces have been introduced and extensively studied from various different perspectives (see [4,13,15,20,23,26,27,30,31]). In particular, in [4], Erné relaxed the concept of sobriety in order to extend the theory of sober spaces and locally hypercompact spaces to situations where directed joins were missing.…”