On topologies defined by irreducible sets

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

“…In fact, in the classical setting, the above corresponding problem is also unsolved, that is, whether the category KBSob of k-bounded sober spaces is a full reflective subcategory of Top 0 . 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.…”

“…The upper topology ν(Q) on Q is the topology generated by sets of the form X− ↓ x for x ∈ Q, where ↓ x = {a ∈ Q | a ≤ x}. Then (Q, ν(Q)) is k-bounded sober, but not bounded sober (see Example 4.14 in [32]). By Proposition 3.13 and 4.9, the stratified Q-cotopological space ω Q (Q) is k-bounded sober, but not bounded sober.…”

“…They proved that the category BSob of all bounded sober spaces is a full reflective subcategory of the category Top 0 of all T 0 spaces. Furthermore, motivated by the definition of the Scott topology on posets, Zhao and Ho (see [32]) introduced a method of deriving a new topology from a given one by using irreducible sets, in a similar way as one derives the Scott topology on a poset from the Alexandroff topology on the poset. This derived topology leads to a weak notion of bounded sobriety, called k-bounded sobriety.…”

On topological Rudin's lemma, well-filtered spaces and sober spaces

“…Clearly, P H (X) = (C(X) \ {∅}, υ(C(X) \ {∅})). So P H (X) is always sober (see, e.g., [17,Corollary 4.10]). The space P H (Irr c (X)), shortly denoted by X s , with the topological embedding η X (= x → {x}) : X −→ P H (Irr c (X)), is the canonical soberification of X (cf.…”

A new convergence inducing the SI-topology

“…The working principle which uses irreducible sets as the topological counterparts of directed sets is now called Zhao-Ho replacement principle in [1]. A subset U of a T 0 space X is SI-open if and only if (i) U is open in X and (ii) if F is irreducible in X then F ∈ U implies F ∩ U ∅ whenever F exists [9]. By observing the fact that Alexandroff-irreducible sets are exactly the directed sets, SI-topology appears to be a proper generalisation of the Scott topology.…”