On the dissonance of some metrizable spaces

Some open problems on well-filtered spaces and sober spaces

“…A topological space X is said to be consonant if for every F ∈ σ(O(X)), there is a family {K i : i ∈ I} ⊆ K(X) with F = K∈A Φ(K) (see, e.g., [3,7,8,22,39]). The consonance is an important topological property (see [2,3,4,8,39]).…”

“…In [8] it is proved that every Céch-complete space -hence, in particular, every completely metrizable space -is consonant. Bouziad [4] (see also [7]) showed that the space of rationals with the subspace topology inherited from the reals is not consonant. It was shown in [6] that quasi-Polish spaces are consonant, and it is known that a separable co-analytic metrizable space is consonant if and only if it is Polish (see [3]).…”

On the commutativity of the powerspace constructions

“…It was shown in [dBSS16] that quasi-Polish spaces are consonant, and it is known that a separable co-analytic metrizable space is consonant if and only if it is Polish [Bou99]. In particular, the space of rationals with the subspace topology inherited from the reals is not consonant [CW98]. Definition 6.7.…”

Variations of consonance and the rational numbers

“…Examples of non-consonant separable metric spaces and first countable spaces, in particular the rational numbers Q with their usual topology, soon followed [1,4,3], and [5]. The result of greatest generality in this direction is: Corollary 3.8]) Every regular first countable space without isolated points, all compact subsets of which are countable, is not F in-trivial (and hence, non-consonant).…”