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Cannot really claim that I have immediate urgent motivation to study this question but it appeared to me long ago, I recalled it now by some reason and decided to ask it here. There is a strong feeli …

According to the Wikipedia article about ${\mathrm T}_1$ spaces your ${\mathrm T}_i$-spaces are called $\it symmetric$ or ${\mathrm R}_0$-spaces. There are several equivalent conditions, my personal f …

As suggested in comments, I turn my comment into an answer here. First of all let me note that in the overwhelming majority of texts I've seen notation is the opposite: $f_*$ from the OP is denoted by …

(Having posted this, I saw that all of it is in the comment by Gro-Tsen) Taking, in the definition of *-space, $C=\text{closure of $O$}$ shows that closure of an open set must be clopen. This is clea …

I believe this is an instance of a semidirect product of one monoid acting on other, construction intermediate in generality between semidirect product of groups and Grothendieck construction/fibratio …

Here is a partial answer. In P. T. Johnstone, "Stone Spaces", pp. 292, 294, it is shown that Scott topologies of continuous posets are precisely all completely distributive complete lattices. Recall t …

(Just noticed - already done by Todd Trimble in a comment:) A proof by Stone duality: the dual question is whether the Boolean algebras $\mathscr P\mathbb N$ and $\mathscr P\mathbb N\otimes\mathscr P …

A self-contained proof is in the book "Continuous lattices and domains" by G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove and D. S. Scott. The particular place you need is Lemma II-1 …

Almost this approach has been initiated by Barr in "Relational algebras" (1970). Recent monograph with lots of references on the subject is "Monoidal topology" by Hofmann, Seal and Tholen. The differ …

An appendix to Denis Nardin's answer: in the wonderful paper "Graduation and dimension in locales" by Isbell (in "Aspects of Topology", London MS Lecture Notes 93 (1985): 195-210), the proof of 1.4 i …

As suggested by @DavidWhite, I am turning my comment into an answer. One class of very naturally appearing non-sober spaces is that of Alexandroff topologies on infinite posets. An irreducible closed …

NB As Qiaochu Yuan explains in the comment below, what follows is not correct: it only captures a very drastic quotient of the isomorphism groupoid of vector bundles; most likely - the groupoid of con …

I am convinced by the answer of Simon Henry completely. This is just an addendum to it, mainly for myself: I want to look at these $I_\kappa$, $T_\kappa$ and $B_\kappa$ in as much detail as possible. …

I believe the paper by Johnstone linked to in the question contains the answer, and it is negative. In that paper, Johnstone constructs a Scott topology that is not sober as a byproduct of answering i …

Judging by the compact regular case, and more generally the spatial case, regular projectivity of locales, resp. regular injectivity of frames, must have something to do with $\neg p\lor\neg\neg p$ an …