In Exercise 5.4.12 of the book, I ask the reader to prove that neither the space of rationals, Q, nor the Sorgenfrey line, Rℓ, is consonant. But the proofs I had in mind were much too simple-minded to stand any chance of succeeding, hence I classified this as important blooper #5 in the list of errata. Good news: Showing that Rℓ is not consonant is not that hard, finally. I will explain the argument in the full post. This will also be an excuse to explain some additional topological properties of Rℓ, an introduction to hereditary Lindelöfness (we will see that Rℓ is hereditarily Lindelöf, although it is not second-countable), and a few additional things in the appendices.
A while back (in March 2019, to be precise), Tomáš Jakl told me that he had a nice, short proof of the fact that the categories of stably compact spaces (and perfect maps) and compact pospaces (and continuous order-preserving maps) are equivalent. He uses an approach through bitopological spaces, and this will give me an opportunity to talk about them. Just as with quasi-uniform spaces, I cannot believe it took me so much time before I mentioned bitopological spaces! Read the full post.
There are three classical powerdomains in domain theory, named after Hoare, Smyth, and Plotkin. The first two are natural and well studied, and the third one is intricate and intriguing. To start with, there are several possible definitions for a Plotkin powerdomain, and while all of them are isomorphic in the nice cases of coherent continuous dcpo, or countably-based continuous dcpos, one may wonder whether they would coincide on all continuous dcpos. That is not the case, and I would like to present a funny counterexample, given in an exercise in Abramsky and Jung’s famous Domain Theory chapter. Let me also call it the hedgehog, because it has spines. Read the full post.
“Quasi-continuous domains and the Smyth powerdomain” is the title of a very nice 2013 paper by Reinhold Heckmann and Klaus Keimel. I will not talk about quasi-continuous domains in this post. Rather, I will mention three pearls that this paper contains: one on so-called supercompact sets, a second one now called the topological Rudin Lemma, and finally a pretty surprising characterization of sober spaces that looks a lot like the definition of well-filteredness. Read the full post.
Formal balls are an extraordinarily useful notion in the study of quasi-metric, and even hemi-metric spaces. Is there any way of extending the notion to the case of quasi-uniform spaces? This is what I would like to start investigating. This is pretty experimental, and I don’t make any guarantee that any of what I am going to say leads to anything of any interest whatsoever! Read the full post.
We embark on the study of notions of completeness for quasi-uniform spaces, and we concentrate on Smyth-completeness. We will see that at least two familiar theorems from the realm of quasi-metric spaces generalize to quasi-uniform spaces: all Smyth-complete quasi-uniform spaces are quasi-sober, and the symcompact quasi-uniform spaces are exactly those that are Smyth-complete and totally bounded. However, and especially for the latter result, the proofs will be trickier. Read the full post.
There is a standard result in the theory of uniform spaces that shows (again) how magical compact Hausdorff space can be: for every compact Hausdorff space X, there is a unique uniformity that induces the topology of X, and its entourages are exactly the neighborhoods of the diagonal. How can we generalize this to stably compact spaces? No, the topology of a stably compact space is not induced by a unique quasi-uniformity… the result has to be a bit more subtle than that. In passing, we will see that every core-compact space, and in particular every locally compact space, has a minimal compatible quasi-uniformity, which has a very simple description. Read the full post.
A uniform space is a natural generalization of the notion of a metric space, on which completeness still makes sense. It is rather puzzling that I managed to avoid the subject of quasi-uniform spaces in something like the 7 years that this blog existed… and it is time that I started. I will only say very classical things, and I will concentrate one a construction due to William Pervin, simplifying an earlier result of Császár, and which shows that every topological space is quasi-uniformizable. Read the full post.
Today I (Jean G.-L.) have the pleasure to have a guest, Aliaume Lopez. We are going to talk about the word topology on X*. In the book, there is a so-called Topological Higman Lemma that says that, if X is a Noetherian space, then X* is Noetherian in the word topology, generalizing a famous theorem due to Graham Higman. However, there used to be no characterization of the word topology as a universal construction, say as a finest topology or a coarsest topology with some properties. Aliaume has managed to find a satisfactory, and simple, answer to this question. We will then discuss the case of infinite words, and we will end with a conjecture which, if true, would provide us with a large set of new Noetherian spaces. Read the full post.
A chain is a totally ordered poset, and a nested space is a topological space whose lattice of open sets is a chain. That may seem like a curious notion, although you might say that the Scott topology on the real line makes it a nested space—so you know that there at least one natural example of the concept. I will show that nested spaces and chains have very strong topological properties. To start with, I will show you why every chain is a continuous poset. I will then tell you how nested spaces arise from the study of so-called minimal T0 and TD topologies, as first explored by R. E. Larson in 1969. And I will conclude with a simple proof of a recent theorem by Mike Mislove. Read the full post.