Prestreams and d-spaces

Posted on July 25, 2021 by jgl

How do you model a topological space with a direction of time? That should seem easy; for example, a topological space with a preordering should be enough. But how do you model the directed circle, where times goes counterclockwise? That should seem harder. Fortunately, there are at least two satisfactory models: Marco Grandis’ d-spaces, and Sanjeevi Krishnan’s (pre)streams, which I will describe. Both appeared in 2009. Although they look rather different, they are connected by an idempotent adjunction, which restricts to an equivalence between so-called complete d-spaces and Haucourt streams, as discovered by Emmanuel Haucourt in 2012. Read the full post.

Posted in Uncategorized | Tagged , | Comments Off on Prestreams and d-spaces

First-countable spaces and their Smyth powerdomain

Posted on June 22, 2021 by jgl

This month, we will look at certain conditions recently found by He, Li, Xi and Zhao in 2019, and then by Xu and Yang in 2021, in order to ensure that the Smyth powerdomain Q(X) (with the Scott topology) of a space X is a dcpo model of X, and in order to ensure that the Scott and upper Vietoris topologies agree on Q(X)… without the usual condition of local compactness, but under some first-countability conditions. I got interested in this while Xiaodong Jia and I explored the Smyth powerdomain of the Sorgenfrey line. Read the full post.

Posted in Uncategorized | Tagged , | Comments Off on First-countable spaces and their Smyth powerdomain

The Sorgenfrey line is not consonant

Posted on May 20, 2021 by jgl

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.

Posted in Uncategorized | Tagged , , , | Comments Off on The Sorgenfrey line is not consonant

Bitopological spaces and stable compactness

Posted on April 22, 2021 by jgl

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.

Posted in Uncategorized | Tagged , , | Comments Off on Bitopological spaces and stable compactness

Plotkin’s powerdomain and the hedgehog

Posted on March 21, 2021 by jgl

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.

Posted in Uncategorized | Tagged , | Comments Off on Plotkin’s powerdomain and the hedgehog

Quasi-continuous domains and the Smyth powerdomain

Posted on February 22, 2021 by jgl

“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.

Posted in Uncategorized | Tagged , | Comments Off on Quasi-continuous domains and the Smyth powerdomain

Quasi-uniform spaces IV: Formal balls—a proposal

Posted on January 23, 2021 by jgl

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.

Posted in Uncategorized | Tagged , | Comments Off on Quasi-uniform spaces IV: Formal balls—a proposal

Quasi-uniform spaces III: Smyth-completeness, symcompactness

Posted on December 20, 2020 by jgl

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.

Posted in Uncategorized | Tagged , | Comments Off on Quasi-uniform spaces III: Smyth-completeness, symcompactness

Quasi-uniform spaces II: Stably compact spaces

Posted on November 21, 2020 by jgl

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.

Posted in Uncategorized | Tagged , | Comments Off on Quasi-uniform spaces II: Stably compact spaces

Quasi-Uniform Spaces I: Pervin Quasi-Uniformities, Pervin Spaces

Posted on October 19, 2020 by jgl

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.

Posted in Uncategorized | Tagged | Comments Off on Quasi-Uniform Spaces I: Pervin Quasi-Uniformities, Pervin Spaces