Non-Hausdorff Topology and Domain Theory

While a topological space is a space of points, a hyperspace is a space of subsets, with a suitable topology.  Examples abound in the literature.  For example, the so-called Smyth powerdomain (Proposition 8.3.25) is one.  To start the series, let me look at the Hoare hyperspace instead.  We shall see that it is a space that has many nice properties.  I won’t say to which purpose it has been put in denotational semantics yet, or that it defines a monad with a very natural inequational theory… all that will have to wait!  Read the full post.

On p.61 of the book, there is a remark that the dcpos are exactly the chain-complete posets.  This is a theorem by George Markowsky (1976).  It is time I explained seriously how this worked.  The first step is Iwamura’s Lemma (1944), which states that every directed subset decomposes as the union of a small chain of small directed subsets.

The reason I did not put the proof of that result in the book is because it rests on using ordinals, and I did not want to introduce ordinals, specially if they served for only one result.  I’m attacking the problem head-front in the full post.

From time to time, we happen to discover that several distinct notions are in fact the same, and this is exactly what happened in 2014, in two papers that appeared about at the same time, with similar discoveries. One is due to Jimmie Lawson and Xiaoyong Xi, the other one is due to Achim Jung and myself.  The theorem says that QRB-domains, QFS-domains, and stably compact, locally finitary spaces, are exactly the same thing.  The first arose in probabilistic domain theory.  See the full post.

I hope you’ve had a Merry Christmas, and wish you a Happy New Year 2015.  To renew with an old habit, I’ve produced a new crosswords puzzle on the occasion (#14).  Sorry, it does not have anything particular related to Christmas, or with any other season-related celebration.  I may do better next time.

Although I am trying to post about every month, I have not posted anything for two months.  My first idea was to talk about synthetic topology, after I read some papers by Weng Kin Ho and by Martín Hötzel Escardó.   I soon realized I would have trouble to match the level of clarity they reach in their papers.  Let me recommend [1], or [2] if you’ve got more time.  Maybe I’ll talk about these one day, who knows.

In the meantime, I realized I’ve never talked about any of my research.  Last year, Achim Jung and myself found a nifty theorem, one of those theorems in mathematics that states that several, apparently unrelated notions are in fact the same.  Jimmie Lawson and Xiaoyong Xi found the same result independently.  This all builds on the related notions of locally finitary compact spaces, and of quasi-continuous dcpos, and I’ll start to introduce you to the latter.  See the full post.

[1] Martín Hötzel Escardó and Weng Kin Ho. An operational domain theory and topology of a sequential programming language. Information and Computation 207(3), p. 411-437, Elsevier, 2009.

[2] Martín Hötzel Escardó. Synthetic topology of data types and classical spaces. ENTCS, Elsevier, volume 87, pages 21-156, November 2004.

Both filter spaces and equilogical spaces form Cartesian-closed categories that contain Top as a full subcategory.  Is there any connection between them?  Very much so, as found by Reinhold Heckmann in 1998, following Martin Hyland (1977).  Read the full post.

A bit more than a month ago, I’ve been reminded of a beautiful construction of yet another Cartesian-closed category of spaces, generalizing the topological spaces: the equilogical spaces of Bauer, Birkedal, and Scott.  Simply put, equilogical spaces are formal (not actual) quotients of topological spaces by equivalence relations.  See the full post.

As promised, let us talk about Wallman compactifications.  They extend the construction of the Stone-Čech compactification ßX, beyond discrete spaces, to all T₄ spaces.  The construction is very similar to what we have seen in Filters IV.  But brace yourself for the finale, which involves Stone duality and Johnstone’s Theorem!  See the full post.

While we’ve got a good theme, let’s hang on to it.  Yes, of course: filters.  It turns out that you can build compactifications using ultrafilters, in special cases.  See the full post.  I’ll talk about more general constructions later.

Not that I’m particulary Irish, but here’s a crossword puzzle for Saint Patrick’s Day.  (Also playable online.)  Meant to be the shape of a cloverleaf… Don’t omit the other 12 puzzles, too.