Non-Hausdorff Topology and Domain Theory

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.

While listening to some talks at the special session on categorical topology last Saturday, I realized that I had said something wrong at the end of part II: the filter spaces satisfying the property that any intersection of filters that converge to a point x must also converge to x are not the topological filter spaces, but the pretopological filter spaces.  See the full post for an in-depth discussion of the matter.

I’m currently at the Joint Mathematics Meeting in Baltimore, MD, USA.  This is a huge conference.  The program alone is a 250+ page booklet!

My main purpose there is to participate in the special session on categorical topology on Saturday, January 18th.  I’m going to talk about the Escardò-Lawson-Simpson construction (Section 5.6 in the book).  Nicely enough, this generalizes outside of pure topology, by the mere virtue of so-called topological functors.  I’m also applying all that to Sanjeevi Krishnan’s notion of streams, an incredibly nice model for directed algebraic topology.  You can find all this in my slides.  If you are brave, you can also read the paper.

Next week, I’ll visit Frédéric Mynard at Georgia Southern University, Statesboro, GA, USA.  On Tuesday, January 21st, I’ll give a related talk that is meant as an introduction to directed algebraic topology and streams.

The Christmas 2013 puzzle is out!  (also playable online.)  Note the black squares make a kind of Christmas cracker shape… Don’t omit the other 11 puzzles, too.

I said earlier that what convergence was the starting point of topology.  Why not take this seriously and replace topological spaces by spaces that would be defined in terms of notions of convergence directly, instead of through opens?

One bonus is that we shall obtain Cartesian-closed categories, and this will be easy!  As usual, while nets will allow us to grasp the concept, this will actually work best with filters.  See the full post.