Author Archives: jgl

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 … Continue reading →

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 … Continue reading →

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 … Continue reading →

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 … Continue reading →

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. … Continue reading →

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 … Continue reading →

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 … Continue reading →

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.