Ennui Spaces
November 30th, 2007 by WaltI was browsing through Wikipedia today when I came across the definition of pretopological space. The notion seemed very exotic until I thought of a family of examples, which I’m christening ennui spaces.
A pretopological space prescribes for each point the set of (not-necessarily open) neighborhoods of that point. The set of neighborhoods of a given point are required to satisfy some natural axioms, but neighborhoods of one point can be completely unrelated to neighborhoods of another point. A sequence in a pretopological space converges if for any neighborhood, the sequence eventually enters that neighborhood and never leaves it again. A topological space can be turned into a pretopological space by taking as the set of neighborhoods of a point to be all sets that contain an open set that contain that point. You can try to reverse the process by borrowing the characterization of the closure of a set in terms of sequences (or nets), but usually the topological space you construct will have a coarser notion of convergence than the pretopological space.
An ennui space has the same underlying set as a metric space. A neighborhood of a point is any set that contains the unit ball around that point. A sequence in an ennui space converges to a point if is guaranteed to be eventually within one unit of the point. The mental image I have is that the sequence gets close to its destination, but then gets bored. If you try to construct a topology out of this space, you get the indiscrete topology, where all sequences converge to all points. Essentially, all information about the convergence properties of the ennui space are lost.
A practical example of an ennui space would be your computer whenever it simulates a convergent sequence of operations, such as numerical integration or Newton’s method. The computer gets within machine precision of the correct answer, and then stops to light up a Gauloise and discuss L’Être et le néant in a cafe.
December 1st, 2007 at 12:37 am
What a stultifying idea!
December 1st, 2007 at 6:44 am
Morphological dilation form image processing is a preclosure operator.
December 1st, 2007 at 9:15 am
[...] First, my computer doesn’t smoke any of that French crap. When my computer is finished with a long day of numerical optimization, it lights up a Marlboro. [...]
December 2nd, 2007 at 8:47 am
Joy-of-home-ownership-spaces:
Drove around town and eventually went home and putzed around the house all night…
Resets every 24 hours.
In this case, the house is the whole neighborhood, but how often do you see your neighbors anyway?
December 2nd, 2007 at 9:01 am
Likewise life-sentence spaces:
Naughty sequences converge to Folsom Prison, but at least they get out of their cells for an hour every day!
December 2nd, 2007 at 9:09 am
Obsession spaces:
I used to think about all sorts of things, but now it’s 9/11 conspiracy theories 24/7.
Did you ever see the “nose-out” video where the second plane apparently passes INTACT through the WTC and you can see the nose of it come out the other side but after a couple of hours the networks covered it up with a chintzy CGI to match the “airplane” that was pasted over the real missile which most of the non-network-affiliated witness heard whistling just before “impact” or detonation depending on…