ToL: This ONE WEIRD Trick to make you a GENIUS at Topology!

[Hazard]

(These are the touched up notes from a class I took with CMU’s Kevin Kelly this past semester on the Topology of Learning. Only partially optimized for legibility)

Time to introduce some new Topological terms. We’re going to create some good intuitions around the concepts of interior, exterior, boundary, closure, and frontier. These are all operators in the sense that if you have a set $S$, then $\text{Int}\left(S\right)$ is me using the $$int$$ operator to create a new set that we call “the interior of $S$”. Ext, Bndr, Cl, and Frnt are the shorthand I will use for these operators.

Before talking about these operators in a topological sense, I want to talk about them in a metric space sense. A metric space is just some mathematical space where you have a way to specify the distance between any two points, according to a specific definition of distance. In the real line, the distance between any two numbers can just be the absolute value of their difference. In n-dimensional euclidean space, distance is given by the n-dimensional version of the Pythagorean Theorem. I want to start talking about interiors and boundaries and such from a metric point of view in order to contrast the way it’s different from the topological view. I found that when I was trying to wrap my head around these concepts, I was implicitly assuming a metric space world view, because literally every math space I’d interacted with up to that point was a metric space.

Let’s start with this picture:

The squiggly loop is our set $S$. In a metric space, a point is in the interior of a set if you can “draw a circle” around it, such that the circle only contains other points that are in $S$ (formally, you talk about “balls” instead of circles. An $r$-ball around $x$ is the set of all points $z$ st $d(z,x)>r$). You can clearly see that I can draw a circle around $x$, where the circle only contains points in $S$, so $x$ is in the interior of $S$. A boundary point like $y$ is a point where no matter how small a circle you draw around it, the circle will contain some points in $S$, and some points not in $S$.

Likewise, the exterior of $S$ consists of all points that you can draw a circle around such that the circle only contains points not in $S$. Here are definitions of our two other operators:

$ClS=S\cup BndrS$

$FrntS=BndrS\setminus S$

Metrical to Topological

Now, here’s where we shift from the metric perspective to the topological perspective. Let’s think back to trying to decide if a point is in the interior of $S$. Re-frame this task as us trying to take a “measurement” around $x$. “Can I make a measurement that would include $x$ and not include anything from $\neg S$?” You can see how this is a more general question. We were asking the same question in the metric space context, it’s just that our “measurements” were circles of arbitrarily small radii. A general metric space abstracts the idea of measuring with a circle to measuring with a ball of arbitrarily small radius. Topology abstracts one step further and says, “we don’t even care about distance, we just want to see if you can make some abstract measurement on space that would show $x$ to be surrounded by $S$.”

So what are the “measurements” on a topological space? Its open sets! Remember, a topological space is a set accompanied by a set of things called “open sets” which are subsets of the original set, subject to various axioms. For us, using possible world semantics and the verifiability-topology, we can think of the opens sets (which are all the verifiable propositions) as “measurements” you could take. How does this translate for our topological operators?$x$ is in the topological interior of $S$ if there exists an open set (verifiable proposition) that includes $x$, and all other members of that open set are members of $S$. In math, $x\in intS⟺\exists O\in \tau :x\in O\wedge O\subseteq S$

For the rest of this sequence, I’m often going to talk in terms of measurements instead of talking about open sets in the topology. Just know that if you ever get confused, all my statements about measurements should cached out as some statements about open sets, which cache out as statements about the information basis.

At this point, you could re-examine the definitions of the topological operators, swapping out notions of drawing circles with the existance of measurements. Or.… you could check out this one WEIRD PICTURE that will make you A GENIUS at topology!

(Note: I used an upside down “P” because I was going for a mirror symmetry aesthetic, but it didn’t work. Just consider the upside down “P” to be $P^c$ or $\neg P$ (they’re the same in possible world semantics, remember?). I already made all the images and don’t want to change them)

For motivating this picture, consider the open sets of our topology to correspond to rectangles that don’t get small enough to cleanly fit into the squiggly boundary and only cover $P$ or $P^c$.

If you want you can just stare at this picture until you become enlightened. You can also keep reading as I walk through examples.

Decidability

If $BndrP=\varnothing$ then your problem is decidable. The only possible worlds are ones where you can cleanly measure whether $P$ or $\neg P$ is true.

Verifiable

If $Frnt\neg P=\varnothing$ then $P$ is verifiable: if $P$ is true, it is true in a way that lets us get a clean measurement showing it’s true. If it’s not true, maybe we get a clean measurement of $\neg P$, maybe we don’t. Note that a problem that is decidable is also verifiable.

Refutable

If $FrntP=\varnothing$ then $P$ is refutable: if $P$ is false, you can get a clean measurement showing it’s false. If it’s true, maybe you get a clean measurement of $P$, maybe you don’t.

These pictures help a lot with being able to see what problem statements are duals of each other, and also for translating problem statements into topological statements. See if you can match these problem statements to the corresponding topological ones:

1. $P$ is “strictly” verifiable (verifiable and not decidable)
2. $\neg P$ poses the problem of metaphysics for $P$
3. $P$ is “strictly” refutable (refutable and not decidable)
4. $\neg P$ poses the problem of induction for $P$
5. $\neg P$ is “strictly” verifiable

6. “You’re fucked”

7. $P$ poses the problem of metaphysics for $\neg P$

There’s lots of other fun exercises you can do to milk intuition from this image. Feel free to play around with it as much as you want. It will be helpful when thinking about and translating between topological ideas in the future.

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Answers:

1. $Frnt\neg P=ExtP=\varnothing$
2. $ExtP=\varnothing$
3. $IntP=FrntP=\varnothing$
4. $IntP=\varnothing$
5. $IntP=FrntP=\varnothing$
6. $IntP=ExtP=\varnothing$
7. $IntP=\varnothing$

[Pattern]

Errata:

[1] If were were looking at a space with a. [2] A general metric space abstracts the idea of measuring with a circle to measuring with a ball of arbitrarily small radius.

Sentence [1] appears incomplete.