(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)

Now that we've got the basic formalism, it's time to go from our intuitive notion of verifiable to our formal notion of verifiable.

$\text{Definition: A proposition}A\text{is verifiable in}(W,I)⟺\forall w\in A\exists e\in I\left(w\right):e\subseteq A$

This basically lines up. Something is verifiable if no matter how it's true, there exists some information that would tell you it's true..

Now, a series of leading questions with a fun surprise at the end:

Question 1: Can you verify the contradiction?

Yes! There is no world in which contradiction is true, and with verifiability we only care about what we can do in worlds where the proposition is true.

Question 2: Can you verify the tautology?

Yes! This follows from the first axiom of our info basis. For every world, you get at least one info state. An info state is a subset of $P\left(W\right)$, so no matter what the info state we have, it's a subset of $W$

Question 3: Can you verify the arbitrary finite union of verifiable propositions?

Yes! Takes a little more thought.

Question 4: Can you verify the intersection of any finite number of verifiable propositions?

Okay, get ready for this, here's The Sixth Sense "He was dead the whole time!" plot twist...

(clipped from wikipedia)

... crazy right?

If we take $W$ to be $X$, and $\tau$ to be "The set of all verifiable propositions in $W$", then what we just proved matches up exactly with this definition. So we get to claim that verifiable propositions in $W$ are open sets, and that they form a topology on our set of possible worlds.

What does this mean? From a practical stance, it means that we can now use all of topology, a rich and developed branch of math, to think and talk about verifiability. For any theorem that proves something about opens sets, we can make the same conclusion about verifiable propositions. Before we get into any of that, let's expand our mapping from verifiability concepts to topological concepts.

$\text{Definition:}A\text{is Closed}⟺A^c\text{is open}$

Okay, so we know that open is verifiable, let's see if we can connect an intuitive notion to closed. Let's look at the Halting Problem. In CS, you'd say the halting problem is semi-decidable. If it's going to halt, you'll see it happen, but if it doesn't you'll never know, maybe it loops forever. Semi-decidable lines up exactly with our notion of verifiability. So what's the negation of the halting problem?

$\neg \text{HALTS}=\left\{\right(M,x):M\text{doesn't halt on}x\}$

Now we've got a situation where if it's true ($M$ doesn't halt), we can't be guaranteed to know). But if it's false ($M$ does halt) we will find out.

Oh shit, this is refutability! To be open is to be verifiable, and to be closed is to be refutable. You can check for yourself, the negation of any refutable proposition is verifiable, and vice verse. Just like how the compliment of any open set is closed, and vice versa.