How to correct description of Gustin's sequence space S122

[StevenClontz]

Blocks #1034, #1035, #1036, #1037, #1038

Months ago I wrote and forgot about https://math.stackexchange.com/questions/4895698/what-separation-axioms-are-held-by-gustins-sequence-space/4896300#4896300 At that time, I claimed we had errata on $\pi$-Base for S122's description, and attempted to write a new characterization of the space.

@Moniker1998 pointed out a few days ago that my recharacterization was flawed - I needed to specify that sequences had even length.

I think I've fixed my recharacterization on MSE (with actually a bit more care than Counterexamples about ensuring that neighborhoods only contain even-length sequences). But I'd like feedback on whether we should just fix the characterization we have currently and keep it similar to what is used in Counterexamples and used by @Moniker1998 in his MSE answer, or whether my reformulation/answer on MSE is actually preferrable and we should use it on $\pi$-Base instead.

[Moniker1998]

I think this boils down to how we define $V_i(n, w)$ and I think either is fine. In the book they defined what $U_i(\alpha)$ is for any sequence of finite length $\alpha$. If we have only defined that for even length sequences then sure, we could go with your formulation.... although it is a bit unhandy, I'd rather refer to $U_i(\alpha q(n, w))$ than $\bigcup_{w\in W, j\geq i} U_i(\alpha q(n, w)j)$.

Note that I was aware of this when I wrote those issues about Gustin's sequence space - the arguments are all valid.

I personally prefer the original notation for the sake of simplicity of referring to it - sure, some reader can get confused when it comes to the definition of the space, so it needs to be stated in a precise manner.

Perhaps in the definition, we could mention that $U_i(\alpha q(n, w)) = \bigcup_{j\geq i} U_i(\alpha q(n, w)j)$ as to make sure that people properly understand the notation.

[Moniker1998]

Other than this - I am not sure what you corrected, it seems to me that your definition and the definition in Counterexamples is precisely the same. Are you referring to the choice of function $q$? Sure, the property of being prime seems to not be needed - although it would be nice to argue somehow that such function $q$ exists (be it like in Counterexamples).

Sure, the definition of $q$ could be more general, I agree with that change, as long as we provide that such function $q$ can be chosen.

[StevenClontz]

Perhaps "recharacterization" isn't the right word; I just prefer to use standard notation for things like $(\omega^2)^{<\omega}$ for a finite sequence of pairs (isomorphic to finite even-length sequence) to help the reader parse things more quickly than writing out definitions in words.

The "correction" needs to be to our S122, which does not specify even length. But I don't like that $U_i(w)$ isn't a subspace of $Y$: it contains sequences of odd length greater than $|w|$, though I guess you can redefine $U_i(w)'=U_i(w)$ and things are fine. I agree that $U_i(\alpha^\frown\langle q(n,w)\rangle)$ is cleaner than the union of $U_i(\alpha^\frown\langle q(n,w),j\rangle)$. (Note: I really don't like the weak typing concatenating sequences of ordinals with bare ordinals like $\alpha q(n,w) j$ .)

I guess my other pet peeve is when authors make people think about number theory when they just want a nice injection.

(If no one else has opinions here, then I think the tie goes towards the status quo; fix the error by requiring even length sequences, and start implementing your observations.)

[Moniker1998]

Perhaps "recharacterization" isn't the right word; I just prefer to use standard notation for things like ( ω 2 ) < ω for a finite sequence of pairs (isomorphic to finite even-length sequence) to help the reader parse things more quickly than writing out definitions in words.

Writing $\omega^2$ might be confusing, since $\omega^2$ can be interpreted as an ordinal $\omega\cdot \omega$.

The "correction" needs to be to our S122, which does not specify even length. But I don't like that U i ( w ) isn't a subspace of Y : it contains sequences of odd length greater than | w | , though I guess you can redefine U i ( w ) ′ = U i ( w ) and things are fine. I agree that U i ( α ⌢ ⟨ q ( n , w ) ⟩ ) is cleaner than the union of U i ( α ⌢ ⟨ q ( n , w ) , j ⟩ ) . (Note: I really don't like the weak typing concatenating sequences of ordinals with bare ordinals like α q ( n , w ) j .)

$U_i(\gamma)$ is always a subspace of $Y$ as per my definitions, and the definitions in Counterexamples! Its a set of those $\alpha\in Y$ that ...

About the notation $\alpha\beta$... I find it easier on the eyes than the $\frown$

[Moniker1998]

@StevenClontz Can I start working on Gustin's sequence space, and if so, are you fine with my notation? Or would you rather I use your notation?

[StevenClontz]

No one else chimed in, so the tie goes to the status quo I think. I'll make a PR now to make the minor fix (rather than propose my more major shift in notation).