[note dump] Iterations and Actions

[JmsNxn]

Example in topology take a topological space \(X\), let \(\bullet={0}\) the trivial topological space on the point, \(I=[0,1]\) the real interval and \(S^1=\{(x,y):x^2+y^2=1\}\) the circle.

• Continuous functions \(P:{0}\to X\) are in bijection with points of \(X\). Call them figures of shape \(\bullet\) or just elements of \(X\);
  
• Continuous functions \(\gamma:I\to X\) are in bijection with paths of \(X\). Call them figures of shape \(I\) or just \(I\)-elements of \(X\);
  
• Continuous functions \(\gamma :{}S^1\to X\) are in bijection with loops in \(X\). Call them figures of shape \(S^1\) or just \(I\)-elements of \(X\);
  

In this sense, we can refer to a sequence \((x_n,y_n) \to 0\) of complex numbers \(x_n,y_n \in \mathbb{C}\) are in bijection with holomorphic functions \(f(z)\) for \(z\) in a neighborhood of zero. Though we have to have a certain decay condition \(x_n = \mathcal{O}(1/n^{1+\delta})\) for \(\delta > 0\) (this could probably relaxed though). 

This essentially creates a bijection between accumulation points and sheafs at zero. This would create an equivalence class of sorts, but it fits perfectly in this list, and I think it would work great for this.


I realized I missed a few steps here...

Weierstrass products for:

\[
h(1/x_n) = 1/y_n\\
\]

Always exist, and they exist based (with scrutiny), on the behaviour at infinity. But this creates a complex identity theorem (with a bonus Weierstrass construction); which we can say:

If \((x_n, y_n) \to 0\) and \(f(x_n) = y_n\), then these objects are bijective to each other (The object of sequences, to holomorphic functions at zero). And all we're asking is an infinite occurrences of \(\bullet\) that converge....

Damn, Mphlee, I hope this makes sense....

Accumulation points \(\mathcal{A}\), given as \((x_n,y_n)\), maps exactly to sheafs at zero...

Remember when I say Weierstrass, we need to add that:

\[
\sum_{n=0}^\infty |x_n|<\infty\\
\]

So just keep that result, and you can make this identification with \(\bullet\) (infinite point wise operations, as sequences) across \(\mathbb{C}\) to make holomorphy at \(0\).

Also, when you talk about Weierstrass about zero, you don't talk about Weierstrass' factorization theorem. Where as the latter factorization theorem is designed for \(\mathbb{C}\), the former is used for simply connected domains. But it is still ultimately, Weierstrass.