What's the limit of a recursive superset operation called?

Question

What's the limit of a recursive superset operation called?

I'm tentatively thinking it's a colimit or an inverse limit but I don't know the correct terminology.

For example:

let $X=2\Bbb N-1$, i.e. the odd, positive integers.

Let $f(X)=\{1,2\}\cdot X$

Where the dot product is defined as $X\cdot Y=\{xy:x\in X, y\in Y\}$, i.e. the union of the products of every pair of elements drawn one from each set.

Then we have $f^{n-1}(X)\subsetneq f^{n}(X)$ and furthermore in this example $\displaystyle\lim_{n\to\infty}f^n(X)=\Bbb N$

where $n$ indicates the number of compositions of $f$.

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Surely there is some terminology for this type of limit?

Answer 1

In your case you're just talking about the union: $\mathbb N=\bigcup_n f^n(X)$.

However, this can also be thought of as a supremum, $\mathbb N = \sup_n f^n(X)$, since indeed, in the partially ordered set $\mathcal P(\mathbb N)$, $\mathbb N$ is the least upper bound of the family $(f^n(X))_n$. In measure theory, you often see the terminology:

$$\limsup_n A_n=\bigcap_{k\geq n}\left(\bigcup_n A_n\right)$$ $$\liminf_n A_n=\bigcup_{k\geq n}\left(\bigcap_n A_n\right)$$

which are just the direct transpositions of the usual definitions of $\limsup$ and $\liminf$ from the poset $\mathbb R$ to the poset $\mathcal P(\mathbb N)$. Notice that these two sets have nice set-theoretic interpretations: the $\limsup$ is the set of points that are in infinitely many of the $A_i$, and the $\liminf$ is the set of points that are in all of the $A_i$ past a certain point. If these two sets are equal, then nothing is stopping you from calling that common value $\lim A_n$, of course, and in your case we will indeed have $\lim f^n(X)=\mathbb N$, as you can check yourself. This is due to a proposition which should reinforce the analogy with limits in $\mathbb R$ even further:

If $A_n$ is an increasing (with respect to inclusion) sequence of sets, then it always has a limit and $\lim A_n=\sup A_n=\bigcup A_n$.

The reason for this analogy between the "topology" of $\mathcal P(\Omega)$ (for some set $\Omega$) and $\mathbb R$ is because, like $\mathbb R$, the poset $\mathcal P(\Omega)$ is a lattice satisfying the crucial least upper bound property.