Most general result for intersecting a family of structures such as ideals and topologies to produce a new structure

Question

I sometimes encounter a kind of repetitive proof in which you take an arbitrary family of some kind of structure and show that its intersection also has this structure.

The proof goes something like this.

Fix a ring $R$. Let $F$ be a nonempty family of ideals in $R$. Consider $\cap F$. We now verify that $\cap F$ is an ideal by running back to the definition.

1. $0$ is in $\cap F$.
2. If $a$ is in $\cap F$, then $a$ is in every $I \in F$ and thus $ra$ is in $\cap F$ for each ring element $r$.
3. If $a-b$ is in $\cap F$, then $a$ and $b$ are in every $I \in F$ and thus $a-b$ is in every $I$ and thus $a-b$ is in $\cap F$.

The proof that the intersection of topologies (sets of open sets satisfying some rules) over a set of points $X$ is again a topology is also quite similar.

Fix a set of points $X$. Let $F$ be a nonempty family of topologies on $X$, i.e. $(X, \tau)$ is a topological space for each $\tau$ in $F$.

1. $\varnothing$ is in $\cap F$.
2. $\varnothing^c$ is in $\cap F$.
3. Let $W$ be an arbitrary subset of $\cap F$. $W$ is a subset of each $\tau$ in $F$. Thus $\cup W$ is a subset of each $\tau$ in $F$. Thus $\cup W$ is in $\cap F$.
4. Let $a$ and $b$ be elements of $\cap F$. Thus $a$ and $b$ are elements of each $\tau$ in $F$. Thus $a \cap b$ is an element of each $\tau$ in $F$. Thus $a \cap b$ is an element of $\cap F$.

I'm curious just how much we can generalize this.

Here's what I've come up with.

• Let $U$ be a nonempty set.
• Let $\lambda$ be a nonzero ordinal. Let $\text{Seq}[\bigcirc]$ denote $\lambda \to \bigcirc$.
• Let $\mathcal{F}$ be a family of functions each of which has the form $\text{Seq}[U] \to 2^U$.
• Let $f(x)$ be defined to be $\varnothing$ if $f$ is a function in $\mathcal{F}$ and $x$ is outside the domain of $x$.

If $X$ is a set, say that $X$ is closed under a function $f : \text{Seq}[U] \to 2^U$ if and only if it holds for all sequences $\vec{v}$ that $f(\vec{v})$ is a subset of $X$.

Say that $X$ is closed under a family of functions $\mathcal{F}$ precisely when $X$ is closed under $f$ for each $f$ in $\mathcal{F}$.

Lemma: Let $R$ be a set. Let $\mathcal{G}$ be a family of functions. Let $W$ be a nonempty family of subsets of $R$. Suppose each $V$ in $W$ is closed under $\mathcal{G}$. Then, $\cap V$ is closed under $\mathcal{G}$ as well.

Proof.

If $\cap W$ is empty, then we're done. Assume that it's not empty.

1. Choose a function $f$ in $\mathcal{G}$ arbitrarily.
2. Choose a sequence $\vec{v}$ with elements among $\cap W$ arbitrarily.
3. $\vec{v}$ is in every $V$ in $W$.
4. $f(\vec{v})$ is in every $V$ in $W$.
5. $f(\vec{v})$ is in $\cap W$.

Since $f$ was chosen arbitrarily, we're done.

Here's how you prove the ideal case and the topology case using this lemma.

As a technical detail, a nullary operation ignores all of its arguments, a unary operation ignores all but its first argument, a binary operation ignores all but its first two arguments, and so on. Functions that are well-defined return a singleton set containing only their result. That's how you shoehorn something like $+$ into $\text{Seq}[U] \to 2^U$.

1. For ideals, choose as our set of functions $0, +, -$ from the ring $R$ that all of our ideals are ideals of, as well as multiplication by $r$ for each element of $R$.
2. For a family of topologies on a topological space $X$, set $\lambda = \min(|X|, \omega)$. Our operations are the nullary operations $\varnothing$ and $\varnothing^c$, the binary operation $\cap$, and arbitrary unions on sequences $\cup(\vec{v})$.

We can also prove that the intersection of a family of subfields is a subfield by choosing as our set of functions, $0, +, -, *$ and the unary partial function $\bigcirc^{-1}$.

Answer 1

You don't need to exclude the empty family. The empty family of X'es always has the largest X as its intersection. So for example the intersection of the empty family of ideals of $R$ is the ideal $R$, and the intersection of the empty family of topologies is the discrete topology. I highly recommend to read MO/45951 for these things.

A family of sets closed under arbitrary intersections is called a Moore family. These correspond to closure operators. The well-known generalization of the examples you mentioned is the following:

Let $X$ be a set and let $(f_s : X^{n_s} \dashrightarrow X)_s$ be any family of operations on $X$. The arities $n_s$ can be any sets, not necessarily finite. And the operations can also be partially defined. Let us call $T \subseteq X$ closed with respect to this family if for all operations $f_s$ we have: If $x \in \mathrm{dom}(f_s)$ and $x_i \in T $ for all $i \in n_s$, then $f_s(x) \in T$. Then the set of closed subsets is a Moore family. The corresponding closure operator maps a set $T \subseteq X$ to the smallest closed subset $T' \subseteq X$ with $T \subseteq T$'. It can be constructed with a transfinite recursion, applying the $f_s$ to the elements of $T$ again and again.

We can also apply this, for example, to prove that $\sigma$-algebras on a set $\Omega$ are closed under arbitrary intersections. Notice that here the disjoint union operator $\sqcup : P(\Omega)^{\mathbb{N}} \dashrightarrow P(\Omega)$ is only partially defined, namely on the disjoint families.