I understand what it means for a set to be closed under an operation, like addition or multiplication. But what does it mean for a set $T$ to be closed under subsets? Is subset an operation? I thought subsets were just sets with a certain property…
What does “closed under subsets” mean
elementary-set-theoryterminology
Related Solutions
In general, "closed under operation * on the set $\Sigma$" means that, if you apply operation * (in this case, taking countable intersections) to elements of the set $\Sigma$ (in this case, the set is the $\sigma$-algebra of subsets of a given set $X$), then the result is still an element of $\Sigma$.
Another example which may be more familiar to you: a vector space is closed under addition (addition of vectors is still a vector)
More precisely, in this case, if $\Sigma$ is a $\sigma$-algebra of subsets of $X$, then if $\{A_i\}_{i=1}^\infty \subset \Sigma$, i.e. $A_1, A_2,\dots, A_n, \dots \in \Sigma$ then $\bigcap_{i=1}^\infty A_i = A_1 \cap A_2 \cap \dots \cap A_n \cap \dots \in \Sigma$.
As I said in a comment above, by allowing countable intersections you get finite intersections too. If $A_1,\dots,A_n \in \Sigma$, then because $X\in \Sigma$ too by definition of $\sigma$-algebra, then using the fact $\Sigma$ is closed under countable intersections, $A_1\cap \dots \cap A_n = A_1 \cap \dots \cap A_n \cap X \cap X \cap X \cap \dots \in \Sigma$.
To answer to your addendum:
There seems to be a bit of confusion. If $X=\{x,y,z\}$, then $\mathcal{P}(X)=\{\emptyset,\{x\},\{y\},\{z\},\{x,y\},\{x,z\},\{y,z\},\{x,y,z\}\}$ is the power set of $X$, and a $\sigma$-algebra is a subset of the power set with some additional properties.
So, if $\Sigma=\{\emptyset,\{x\},\{y,z\},\{x,y,z\}\} \subset \mathcal{P}(X)$, then it is a $\sigma$-algebra of subsets of $X$.
To check item 2, what "closed under complementation" means is that, whenever $A\in \Sigma$, we have $A^c=X\setminus A \in \Sigma$. So, we check that for all four elements of $\Sigma$:
$\emptyset \in \Sigma \Rightarrow \{x,y,z\} \in \Sigma$.
$\{x\} \in \Sigma \Rightarrow \{y,z\} \in \Sigma$.
$\{y,z\} \in \Sigma \Rightarrow \{x\} \in \Sigma$.
$\{x,y,z\} \in \Sigma \Rightarrow \emptyset \in \Sigma$.
All these sentences are true, so item 2 is checked.
Also, as a side note, be careful, do not confuse $\emptyset$ with $\{\emptyset\}$.
In an ordinary mathematical context I would never say that an operation $\mathcal O$ is closed over a set $A$; I consider that an error for ‘$A$ is closed under $\mathcal O$’. The latter terminology can be properly applied very generally. For starters, if $A \subseteq S$, $n \in \omega$, and $f:S^n \to S$, ‘$A$ is closed under $f$’ means precisely that for every $\langle a_1,\dots,a_n \rangle \in A^n$, $f(a_1,\dots,a_n) \in A$.
This is the most common algebraic usage, I think, but it’s just the beginning. For instance, the non-negative integers $n$ can be replaced by any ordinal $\alpha$, and the operation need not be defined on all members of $S^\alpha$. Suppose that $X$ is a topological space and $A \subseteq X$. ‘$A$ is closed under (the operation of taking) limits of convergent sequences’ then means that if $\langle a_n:n \in \omega \rangle$ is a sequence of points in $A$ that converges as a sequence in the space $X$, its limit point is actually in $A$. Here $\alpha = \omega$, and the operation is defined only for those elements of $X^\omega$ that converge in $X$. This example at least starts to show the relationship between the general notion and that of topological closedness.
Moreover, the terminology is still used when the input to the operation isn’t ordered: it’s perfectly correct to say that a family $\mathcal A$ of sets is closed under (taking) finite intersections, for instance, meaning that if $\mathcal F$ is any finite subfamily of $\mathcal A$, $\bigcap \mathcal F \in \mathcal A$. If $S$ is the underlying set, the operation is $\mathcal O:[(\mathcal P(S)]^{< \omega} \to \mathcal P(S):\mathcal F \mapsto \bigcap \mathcal F$, and $\mathcal A \subseteq \mathcal P(S)$ is closed under it because $\mathcal O$ maps $[\mathcal A]^{< \omega}$ into $\mathcal A$. (Here $[X]^{< \omega}$ denotes the set of finite subsets of $X$.)
It would be a bit difficult to formulate an exhaustive formal definition of the usage, and I’m not at all sure that it would be particularly helpful; it seems more useful to present a variety of examples showing the flexibility of the usage.
Best Answer
If $T$ is a collection of sets, $T$ is closed under subsets (or closed under taking subsets) if it has the following property: if $s\subseteq t\in T$, then $s\in T$. That is, every subset of a member of $T$ is also a member of $T$.