This a just a small question about the definition of a sigma-algebra and what is eaxactly meant by countable? Would be grateful for any clarification on this. Most texts define a sigma-algebra ($\mathcal{A}$) on a set $X$ as a collection of subsets of $X$ that satisfy the following properties
(i) $\emptyset\in\mathcal{A}$
(ii) $A\in\mathcal{A}\Rightarrow X\backslash A\in\mathcal{A}$
(iii) If $A_1,A_2...\in\mathcal{A}$ then $\bigcup\limits_{i=1}^{\infty}A_i\in\mathcal{A}$, i.e. $\mathcal{A}$ is closed under countable unions.
If a set (say $A$) is countable, I understand it to mean that $|A|\leq\aleph_0$, so finite sets are countable (or do some texts just mean infinitely countable by countable), so my 1st question is why do some texts prove that sigma-algebras are closed under finite intersections, by defining for a collection of n sets the $n+k$ set, ($k\in\mathbb{N}$) to be the empty set, as surley if we take countable to be possibly finite then by (iii) we must have $((\mathcal{A}'\subset\mathcal{A}\space$ & $|\mathcal{A}'|<\aleph_0)\Rightarrow\bigcup\mathcal{A}'\in\mathcal{A})$. So then wouldn't (i) be redundant as $\emptyset$ is a countable (finite) subset of any collection of sets and $\bigcup\emptyset=\emptyset$. Any feedback to clarify this is most welcome. Thank You