Sigma algebra and algebra difference

1. An algebra is a collection of subsets closed under finite unions and intersections.

2. A sigma algebra is a collection closed under countable unions and intersections.

Whats the difference between finite and countable unions and intersections? Does "countable" mean it implies there can be infinitely many unions and intersections?

Secondly, I was reading a definition

For an algebra on a set: By De Morgan's law, $A \cap B = (A^c \cup B^c)^c$, thus an algebra is a collection of subsets closed under finite unions and intersections.

What law are they using here to get $A \cap B = (A^c \cup B^c)^c$? I thought de morgan's law was $(A\cap B)^c = A^c \cup B^c$?

Finally, what exactly do they mean by "closed under finite unions and intersections?

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Take your version of the De Morgan law, then take the complement of both sides. –  André Nicolas May 27 '12 at 22:33
Ahh okay, thanks! –  Steven May 27 '12 at 22:51

A common case where this might come up is with respect to open and closed sets. A finite union of closed sets is closed. But an infinite union of closed sets might not be closed. For example, if we consider the sets $I_n = [\frac{1}{n}, 1 - \frac{1}{n}]$, then each $I_n$ is closed. But $\cup_{n \in \mathbb{N}} I_n = (0,1)$, an open set.
With respect to your De Morgan's law question: It is a fundamental fact that $A = B \iff A^c = B^c$, and that $(A^c)^c = A$. So they complemented your De Morgan's law to get that statement.