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I know the definitions of $\sigma$-algebra, algebra and topology, but why countable/finite union, as in $\sigma$-algebra/algebra, and finite intersection, arbitrary union as in topology? What inspire or motivate these definitions? What will happen if we shuffle these words: finite/countable/arbitrary?

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Because in Euclidean spaces arbitrary union of open sets is open, and infinite intersection of open sets may fail to be so. That is why we have finite intersection and arbitrary union in topology. But for measure theory you need $\mu(\cup_{n=1}^{\infty}E_n)=\sum_{n=1}^{\infty}\mu(E_n)$, so as least $\cup_{n=1}^{\infty}E_n$ should be measurable. That is why we need countable union for $\sigma$-algebras. –  Hui Yu Nov 11 '12 at 3:54
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