Complexity classes closed under finite union and intersection, why not infinite union and intersection?
Consider the "singleton" languages: $L=\{w\}$ for some fixed word $w$.  (e.g., the language $L=\{00010100\}$ would be a singleton language)  Most "nice" complexity classes include the singleton languages.

Every language can be written as an infinite union of singleton languages. So, if all singleton languages are in the class, and if the class is closed under infinite union, then all languages are in the class.

Thus, if there is any language at all that isn't in such a complexity class yet it does contain all the singleton languages, then the class cannot be closed under infinite union.