One interesting infinitary operation that hasn’t been mentioned is the Suslin operation, sometimes (following Suslin) called operation A, which is important in descriptive set theory. If $\mathscr{F}=\{F_\sigma:\sigma\in^{&lt;\omega}\omega\}$ is a family of sets indexed by the set of finite sequences of natural numbers, the result of applying the Suslin operation to $\mathscr{F}$ is $$\bigcup_{\sigma\in^\omega\omega}\;\;\bigcap_{n\in\omega}\;F_{\sigma\upharpoonright n}\;.$$

Added: I should probably note that in terms of the notation in the question, $I$ here is $^{&lt;\omega}\omega$, and $A$ is some family of sets, e.g., the family of closed sets in a metric space.

Non-trivial examples of operations.
I haven't driven my car in about a week, and when I went to try and start it today, it would not start. It has enough battery to play the radio and whatnot, so I'm not sure if it's a battery related issue. 

When I turn the key to start the car, the check engine light flashes as it's being turned, and the engine produces a sound similar to a machine gun.

Incase my description isn't good enough, here's a video of what happens when I attempt to start it.
