These are precisely the relations obeyed by the ladder operators for an infinite set of independent quantum harmonic oscillators, one for each single-particle state.

Many theories of interest include an infinite set of axioms, however.

However, the product of an infinite set of metric spaces may not exist, because the distances in the base spaces may not have a supremum.

Discrete random variables can take on either a finite or at most a countably infinite set of discrete values (for example, the integers).

A context free grammar that defines the rules for a second grammar can yield an effectively infinite set of rules for the derived grammar.