What is a "closed set" in context of AI? How is it different from an open set or a frontier? I am reading some properties on the A* search algorithm on Wikipedia, specifically:

"If the heuristic function h is admissible, meaning that it never overestimates the actual minimal cost of reaching the goal, then A* is itself admissible (or optimal) if we do not use a closed set. If a closed set is used, then h must also be monotonic (or consistent) for A* to be optimal."

But I am not sure I understand what a "closed set" refers to. Is it different than an open set / frontier? If so, why? And why do we use a "closed set" as opposed to an "open set" for A*? Is a priority queue a closed set?
There are two variants of the A* algorithm:


The standard algorithm maintains two sets of vertices.  One called the "open set", and the other is called the "closed set".
An alternative version maintains just an open set, and doesn't use a closed set.  This requires that the heuristic function satisfy some some additional conditions.


See any standard treatment of A* for explanation of how it works and what the closed set is; e.g., A* search.