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Out of idle curiosity, I'm wondering about all the various idempotent constructions we have in mathematics (they seem to be generally referred to as a "closure" or "completion"), and how some of them are related (e.g., the radical of an ideal and the closure of a subset of $k^n$ in the Zariski topology, via the Nullstellensatz - the radical and the topological closure both being idempotent).

So, one answer per post, but if you have two concepts which are related, I guess it'd be okay to put them together. For the sake of the completeness (ha ha) of this list, I'll add "radical" and "topological closure".

EDIT: My bad - I should have looked around more first. There's this list at Wikipedia and this list at nLab. Well, I'm sure there's plenty more concepts out there, so if you think of any more, feel free to add them. But let's focus on how some of these concepts are related - e.g., is the completion does one kind of a metric space it's order completion arise in terms of another? What are some ordergeneral ways in which completions and closures arise?

4 added 61 characters in body

Out of idle curiosity, I'm wondering about all the various idempotent constructions we have in mathematics (they seem to be generally referred to as a "closure" or "completion"), and how some of them are related (e.g., the radical of an ideal and the closure of a subset of $k^n$ in the Zariski topology, via the Nullstellensatz - the radical and the topological closure both being idempotent).

So, one answer per post, but if you have two concepts which are related, I guess it'd be okay to put them together. For the sake of the completeness (ha ha) of this list, I'll add "radical" and "topological closure".

EDIT: After looking My bad - I should have looked around a bit more , I found first. There's this list at nLab. Perhaps that's why someone voted Wikipedia and this downlist at nLab. Well, I'm sure there's plenty more concepts out there, so if you think of any more, feel free to add them. And again, I'm interested in But let's focus on how some of these concepts are related - e.g., is the completion of a metric space it's order completion in some order?

3 added 441 characters in body

Out of idle curiosity, I'm wondering about all the various idempotent constructions we have in mathematics (they seem to be generally referred to as a "closure" or "completion"), and how some of them are related (e.g., the radical of an ideal and the closure of a subset of $k^n$ in the Zariski topology, via the Nullstellensatz - the radical and the topological closure both being idempotent).

So, one answer per post, but if you have two concepts which are related, I guess it'd be okay to put them together. For the sake of the completeness (ha ha) of this list, I'll add "radical" and "topological closure".

EDIT: After looking around a bit more, I found this list at nLab. Perhaps that's why someone voted this down. Well, I'm sure there's plenty more concepts out there, so if you think of any more, feel free to add them. And again, I'm interested in how these concepts are related - e.g., is the completion of a metric space it's order completion in some order?

2 added 12 characters in body