Completely distributive lattices are a special type of lattice where every subset has both a join (least upper bound) and a meet (greatest lower bound) that are compatible with the lattice operations. This means that the distributive property holds for arbitrary joins and meets, not just finite ones, making them a robust structure in order theory. They ensure that the operations of taking joins and meets can be interchanged freely, leading to many powerful results and applications in both algebra and topology.
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