TopologicalSpace.Clopens.exists_finset_eq_sup_prod

theorem TopologicalSpace.Clopens.exists_finset_eq_sup_prod : ∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] [inst_2 : CompactSpace Y] [inst_3 : CompactSpace X] (W : TopologicalSpace.Clopens (X × Y)), ∃ I, W = I.sup fun i => i.1 ×ˢ i.2

This theorem states that for any clopen set in the cartesian product of two compact spaces, there exists a finite set such that the clopen set can be expressed as the supremum of the cartesian product of elements from this finite set. In other words, any clopen set in a product of two compact spaces can be represented as a union of finitely many clopen boxes, where each box is the cartesian product of two sets, one from each of the original spaces. This is an important result in topology, which often needs to break down complex sets into simpler, 'box'-like subsets for analysis.

More concisely: Every clopen set in the product of two compact spaces can be expressed as the supremum of a finite number of cartesian products of clopen sets from each factor space.