TopCat.nonempty_limitCone_of_compact_t2_cofiltered_system

theorem TopCat.nonempty_limitCone_of_compact_t2_cofiltered_system : ∀ {J : Type u} [inst : CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J TopCatMax) [inst_1 : CategoryTheory.IsCofilteredOrEmpty J] [inst_2 : ∀ (j : J), Nonempty ↑(F.obj j)] [inst_3 : ∀ (j : J), CompactSpace ↑(F.obj j)] [inst_4 : ∀ (j : J), T2Space ↑(F.obj j)], Nonempty ↑(TopCat.limitCone F).pt

This theorem states that for any system of compact Hausdorff topological spaces indexed by a small, possibly empty, cofiltered category `J` (via a functor `F : J → TopCatMax`), the limit of this system (if it exists) is a nonempty topological space. In other words, if we have a diagram of topological spaces where each space is compact and Hausdorff, and the indexing category is cofiltered or empty (allowing us to form limits), then the limit of this diagram is not empty. The conditions of compactness and the T2 separation axiom (Hausdorff property) are required for each space in the system (for each object `j` in the category `J`).

More concisely: For any small, cofiltered category `J` and a functor `F : J → TopCatMax` mapping it to the category of compact Hausdorff topological spaces, the limit of the system (if it exists) is a nonempty topological space.