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Theorem List for Metamath Proof Explorer - 39301-39400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremntrk2imkb 39301* If an interior function is contracting, the interiors of disjoint sets are disjoint. Kuratowski's K2 axiom implies KB. Interior version. (Contributed by RP, 9-Jun-2021.)
(∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 → ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅))

Theoremntrkbimka 39302* If the interiors of disjoint sets are disjoint, then the interior of the empty set is the empty set. (Contributed by RP, 14-Jun-2021.)
(∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) → (𝐼‘∅) = ∅)

Theoremntrk0kbimka 39303* If the interiors of disjoint sets are disjoint and the interior of the base set is the base set, then the interior of the empty set is the empty set. Obsolete version of ntrkbimka 39302. (Contributed by RP, 12-Jun-2021.)
((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)) → (𝐼‘∅) = ∅))

Theoremclsk3nimkb 39304* If the base set is not empty, axiom K3 does not imply KB. A concrete example with a pseudo-closure function of 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏𝑥)) is given. (Contributed by RP, 16-Jun-2021.)
¬ ∀𝑏𝑘 ∈ (𝒫 𝑏𝑚 𝒫 𝑏)(∀𝑠 ∈ 𝒫 𝑏𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡)) → ∀𝑠 ∈ 𝒫 𝑏𝑡 ∈ 𝒫 𝑏((𝑠𝑡) = 𝑏 → ((𝑘𝑠) ∪ (𝑘𝑡)) = 𝑏))

Theoremclsk1indlem0 39305 The ansatz closure function (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) has the K0 property of preserving the nullary union. (Contributed by RP, 6-Jul-2021.)
𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))       (𝐾‘∅) = ∅

Theoremclsk1indlem2 39306* The ansatz closure function (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) has the K2 property of expanding. (Contributed by RP, 6-Jul-2021.)
𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))       𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝐾𝑠)

Theoremclsk1indlem3 39307* The ansatz closure function (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) has the K3 property of being sub-linear. (Contributed by RP, 6-Jul-2021.)
𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))       𝑠 ∈ 𝒫 3o𝑡 ∈ 𝒫 3o(𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡))

Theoremclsk1indlem4 39308* The ansatz closure function (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) has the K4 property of idempotence. (Contributed by RP, 6-Jul-2021.)
𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))       𝑠 ∈ 𝒫 3o(𝐾‘(𝐾𝑠)) = (𝐾𝑠)

Theoremclsk1indlem1 39309* The ansatz closure function (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) does not have the K1 property of isotony. (Contributed by RP, 6-Jul-2021.)
𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))       𝑠 ∈ 𝒫 3o𝑡 ∈ 𝒫 3o(𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡))

Theoremclsk1independent 39310* For generalized closure functions, property K1 (isotony) is independent of the properties K0, K2, K3, K4. This contradicts a claim which appears in preprints of Table 2 in Bärbel M. R. Stadler and Peter F. Stadler. "Generalized Topological Spaces in Evolutionary Theory and Combinatorial Chemistry." J. Chem. Inf. Comput. Sci., 42:577-585, 2002. Proceedings MCC 2001, Dubrovnik. The same table row implying K1 follows from the other four appears in the supplemental materials Bärbel M. R. Stadler and Peter F. Stadler. "Basic Properties of Closure Spaces" 2001 on page 12. (Contributed by RP, 5-Jul-2021.)
(𝜑 ↔ (𝑘‘∅) = ∅)    &   (𝜓 ↔ ∀𝑠 ∈ 𝒫 𝑏𝑡 ∈ 𝒫 𝑏(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡)))    &   (𝜒 ↔ ∀𝑠 ∈ 𝒫 𝑏𝑠 ⊆ (𝑘𝑠))    &   (𝜃 ↔ ∀𝑠 ∈ 𝒫 𝑏𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡)))    &   (𝜏 ↔ ∀𝑠 ∈ 𝒫 𝑏(𝑘‘(𝑘𝑠)) = (𝑘𝑠))        ¬ ∀𝑏𝑘 ∈ (𝒫 𝑏𝑚 𝒫 𝑏)(((𝜑𝜒) ∧ (𝜃𝜏)) → 𝜓)

Theoremneik0pk1imk0 39311* Kuratowski's K0' and K1 axioms imply K0. Neighborhood version. (Contributed by RP, 3-Jun-2021.)
(𝜑𝐵𝑉)    &   (𝜑𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵))    &   (𝜑 → ∀𝑥𝐵 (𝑁𝑥) ≠ ∅)    &   (𝜑 → ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)))       (𝜑 → ∀𝑥𝐵 𝐵 ∈ (𝑁𝑥))

Theoremisotone1 39312* Two different ways to say subset relation persists across applications of a function. (Contributed by RP, 31-May-2021.)
(∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝑎𝑏 → (𝐹𝑎) ⊆ (𝐹𝑏)) ↔ ∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ (𝐹‘(𝑎𝑏)))

Theoremisotone2 39313* Two different ways to say subset relation persists across applications of a function. (Contributed by RP, 31-May-2021.)
(∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝑎𝑏 → (𝐹𝑎) ⊆ (𝐹𝑏)) ↔ ∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎𝑏)) ⊆ ((𝐹𝑎) ∩ (𝐹𝑏)))

Theoremntrk1k3eqk13 39314* An interior function is both monotone and sub-linear if and only if it is finitely linear. (Contributed by RP, 18-Jun-2021.)
((∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡))) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)))

Theoremntrclsf1o 39315* If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator we may characterize the relation as part of a 1-to-1 onto function. (Contributed by RP, 29-May-2021.)
𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)       (𝜑𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵))

Theoremntrclsnvobr 39316* If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then they are related the opposite way. (Contributed by RP, 21-May-2021.)
𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)       (𝜑𝐾𝐷𝐼)

Theoremntrclsiex 39317* If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then those functions are maps of subsets to subsets. (Contributed by RP, 21-May-2021.)
𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)       (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))

Theoremntrclskex 39318* If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then those functions are maps of subsets to subsets. (Contributed by RP, 21-May-2021.)
𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)       (𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))

Theoremntrclsfv1 39319* If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is a functional relation between them (Contributed by RP, 28-May-2021.)
𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)       (𝜑 → (𝐷𝐼) = 𝐾)

Theoremntrclsfv2 39320* If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is a functional relation between them (Contributed by RP, 28-May-2021.)
𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)       (𝜑 → (𝐷𝐾) = 𝐼)

Theoremntrclselnel1 39321* If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is an equivalence between membership in the interior of a set and non-membership in the closure of the complement of the set. (Contributed by RP, 28-May-2021.)
𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)    &   (𝜑𝑋𝐵)    &   (𝜑𝑆 ∈ 𝒫 𝐵)       (𝜑 → (𝑋 ∈ (𝐼𝑆) ↔ ¬ 𝑋 ∈ (𝐾‘(𝐵𝑆))))

Theoremntrclselnel2 39322* If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is an equivalence between membership in interior of the complement of a set and non-membership in the closure of the set. (Contributed by RP, 28-May-2021.)
𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)    &   (𝜑𝑋𝐵)    &   (𝜑𝑆 ∈ 𝒫 𝐵)       (𝜑 → (𝑋 ∈ (𝐼‘(𝐵𝑆)) ↔ ¬ 𝑋 ∈ (𝐾𝑆)))

Theoremntrclsfv 39323* The value of the interior (closure) expressed in terms of the closure (interior). (Contributed by RP, 25-Jun-2021.)
𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)    &   (𝜑𝑆 ∈ 𝒫 𝐵)       (𝜑 → (𝐼𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑆))))

Theoremntrclsfveq1 39324* If interior and closure functions are related then specific function values are complementary. (Contributed by RP, 27-Jun-2021.)
𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)    &   (𝜑𝑆 ∈ 𝒫 𝐵)    &   (𝜑𝐶 ∈ 𝒫 𝐵)       (𝜑 → ((𝐼𝑆) = 𝐶 ↔ (𝐾‘(𝐵𝑆)) = (𝐵𝐶)))

Theoremntrclsfveq2 39325* If interior and closure functions are related then specific function values are complementary. (Contributed by RP, 27-Jun-2021.)
𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)    &   (𝜑𝑆 ∈ 𝒫 𝐵)    &   (𝜑𝐶 ∈ 𝒫 𝐵)       (𝜑 → ((𝐼‘(𝐵𝑆)) = 𝐶 ↔ (𝐾𝑆) = (𝐵𝐶)))

Theoremntrclsfveq 39326* If interior and closure functions are related then equality of a pair of function values is equivalent to equality of a pair of the other function's values. (Contributed by RP, 27-Jun-2021.)
𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)    &   (𝜑𝑆 ∈ 𝒫 𝐵)    &   (𝜑𝑇 ∈ 𝒫 𝐵)       (𝜑 → ((𝐼𝑆) = (𝐼𝑇) ↔ (𝐾‘(𝐵𝑆)) = (𝐾‘(𝐵𝑇))))

Theoremntrclsss 39327* If interior and closure functions are related then a subset relation of a pair of function values is equivalent to subset relation of a pair of the other function's values. (Contributed by RP, 27-Jun-2021.)
𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)    &   (𝜑𝑆 ∈ 𝒫 𝐵)    &   (𝜑𝑇 ∈ 𝒫 𝐵)       (𝜑 → ((𝐼𝑆) ⊆ (𝐼𝑇) ↔ (𝐾‘(𝐵𝑇)) ⊆ (𝐾‘(𝐵𝑆))))

Theoremntrclsneine0lem 39328* If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that at least one (pseudo-)neighborbood of a particular point exists hold equally. (Contributed by RP, 21-May-2021.)
𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)    &   (𝜑𝑋𝐵)       (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐾𝑠)))

Theoremntrclsneine0 39329* If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that for every point, at least one (pseudo-)neighborbood exists hold equally. (Contributed by RP, 21-May-2021.)
𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)       (𝜑 → (∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑥 ∈ (𝐼𝑠) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵 ¬ 𝑥 ∈ (𝐾𝑠)))

Theoremntrclscls00 39330* If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 1-Jun-2021.)
𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)       (𝜑 → ((𝐼𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅))

Theoremntrclsiso 39331* If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that either is isotonic hold equally. (Contributed by RP, 3-Jun-2021.)
𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐾𝑠) ⊆ (𝐾𝑡))))

Theoremntrclsk2 39332* An interior function is contracting if and only if the closure function is expansive. (Contributed by RP, 9-Jun-2021.)
𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐾𝑠)))

Theoremntrclskb 39333* The interiors of disjoint sets are disjoint if and only if the closures of sets that span the base set also span the base set. (Contributed by RP, 10-Jun-2021.)
𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = 𝐵 → ((𝐾𝑠) ∪ (𝐾𝑡)) = 𝐵)))

Theoremntrclsk3 39334* The intersection of interiors of a every pair is a subset of the interior of the intersection of the pair if an only if the closure of the union of every pair is a subset of the union of closures of the pair. (Contributed by RP, 19-Jun-2021.)
𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡))))

Theoremntrclsk13 39335* The interior of the intersection of any pair is equal to the intersection of the interiors if and only if the closure of the unions of any pair is equal to the union of closures. (Contributed by RP, 19-Jun-2021.)
𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠𝑡)) = ((𝐾𝑠) ∪ (𝐾𝑡))))

Theoremntrclsk4 39336* Idempotence of the interior function is equivalent to idempotence of the closure function. (Contributed by RP, 10-Jul-2021.)
𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼𝑠)) = (𝐼𝑠) ↔ ∀𝑠 ∈ 𝒫 𝐵(𝐾‘(𝐾𝑠)) = (𝐾𝑠)))

Theoremntrneibex 39337* If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then the base set exists. (Contributed by RP, 29-May-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑𝐵 ∈ V)

Theoremntrneircomplex 39338* The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 26-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)

Theoremntrneif1o 39339* If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, we may characterize the relation as part of a 1-to-1 onto function. (Contributed by RP, 29-May-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵))

Theoremntrneiiex 39340* If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then the interior function exists. (Contributed by RP, 29-May-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))

Theoremntrneinex 39341* If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then the neighborhood function exists. (Contributed by RP, 29-May-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵))

Theoremntrneicnv 39342* If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then converse of 𝐹 is known. (Contributed by RP, 29-May-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑𝐹 = (𝐵𝑂𝒫 𝐵))

Theoremntrneifv1 39343* If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then the function value of 𝐹 is the neighborhood function. (Contributed by RP, 29-May-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (𝐹𝐼) = 𝑁)

Theoremntrneifv2 39344* If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then the function value of converse of 𝐹 is the interior function. (Contributed by RP, 29-May-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (𝐹𝑁) = 𝐼)

Theoremntrneiel 39345* If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then there is an equivalence between membership in the interior of a set and non-membership in the closure of the complement of the set. (Contributed by RP, 29-May-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)    &   (𝜑𝑋𝐵)    &   (𝜑𝑆 ∈ 𝒫 𝐵)       (𝜑 → (𝑋 ∈ (𝐼𝑆) ↔ 𝑆 ∈ (𝑁𝑋)))

Theoremntrneifv3 39346* The value of the neighbors (convergents) expressed in terms of the interior (closure) function. (Contributed by RP, 26-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑁𝑋) = {𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠)})

Theoremntrneineine0lem 39347* If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that for every point, at least one (pseudo-)neighborbood exists hold equally. (Contributed by RP, 29-May-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)    &   (𝜑𝑋𝐵)       (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠) ↔ (𝑁𝑋) ≠ ∅))

Theoremntrneineine1lem 39348* If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that for every point, at not all subsets are (pseudo-)neighborboods hold equally. (Contributed by RP, 1-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)    &   (𝜑𝑋𝐵)       (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐼𝑠) ↔ (𝑁𝑋) ≠ 𝒫 𝐵))

Theoremntrneifv4 39349* The value of the interior (closure) expressed in terms of the neighbors (convergents) function. (Contributed by RP, 26-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)    &   (𝜑𝑆 ∈ 𝒫 𝐵)       (𝜑 → (𝐼𝑆) = {𝑥𝐵𝑆 ∈ (𝑁𝑥)})

Theoremntrneiel2 39350* Membership in iterated interior of a set is equivalent to there existing a particular neighborhood of that member such that points are members of that neighborhood if and only if the set is a neighborhood of each of those points. (Contributed by RP, 11-Jul-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)    &   (𝜑𝑋𝐵)    &   (𝜑𝑆 ∈ 𝒫 𝐵)       (𝜑 → (𝑋 ∈ (𝐼‘(𝐼𝑆)) ↔ ∃𝑢 ∈ (𝑁𝑋)∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦))))

Theoremntrneineine0 39351* If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that for every point, at least one (pseudo-)neighborbood exists hold equally. (Contributed by RP, 29-May-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑥 ∈ (𝐼𝑠) ↔ ∀𝑥𝐵 (𝑁𝑥) ≠ ∅))

Theoremntrneineine1 39352* If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that for every point, at not all subsets are (pseudo-)neighborboods hold equally. (Contributed by RP, 1-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑥𝐵𝑠 ∈ 𝒫 𝐵 ¬ 𝑥 ∈ (𝐼𝑠) ↔ ∀𝑥𝐵 (𝑁𝑥) ≠ 𝒫 𝐵))

Theoremntrneicls00 39353* If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 2-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → ((𝐼𝐵) = 𝐵 ↔ ∀𝑥𝐵 𝐵 ∈ (𝑁𝑥)))

Theoremntrneicls11 39354* If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that the interior of the empty set is the empty set hold equally. (Contributed by RP, 2-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → ((𝐼‘∅) = ∅ ↔ ∀𝑥𝐵 ¬ ∅ ∈ (𝑁𝑥)))

Theoremntrneiiso 39355* If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that the interior function is isotonic hold equally. (Contributed by RP, 3-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))

Theoremntrneik2 39356* An interior function is contracting if and only if all the neighborhoods of a point contain that point. (Contributed by RP, 11-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁𝑥) → 𝑥𝑠)))

Theoremntrneix2 39357* An interior (closure) function is expansive if and only if all subsets which contain a point are neighborhoods (convergents) of that point. (Contributed by RP, 11-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐼𝑠) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵(𝑥𝑠𝑠 ∈ (𝑁𝑥))))

Theoremntrneikb 39358* The interiors of disjoint sets are disjoint if and only if the neighborhoods of every point contain no disjoint sets. (Contributed by RP, 11-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)) → (𝑠𝑡) ≠ ∅)))

Theoremntrneixb 39359* The interiors (closures) of sets that span the base set also span the base set if and only if the neighborhoods (convergents) of every point contain at least one of every pair of sets that span the base set. (Contributed by RP, 11-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = 𝐵 → ((𝐼𝑠) ∪ (𝐼𝑡)) = 𝐵) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))

Theoremntrneik3 39360* The intersection of interiors of any pair is a subset of the interior of the intersection if and only if the intersection of any two neighborhoods of a point is also a neighborhood. (Contributed by RP, 19-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)) → (𝑠𝑡) ∈ (𝑁𝑥))))

Theoremntrneix3 39361* The closure of the union of any pair is a subset of the union of closures if and only if the union of any pair belonging to the convergents of a point implies at least one of the pair belongs to the the convergents of that point. (Contributed by RP, 19-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))

Theoremntrneik13 39362* The interior of the intersection of any pair equals intersection of interiors if and only if the intersection of any pair belonging to the neighborhood of a point is equivalent to both of the pair belonging to the neighborhood of that point. (Contributed by RP, 19-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)))))

Theoremntrneix13 39363* The closure of the union of any pair is equal to the union of closures if and only if the union of any pair belonging to the convergents of a point if equivalent to at least one of the pain belonging to the convergents of that point. (Contributed by RP, 19-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))

Theoremntrneik4w 39364* Idempotence of the interior function is equivalent to saying a set is a neighborhood of a point if and only if the interior of the set is a neighborhood of a point. (Contributed by RP, 11-Jul-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼𝑠)) = (𝐼𝑠) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁𝑥) ↔ (𝐼𝑠) ∈ (𝑁𝑥))))

Theoremntrneik4 39365* Idempotence of the interior function is equivalent to stating a set, 𝑠, is a neighborhood of a point, 𝑥 is equivalent to there existing a special neighborhood, 𝑢, of 𝑥 such that a point is an element of the special neighborhood if and only if 𝑠 is also a neighborhood of the point. (Contributed by RP, 11-Jul-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼𝑠)) = (𝐼𝑠) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁𝑥) ↔ ∃𝑢 ∈ (𝑁𝑥)∀𝑦𝐵 (𝑦𝑢𝑠 ∈ (𝑁𝑦)))))

Theoremclsneibex 39366 If (pseudo-)closure and (pseudo-)neighborhood functions are related by the composite operator, 𝐻, then the base set exists. (Contributed by RP, 4-Jun-2021.)
𝐷 = (𝑃𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)       (𝜑𝐵 ∈ V)

Theoremclsneircomplex 39367 The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 26-Jun-2021.)
𝐷 = (𝑃𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)       (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)

Theoremclsneif1o 39368* If a (pseudo-)closure function and a (pseudo-)neighborhood function are related by the 𝐻 operator, then the operator is a one-to-one, onto mapping. (Contributed by RP, 5-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)       (𝜑𝐻:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵))

Theoremclsneicnv 39369* If a (pseudo-)closure function and a (pseudo-)neighborhood function are related by the 𝐻 operator, then the converse of the operator is known. (Contributed by RP, 5-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)       (𝜑𝐻 = (𝐷 ∘ (𝐵𝑂𝒫 𝐵)))

Theoremclsneikex 39370* If closure and neighborhoods functions are related, the closure function exists. (Contributed by RP, 27-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)       (𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))

Theoremclsneinex 39371* If closure and neighborhoods functions are related, the neighborhoods function exists. (Contributed by RP, 27-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)       (𝜑𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵))

Theoremclsneiel1 39372* If a (pseudo-)closure function and a (pseudo-)neighborhood function are related by the 𝐻 operator, then membership in the closure of a subset is equivalent to the complement of the subset not being a neighborhood of the point. (Contributed by RP, 7-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)    &   (𝜑𝑋𝐵)    &   (𝜑𝑆 ∈ 𝒫 𝐵)       (𝜑 → (𝑋 ∈ (𝐾𝑆) ↔ ¬ (𝐵𝑆) ∈ (𝑁𝑋)))

Theoremclsneiel2 39373* If a (pseudo-)closure function and a (pseudo-)neighborhood function are related by the 𝐻 operator, then membership in the closure of the complement of a subset is equivalent to the subset not being a neighborhood of the point. (Contributed by RP, 7-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)    &   (𝜑𝑋𝐵)    &   (𝜑𝑆 ∈ 𝒫 𝐵)       (𝜑 → (𝑋 ∈ (𝐾‘(𝐵𝑆)) ↔ ¬ 𝑆 ∈ (𝑁𝑋)))

Theoremclsneifv3 39374* Value of the neighborhoods (convergents) in terms of the closure (interior) function. (Contributed by RP, 27-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑁𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ (𝐾‘(𝐵𝑠))})

Theoremclsneifv4 39375* Value of the closure (interior) function in terms of the neighborhoods (convergents) function. (Contributed by RP, 27-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)    &   (𝜑𝑆 ∈ 𝒫 𝐵)       (𝜑 → (𝐾𝑆) = {𝑥𝐵 ∣ ¬ (𝐵𝑆) ∈ (𝑁𝑥)})

Theoremneicvgbex 39376 If (pseudo-)neighborhood and (pseudo-)convergent functions are related by the composite operator, 𝐻, then the base set exists. (Contributed by RP, 4-Jun-2021.)
𝐷 = (𝑃𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)       (𝜑𝐵 ∈ V)

Theoremneicvgrcomplex 39377 The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 26-Jun-2021.)
𝐷 = (𝑃𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)       (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)

Theoremneicvgf1o 39378* If neighborhood and convergent functions are related by operator 𝐻, it is a one-to-one onto relation. (Contributed by RP, 11-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐺 = (𝐵𝑂𝒫 𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)       (𝜑𝐻:(𝒫 𝒫 𝐵𝑚 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵))

Theoremneicvgnvo 39379* If neighborhood and convergent functions are related by operator 𝐻, it is its own converse function. (Contributed by RP, 11-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐺 = (𝐵𝑂𝒫 𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)       (𝜑𝐻 = 𝐻)

Theoremneicvgnvor 39380* If neighborhood and convergent functions are related by operator 𝐻, the relationship holds with the functions swapped. (Contributed by RP, 11-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐺 = (𝐵𝑂𝒫 𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)       (𝜑𝑀𝐻𝑁)

Theoremneicvgmex 39381* If the neighborhoods and convergents functions are related, the convergents function exists. (Contributed by RP, 27-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐺 = (𝐵𝑂𝒫 𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)       (𝜑𝑀 ∈ (𝒫 𝒫 𝐵𝑚 𝐵))

Theoremneicvgnex 39382* If the neighborhoods and convergents functions are related, the neighborhoods function exists. (Contributed by RP, 27-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐺 = (𝐵𝑂𝒫 𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)       (𝜑𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵))

Theoremneicvgel1 39383* A subset being an element of a neighborhood of a point is equivalent to the complement of that subset not being a element of the convergent of that point. (Contributed by RP, 12-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐺 = (𝐵𝑂𝒫 𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)    &   (𝜑𝑋𝐵)    &   (𝜑𝑆 ∈ 𝒫 𝐵)       (𝜑 → (𝑆 ∈ (𝑁𝑋) ↔ ¬ (𝐵𝑆) ∈ (𝑀𝑋)))

Theoremneicvgel2 39384* The complement of a subset being an element of a neighborhood at a point is equivalent to that subset not being a element of the convergent at that point. (Contributed by RP, 12-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐺 = (𝐵𝑂𝒫 𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)    &   (𝜑𝑋𝐵)    &   (𝜑𝑆 ∈ 𝒫 𝐵)       (𝜑 → ((𝐵𝑆) ∈ (𝑁𝑋) ↔ ¬ 𝑆 ∈ (𝑀𝑋)))

Theoremneicvgfv 39385* The value of the neighborhoods (convergents) in terms of the the convergents (neighborhoods) function. (Contributed by RP, 27-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐺 = (𝐵𝑂𝒫 𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑁𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵𝑠) ∈ (𝑀𝑋)})

Theoremntrrn 39386 The range of the interior function of a topology a subset of the open sets of the topology. (Contributed by RP, 22-Apr-2021.)
𝑋 = 𝐽    &   𝐼 = (int‘𝐽)       (𝐽 ∈ Top → ran 𝐼𝐽)

Theoremntrf 39387 The interior function of a topology is a map from the powerset of the base set to the open sets of the topology. (Contributed by RP, 22-Apr-2021.)
𝑋 = 𝐽    &   𝐼 = (int‘𝐽)       (𝐽 ∈ Top → 𝐼:𝒫 𝑋𝐽)

Theoremntrf2 39388 The interior function is a map from the powerset of the base set to itself. (Contributed by RP, 22-Apr-2021.)
𝑋 = 𝐽    &   𝐼 = (int‘𝐽)       (𝐽 ∈ Top → 𝐼:𝒫 𝑋⟶𝒫 𝑋)

Theoremntrelmap 39389 The interior function is a map from the powerset of the base set to itself. (Contributed by RP, 22-Apr-2021.)
𝑋 = 𝐽    &   𝐼 = (int‘𝐽)       (𝐽 ∈ Top → 𝐼 ∈ (𝒫 𝑋𝑚 𝒫 𝑋))

Theoremclsf2 39390 The closure function is a map from the powerset of the base set to itself. This is less precise than clsf 21264. (Contributed by RP, 22-Apr-2021.)
𝑋 = 𝐽    &   𝐾 = (cls‘𝐽)       (𝐽 ∈ Top → 𝐾:𝒫 𝑋⟶𝒫 𝑋)

Theoremclselmap 39391 The closure function is a map from the powerset of the base set to itself. (Contributed by RP, 22-Apr-2021.)
𝑋 = 𝐽    &   𝐾 = (cls‘𝐽)       (𝐽 ∈ Top → 𝐾 ∈ (𝒫 𝑋𝑚 𝒫 𝑋))

Theoremdssmapntrcls 39392* The interior and closure operators on a topology are duals of each other. See also kur14lem2 31792. (Contributed by RP, 21-Apr-2021.)
𝑋 = 𝐽    &   𝐾 = (cls‘𝐽)    &   𝐼 = (int‘𝐽)    &   𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))))    &   𝐷 = (𝑂𝑋)       (𝐽 ∈ Top → 𝐼 = (𝐷𝐾))

Theoremdssmapclsntr 39393* The closure and interior operators on a topology are duals of each other. See also kur14lem2 31792. (Contributed by RP, 22-Apr-2021.)
𝑋 = 𝐽    &   𝐾 = (cls‘𝐽)    &   𝐼 = (int‘𝐽)    &   𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))))    &   𝐷 = (𝑂𝑋)       (𝐽 ∈ Top → 𝐾 = (𝐷𝐼))

20.28.4.3  Generic Neighborhood Spaces

Any neighborhood space is an open set topology and any open set topology is a neighborhood space. Seifert and Threlfall define a generic neighborhood space which is a superset of what is now generally used and related concepts and the following will show that those definitions apply to elements of Top.

Seifert and Threlfall do not allow neighborhood spaces on the empty set while sn0top 21215 is an example of a topology with an empty base set. This divergence is unlikely to pose serious problems.

Theoremgneispa 39394* Each point 𝑝 of the neighborhood space has at least one neighborhood; each neighborhood of 𝑝 contains 𝑝. Axiom A of Seifert and Threlfall. (Contributed by RP, 5-Apr-2021.)
𝑋 = 𝐽       (𝐽 ∈ Top → ∀𝑝𝑋 (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝𝑛))

Theoremgneispb 39395* Given a neighborhood 𝑁 of 𝑃, each subset of the neighborhood space containing this neighborhood is also a neighborhood of 𝑃. Axiom B of Seifert and Threlfall. (Contributed by RP, 5-Apr-2021.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → ∀𝑠 ∈ 𝒫 𝑋(𝑁𝑠𝑠 ∈ ((nei‘𝐽)‘{𝑃})))

Theoremgneispace2 39396* The predicate that 𝐹 is a (generic) Seifert and Threlfall neighborhood space. (Contributed by RP, 15-Apr-2021.)
𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝑉 → (𝐹𝐴 ↔ (𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝))))))

Theoremgneispace3 39397* The predicate that 𝐹 is a (generic) Seifert and Threlfall neighborhood space. (Contributed by RP, 15-Apr-2021.)
𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝑉 → (𝐹𝐴 ↔ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) ∧ ∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝))))))

Theoremgneispace 39398* The predicate that 𝐹 is a (generic) Seifert and Threlfall neighborhood space. (Contributed by RP, 14-Apr-2021.)
𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝑉 → (𝐹𝐴 ↔ (Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)))))))

Theoremgneispacef 39399* A generic neighborhood space is a function with a range that is a subset of the powerset of the powerset of its domain. (Contributed by RP, 15-Apr-2021.)
𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝐴𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))

Theoremgneispacef2 39400* A generic neighborhood space is a function with a range that is a subset of the powerset of the powerset of its domain. (Contributed by RP, 15-Apr-2021.)
𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝐴𝐹:dom 𝐹⟶𝒫 𝒫 dom 𝐹)

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