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Type | Label | Description |
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Statement | ||
Theorem | ntrk0kbimka 42301* | 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 42300. (Contributed by RP, 12-Jun-2021.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (((𝐼‘𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅)) → (𝐼‘∅) = ∅)) | ||
Theorem | clsk3nimkb 42302* | 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 ↦ (𝑏 ∖ 𝑥)) is given. (Contributed by RP, 16-Jun-2021.) |
⊢ ¬ ∀𝑏∀𝑘 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏)(∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) → ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏)) | ||
Theorem | clsk1indlem0 42303 | 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‘∅) = ∅ | ||
Theorem | clsk1indlem2 42304* | 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 ⊆ (𝐾‘𝑠) | ||
Theorem | clsk1indlem3 42305* | 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 3o∀𝑡 ∈ 𝒫 3o(𝐾‘(𝑠 ∪ 𝑡)) ⊆ ((𝐾‘𝑠) ∪ (𝐾‘𝑡)) | ||
Theorem | clsk1indlem4 42306* | 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‘(𝐾‘𝑠)) = (𝐾‘𝑠) | ||
Theorem | clsk1indlem1 42307* | 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 3o∃𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡)) | ||
Theorem | clsk1independent 42308* | 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 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡))) & ⊢ (𝜒 ↔ ∀𝑠 ∈ 𝒫 𝑏𝑠 ⊆ (𝑘‘𝑠)) & ⊢ (𝜃 ↔ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡))) & ⊢ (𝜏 ↔ ∀𝑠 ∈ 𝒫 𝑏(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠)) ⇒ ⊢ ¬ ∀𝑏∀𝑘 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏)(((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜓) | ||
Theorem | neik0pk1imk0 42309* | Kuratowski's K0' and K1 axioms imply K0. Neighborhood version. (Contributed by RP, 3-Jun-2021.) |
⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) ≠ ∅) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥))) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥)) | ||
Theorem | isotone1 42310* | Two different ways to say subset relation persists across applications of a function. (Contributed by RP, 31-May-2021.) |
⊢ (∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝑎 ⊆ 𝑏 → (𝐹‘𝑎) ⊆ (𝐹‘𝑏)) ↔ ∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘(𝑎 ∪ 𝑏))) | ||
Theorem | isotone2 42311* | Two different ways to say subset relation persists across applications of a function. (Contributed by RP, 31-May-2021.) |
⊢ (∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝑎 ⊆ 𝑏 → (𝐹‘𝑎) ⊆ (𝐹‘𝑏)) ↔ ∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎 ∩ 𝑏)) ⊆ ((𝐹‘𝑎) ∩ (𝐹‘𝑏))) | ||
Theorem | ntrk1k3eqk13 42312* | 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 𝐵∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ∧ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∩ 𝑡))) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠 ∩ 𝑡)) = ((𝐼‘𝑠) ∩ (𝐼‘𝑡))) | ||
Theorem | ntrclsf1o 42313* | 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 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) ⇒ ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵)) | ||
Theorem | ntrclsnvobr 42314* | 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 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) ⇒ ⊢ (𝜑 → 𝐾𝐷𝐼) | ||
Theorem | ntrclsiex 42315* | 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 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) ⇒ ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) | ||
Theorem | ntrclskex 42316* | 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 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) ⇒ ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) | ||
Theorem | ntrclsfv1 42317* | 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 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) ⇒ ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾) | ||
Theorem | ntrclsfv2 42318* | 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 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) ⇒ ⊢ (𝜑 → (𝐷‘𝐾) = 𝐼) | ||
Theorem | ntrclselnel1 42319* | 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 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝐼‘𝑆) ↔ ¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆)))) | ||
Theorem | ntrclselnel2 42320* | 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 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑆)) ↔ ¬ 𝑋 ∈ (𝐾‘𝑆))) | ||
Theorem | ntrclsfv 42321* | The value of the interior (closure) expressed in terms of the closure (interior). (Contributed by RP, 25-Jun-2021.) |
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) & ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆)))) | ||
Theorem | ntrclsfveq1 42322* | If interior and closure functions are related then specific function values are complementary. (Contributed by RP, 27-Jun-2021.) |
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) & ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → ((𝐼‘𝑆) = 𝐶 ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ 𝐶))) | ||
Theorem | ntrclsfveq2 42323* | If interior and closure functions are related then specific function values are complementary. (Contributed by RP, 27-Jun-2021.) |
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) & ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → ((𝐼‘(𝐵 ∖ 𝑆)) = 𝐶 ↔ (𝐾‘𝑆) = (𝐵 ∖ 𝐶))) | ||
Theorem | ntrclsfveq 42324* | 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 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) & ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) & ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → ((𝐼‘𝑆) = (𝐼‘𝑇) ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐾‘(𝐵 ∖ 𝑇)))) | ||
Theorem | ntrclsss 42325* | 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 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) & ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) & ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → ((𝐼‘𝑆) ⊆ (𝐼‘𝑇) ↔ (𝐾‘(𝐵 ∖ 𝑇)) ⊆ (𝐾‘(𝐵 ∖ 𝑆)))) | ||
Theorem | ntrclsneine0lem 42326* | 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 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼‘𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐾‘𝑠))) | ||
Theorem | ntrclsneine0 42327* | 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 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∃𝑠 ∈ 𝒫 𝐵𝑥 ∈ (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑥 ∈ (𝐾‘𝑠))) | ||
Theorem | ntrclscls00 42328* | 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 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) ⇒ ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅)) | ||
Theorem | ntrclsiso 42329* | 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 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) ⇒ ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐾‘𝑠) ⊆ (𝐾‘𝑡)))) | ||
Theorem | ntrclsk2 42330* | 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 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) ⇒ ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐾‘𝑠))) | ||
Theorem | ntrclskb 42331* | 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 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) ⇒ ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) = 𝐵 → ((𝐾‘𝑠) ∪ (𝐾‘𝑡)) = 𝐵))) | ||
Theorem | ntrclsk3 42332* | 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 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) ⇒ ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∩ 𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠 ∪ 𝑡)) ⊆ ((𝐾‘𝑠) ∪ (𝐾‘𝑡)))) | ||
Theorem | ntrclsk13 42333* | 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 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) ⇒ ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠 ∩ 𝑡)) = ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠 ∪ 𝑡)) = ((𝐾‘𝑠) ∪ (𝐾‘𝑡)))) | ||
Theorem | ntrclsk4 42334* | Idempotence of the interior function is equivalent to idempotence of the closure function. (Contributed by RP, 10-Jul-2021.) |
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) ⇒ ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ ∀𝑠 ∈ 𝒫 𝐵(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠))) | ||
Theorem | ntrneibex 42335* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) ⇒ ⊢ (𝜑 → 𝐵 ∈ V) | ||
Theorem | ntrneircomplex 42336* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) ⇒ ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵) | ||
Theorem | ntrneif1o 42337* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) ⇒ ⊢ (𝜑 → 𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵)) | ||
Theorem | ntrneiiex 42338* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) ⇒ ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) | ||
Theorem | ntrneinex 42339* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) ⇒ ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵)) | ||
Theorem | ntrneicnv 42340* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) ⇒ ⊢ (𝜑 → ◡𝐹 = (𝐵𝑂𝒫 𝐵)) | ||
Theorem | ntrneifv1 42341* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) ⇒ ⊢ (𝜑 → (𝐹‘𝐼) = 𝑁) | ||
Theorem | ntrneifv2 42342* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) ⇒ ⊢ (𝜑 → (◡𝐹‘𝑁) = 𝐼) | ||
Theorem | ntrneiel 42343* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝐼‘𝑆) ↔ 𝑆 ∈ (𝑁‘𝑋))) | ||
Theorem | ntrneifv3 42344* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑁‘𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑠)}) | ||
Theorem | ntrneineine0lem 42345* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼‘𝑠) ↔ (𝑁‘𝑋) ≠ ∅)) | ||
Theorem | ntrneineine1lem 42346* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐼‘𝑠) ↔ (𝑁‘𝑋) ≠ 𝒫 𝐵)) | ||
Theorem | ntrneifv4 42347* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) & ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → (𝐼‘𝑆) = {𝑥 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑥)}) | ||
Theorem | ntrneiel2 42348* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝐼‘(𝐼‘𝑆)) ↔ ∃𝑢 ∈ (𝑁‘𝑋)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑆 ∈ (𝑁‘𝑦)))) | ||
Theorem | ntrneineine0 42349* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∃𝑠 ∈ 𝒫 𝐵𝑥 ∈ (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) ≠ ∅)) | ||
Theorem | ntrneineine1 42350* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑥 ∈ (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) ≠ 𝒫 𝐵)) | ||
Theorem | ntrneicls00 42351* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) ⇒ ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥))) | ||
Theorem | ntrneicls11 42352* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) ⇒ ⊢ (𝜑 → ((𝐼‘∅) = ∅ ↔ ∀𝑥 ∈ 𝐵 ¬ ∅ ∈ (𝑁‘𝑥))) | ||
Theorem | ntrneiiso 42353* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) ⇒ ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥)))) | ||
Theorem | ntrneik2 42354* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) ⇒ ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) → 𝑥 ∈ 𝑠))) | ||
Theorem | ntrneix2 42355* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) ⇒ ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵(𝑥 ∈ 𝑠 → 𝑠 ∈ (𝑁‘𝑥)))) | ||
Theorem | ntrneikb 42356* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) ⇒ ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑡 ∈ (𝑁‘𝑥)) → (𝑠 ∩ 𝑡) ≠ ∅))) | ||
Theorem | ntrneixb 42357* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) ⇒ ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) = 𝐵 → ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) = 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) = 𝐵 → (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥))))) | ||
Theorem | ntrneik3 42358* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) ⇒ ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∩ 𝑡)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑡 ∈ (𝑁‘𝑥)) → (𝑠 ∩ 𝑡) ∈ (𝑁‘𝑥)))) | ||
Theorem | ntrneix3 42359* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) ⇒ ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠 ∪ 𝑡)) ⊆ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) ∈ (𝑁‘𝑥) → (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥))))) | ||
Theorem | ntrneik13 42360* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) ⇒ ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠 ∩ 𝑡)) = ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) ∈ (𝑁‘𝑥) ↔ (𝑠 ∈ (𝑁‘𝑥) ∧ 𝑡 ∈ (𝑁‘𝑥))))) | ||
Theorem | ntrneix13 42361* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) ⇒ ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠 ∪ 𝑡)) = ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) ∈ (𝑁‘𝑥) ↔ (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥))))) | ||
Theorem | ntrneik4w 42362* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) ⇒ ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ↔ (𝐼‘𝑠) ∈ (𝑁‘𝑥)))) | ||
Theorem | ntrneik4 42363* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) ⇒ ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ↔ ∃𝑢 ∈ (𝑁‘𝑥)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑠 ∈ (𝑁‘𝑦))))) | ||
Theorem | clsneibex 42364 | 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 = (𝑃‘𝐵) & ⊢ 𝐻 = (𝐹 ∘ 𝐷) & ⊢ (𝜑 → 𝐾𝐻𝑁) ⇒ ⊢ (𝜑 → 𝐵 ∈ V) | ||
Theorem | clsneircomplex 42365 | The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 26-Jun-2021.) |
⊢ 𝐷 = (𝑃‘𝐵) & ⊢ 𝐻 = (𝐹 ∘ 𝐷) & ⊢ (𝜑 → 𝐾𝐻𝑁) ⇒ ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵) | ||
Theorem | clsneif1o 42366* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) & ⊢ 𝐷 = (𝑃‘𝐵) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ 𝐻 = (𝐹 ∘ 𝐷) & ⊢ (𝜑 → 𝐾𝐻𝑁) ⇒ ⊢ (𝜑 → 𝐻:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵)) | ||
Theorem | clsneicnv 42367* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) & ⊢ 𝐷 = (𝑃‘𝐵) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ 𝐻 = (𝐹 ∘ 𝐷) & ⊢ (𝜑 → 𝐾𝐻𝑁) ⇒ ⊢ (𝜑 → ◡𝐻 = (𝐷 ∘ (𝐵𝑂𝒫 𝐵))) | ||
Theorem | clsneikex 42368* | If closure and neighborhoods functions are related, the closure function exists. (Contributed by RP, 27-Jun-2021.) |
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) & ⊢ 𝐷 = (𝑃‘𝐵) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ 𝐻 = (𝐹 ∘ 𝐷) & ⊢ (𝜑 → 𝐾𝐻𝑁) ⇒ ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) | ||
Theorem | clsneinex 42369* | If closure and neighborhoods functions are related, the neighborhoods function exists. (Contributed by RP, 27-Jun-2021.) |
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) & ⊢ 𝐷 = (𝑃‘𝐵) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ 𝐻 = (𝐹 ∘ 𝐷) & ⊢ (𝜑 → 𝐾𝐻𝑁) ⇒ ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵)) | ||
Theorem | clsneiel1 42370* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) & ⊢ 𝐷 = (𝑃‘𝐵) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ 𝐻 = (𝐹 ∘ 𝐷) & ⊢ (𝜑 → 𝐾𝐻𝑁) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝐾‘𝑆) ↔ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋))) | ||
Theorem | clsneiel2 42371* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) & ⊢ 𝐷 = (𝑃‘𝐵) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ 𝐻 = (𝐹 ∘ 𝐷) & ⊢ (𝜑 → 𝐾𝐻𝑁) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆)) ↔ ¬ 𝑆 ∈ (𝑁‘𝑋))) | ||
Theorem | clsneifv3 42372* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) & ⊢ 𝐷 = (𝑃‘𝐵) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ 𝐻 = (𝐹 ∘ 𝐷) & ⊢ (𝜑 → 𝐾𝐻𝑁) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑁‘𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑠))}) | ||
Theorem | clsneifv4 42373* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) & ⊢ 𝐷 = (𝑃‘𝐵) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ 𝐻 = (𝐹 ∘ 𝐷) & ⊢ (𝜑 → 𝐾𝐻𝑁) & ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → (𝐾‘𝑆) = {𝑥 ∈ 𝐵 ∣ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑥)}) | ||
Theorem | neicvgbex 42374 | 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 = (𝑃‘𝐵) & ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) & ⊢ (𝜑 → 𝑁𝐻𝑀) ⇒ ⊢ (𝜑 → 𝐵 ∈ V) | ||
Theorem | neicvgrcomplex 42375 | The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 26-Jun-2021.) |
⊢ 𝐷 = (𝑃‘𝐵) & ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) & ⊢ (𝜑 → 𝑁𝐻𝑀) ⇒ ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵) | ||
Theorem | neicvgf1o 42376* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) & ⊢ 𝐷 = (𝑃‘𝐵) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵) & ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) & ⊢ (𝜑 → 𝑁𝐻𝑀) ⇒ ⊢ (𝜑 → 𝐻:(𝒫 𝒫 𝐵 ↑m 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵)) | ||
Theorem | neicvgnvo 42377* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) & ⊢ 𝐷 = (𝑃‘𝐵) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵) & ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) & ⊢ (𝜑 → 𝑁𝐻𝑀) ⇒ ⊢ (𝜑 → ◡𝐻 = 𝐻) | ||
Theorem | neicvgnvor 42378* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) & ⊢ 𝐷 = (𝑃‘𝐵) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵) & ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) & ⊢ (𝜑 → 𝑁𝐻𝑀) ⇒ ⊢ (𝜑 → 𝑀𝐻𝑁) | ||
Theorem | neicvgmex 42379* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) & ⊢ 𝐷 = (𝑃‘𝐵) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵) & ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) & ⊢ (𝜑 → 𝑁𝐻𝑀) ⇒ ⊢ (𝜑 → 𝑀 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵)) | ||
Theorem | neicvgnex 42380* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) & ⊢ 𝐷 = (𝑃‘𝐵) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵) & ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) & ⊢ (𝜑 → 𝑁𝐻𝑀) ⇒ ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵)) | ||
Theorem | neicvgel1 42381* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) & ⊢ 𝐷 = (𝑃‘𝐵) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵) & ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) & ⊢ (𝜑 → 𝑁𝐻𝑀) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → (𝑆 ∈ (𝑁‘𝑋) ↔ ¬ (𝐵 ∖ 𝑆) ∈ (𝑀‘𝑋))) | ||
Theorem | neicvgel2 42382* | 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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) & ⊢ 𝐷 = (𝑃‘𝐵) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵) & ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) & ⊢ (𝜑 → 𝑁𝐻𝑀) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → ((𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋) ↔ ¬ 𝑆 ∈ (𝑀‘𝑋))) | ||
Theorem | neicvgfv 42383* | The value of the neighborhoods (convergents) in terms of the convergents (neighborhoods) function. (Contributed by RP, 27-Jun-2021.) |
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) & ⊢ 𝐷 = (𝑃‘𝐵) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵) & ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) & ⊢ (𝜑 → 𝑁𝐻𝑀) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑁‘𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵 ∖ 𝑠) ∈ (𝑀‘𝑋)}) | ||
Theorem | ntrrn 42384 | 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 𝐼 ⊆ 𝐽) | ||
Theorem | ntrf 42385 | 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 𝑋⟶𝐽) | ||
Theorem | ntrf2 42386 | 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) | ||
Theorem | ntrelmap 42387 | 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 ↑m 𝒫 𝑋)) | ||
Theorem | clsf2 42388 | The closure function is a map from the powerset of the base set to itself. This is less precise than clsf 22399. (Contributed by RP, 22-Apr-2021.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (cls‘𝐽) ⇒ ⊢ (𝐽 ∈ Top → 𝐾:𝒫 𝑋⟶𝒫 𝑋) | ||
Theorem | clselmap 42389 | 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 𝑋 ↑m 𝒫 𝑋)) | ||
Theorem | dssmapntrcls 42390* | The interior and closure operators on a topology are duals of each other. See also kur14lem2 33801. (Contributed by RP, 21-Apr-2021.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (cls‘𝐽) & ⊢ 𝐼 = (int‘𝐽) & ⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠)))))) & ⊢ 𝐷 = (𝑂‘𝑋) ⇒ ⊢ (𝐽 ∈ Top → 𝐼 = (𝐷‘𝐾)) | ||
Theorem | dssmapclsntr 42391* | The closure and interior operators on a topology are duals of each other. See also kur14lem2 33801. (Contributed by RP, 22-Apr-2021.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (cls‘𝐽) & ⊢ 𝐼 = (int‘𝐽) & ⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠)))))) & ⊢ 𝐷 = (𝑂‘𝑋) ⇒ ⊢ (𝐽 ∈ Top → 𝐾 = (𝐷‘𝐼)) | ||
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 22349 is an example of a topology with an empty base set. This divergence is unlikely to pose serious problems. | ||
Theorem | gneispa 42392* | 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 → ∀𝑝 ∈ 𝑋 (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝 ∈ 𝑛)) | ||
Theorem | gneispb 42393* | 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 ∧ 𝑃 ∈ 𝑋 ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → ∀𝑠 ∈ 𝒫 𝑋(𝑁 ⊆ 𝑠 → 𝑠 ∈ ((nei‘𝐽)‘{𝑃}))) | ||
Theorem | gneispace2 42394* | 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 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))} ⇒ ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ 𝐴 ↔ (𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))))) | ||
Theorem | gneispace3 42395* | 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 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))} ⇒ ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ 𝐴 ↔ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) ∧ ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))))) | ||
Theorem | gneispace 42396* | 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 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))} ⇒ ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ 𝐴 ↔ (Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))))) | ||
Theorem | gneispacef 42397* | 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 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))} ⇒ ⊢ (𝐹 ∈ 𝐴 → 𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) | ||
Theorem | gneispacef2 42398* | 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 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))} ⇒ ⊢ (𝐹 ∈ 𝐴 → 𝐹:dom 𝐹⟶𝒫 𝒫 dom 𝐹) | ||
Theorem | gneispacefun 42399* | A generic neighborhood space is a function. (Contributed by RP, 15-Apr-2021.) |
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))} ⇒ ⊢ (𝐹 ∈ 𝐴 → Fun 𝐹) | ||
Theorem | gneispacern 42400* | A generic neighborhood space has 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 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))} ⇒ ⊢ (𝐹 ∈ 𝐴 → ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) |
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