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Theorem List for Metamath Proof Explorer - 37301-37400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfrege124 37301 If 𝑋 is a result of an application of the single-valued procedure 𝑅 to 𝑌 and if 𝑀 follows 𝑌 in the 𝑅-sequence, then 𝑀 belongs to the 𝑅-sequence beginning with 𝑋. Proposition 124 of [Frege1879] p. 80. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑀𝑊    &   𝑅𝑆       (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀𝑋((t+‘𝑅) ∪ I )𝑀)))

Theoremfrege125 37302 Lemma for frege126 37303. Proposition 125 of [Frege1879] p. 81. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑀𝑊    &   𝑅𝑆       ((𝑋((t+‘𝑅) ∪ I )𝑀 → (¬ 𝑋(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑋)) → (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀 → (¬ 𝑋(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑋)))))

Theoremfrege126 37303 If 𝑀 follows 𝑌 in the 𝑅-sequence and if the procedure 𝑅 is single-valued, then every result of an application of the procedure 𝑅 to 𝑌 belongs to the 𝑅-sequence beginning with 𝑀 or precedes 𝑀 in the 𝑅-sequence. Proposition 126 of [Frege1879] p. 81. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑀𝑊    &   𝑅𝑆       (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀 → (¬ 𝑋(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑋))))

Theoremfrege127 37304 Communte antecedents of frege126 37303. Proposition 127 of [Frege1879] p. 82. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑀𝑊    &   𝑅𝑆       (Fun 𝑅 → (𝑌(t+‘𝑅)𝑀 → (𝑌𝑅𝑋 → (¬ 𝑋(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑋))))

Theoremfrege128 37305 Lemma for frege129 37306. Proposition 128 of [Frege1879] p. 83. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑀𝑊    &   𝑅𝑆       ((𝑀((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑋 → (¬ 𝑋(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑋))) → (Fun 𝑅 → ((¬ 𝑌(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑌) → (𝑌𝑅𝑋 → (¬ 𝑋(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑋)))))

Theoremfrege129 37306 If the procedure 𝑅 is single-valued and 𝑌 belongs to the 𝑅 -sequence begining with 𝑀 or precedes 𝑀 in the 𝑅-sequence, then every result of an application of the procedure 𝑅 to 𝑌 belongs to the 𝑅-sequence begining with 𝑀 or precedes 𝑀 in the 𝑅-sequence. Proposition 129 of [Frege1879] p. 83. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑀𝑊    &   𝑅𝑆       (Fun 𝑅 → ((¬ 𝑌(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑌) → (𝑌𝑅𝑋 → (¬ 𝑋(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑋))))

Theoremfrege130 37307* Lemma for frege131 37308. Proposition 130 of [Frege1879] p. 84. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
𝑀𝑈    &   𝑅𝑉       ((∀𝑏((¬ 𝑏(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑏) → ∀𝑎(𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑎))) → 𝑅 hereditary (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))) → (Fun 𝑅𝑅 hereditary (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))))

Theoremfrege131 37308 If the procedure 𝑅 is single-valued, then the property of belonging to the 𝑅-sequence begining with 𝑀 or preceeding 𝑀 in the 𝑅-sequence is hereditary in the 𝑅-sequence. Proposition 131 of [Frege1879] p. 85. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
𝑀𝑈    &   𝑅𝑉       (Fun 𝑅𝑅 hereditary (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})))

Theoremfrege132 37309 Lemma for frege133 37310. Proposition 132 of [Frege1879] p. 86. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
𝑀𝑈    &   𝑅𝑉       ((𝑅 hereditary (((t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) → (𝑋(t+‘𝑅)𝑀 → (𝑋(t+‘𝑅)𝑌 → (¬ 𝑌(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑌)))) → (Fun 𝑅 → (𝑋(t+‘𝑅)𝑀 → (𝑋(t+‘𝑅)𝑌 → (¬ 𝑌(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑌)))))

Theoremfrege133 37310 If the procedure 𝑅 is single-valued and if 𝑀 and 𝑌 follow 𝑋 in the 𝑅-sequence, then 𝑌 belongs to the 𝑅-sequence beginning with 𝑀 or precedes 𝑀 in the 𝑅-sequence. Proposition 133 of [Frege1879] p. 86. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑀𝑊    &   𝑅𝑆       (Fun 𝑅 → (𝑋(t+‘𝑅)𝑀 → (𝑋(t+‘𝑅)𝑌 → (¬ 𝑌(t+‘𝑅)𝑀𝑀((t+‘𝑅) ∪ I )𝑌))))

21.26.4  Exploring Topology via Seifert And Threlfall

See Seifert And Threlfall: A Textbook Of Topology (1980) which is an English translation of Lehrbuch der Topologie (1934).

21.26.4.1  Equinumerosity of sets of relations and maps

Because ((2𝑜𝑚 𝐵) ↑𝑚 𝐴) ≈ (2𝑜𝑚 (𝐴 × 𝐵)) ≈ ((2𝑜𝑚 𝐴) ↑𝑚 𝐵) is an instance of the law of exponents: ((𝐶𝑚 𝐵) ↑𝑚 𝐴) ≈ (𝐶𝑚 (𝐴 × 𝐵)) ≈ ((𝐶𝑚 𝐴) ↑𝑚 𝐵) we are led to see that (𝒫 𝐵𝑚 𝐴) ≈ 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐴𝑚 𝐵) is true for any two sets, 𝐴 and 𝐵, and thus there exist one-to-one onto relations between each of these three sets of relations.

Theoremenrelmap 37311 The set of all possible relations between two sets is equinumerous to the set of all mappings from one set to the powerset of the other. See rfovf1od 37320 for a demonstration of an natural one-to-one onto mapping. (Contributed by RP, 27-Apr-2021.)
((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵𝑚 𝐴))

Theoremenrelmapr 37312 The set of all possible relations between two sets is equinumerous to the set of all mappings from one set to the powerset of the other. (Contributed by RP, 27-Apr-2021.)
((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐴𝑚 𝐵))

Theoremenmappw 37313 The set of all mappings from one set to the powerset of the other is equinumerous to the set of all mappings from the second set to the powerset of the first. (Contributed by RP, 27-Apr-2021.)
((𝐴𝑉𝐵𝑊) → (𝒫 𝐵𝑚 𝐴) ≈ (𝒫 𝐴𝑚 𝐵))

Theoremenmappwid 37314 The set of all mappings from the powerset to the powerset is equinumerous to the set of all mappings from the set to the powerset of the powerset. (Contributed by RP, 27-Apr-2021.)
(𝐴𝑉 → (𝒫 𝐴𝑚 𝒫 𝐴) ≈ (𝒫 𝒫 𝐴𝑚 𝐴))

Theoremrfovd 37315* Value of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵. (Contributed by RP, 25-Apr-2021.)
𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥𝑎 ↦ {𝑦𝑏𝑥𝑟𝑦})))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → (𝐴𝑂𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))

Theoremrfovfvd 37316* Value of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵, and relation 𝑅. (Contributed by RP, 25-Apr-2021.)
𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥𝑎 ↦ {𝑦𝑏𝑥𝑟𝑦})))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝑅 ∈ 𝒫 (𝐴 × 𝐵))    &   𝐹 = (𝐴𝑂𝐵)       (𝜑 → (𝐹𝑅) = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑅𝑦}))

Theoremrfovfvfvd 37317* Value of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵, relation 𝑅, and left element 𝑋. (Contributed by RP, 25-Apr-2021.)
𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥𝑎 ↦ {𝑦𝑏𝑥𝑟𝑦})))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝑅 ∈ 𝒫 (𝐴 × 𝐵))    &   𝐹 = (𝐴𝑂𝐵)    &   (𝜑𝑋𝐴)    &   𝐺 = (𝐹𝑅)       (𝜑 → (𝐺𝑋) = {𝑦𝐵𝑋𝑅𝑦})

Theoremrfovcnvf1od 37318* Properties of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵. (Contributed by RP, 27-Apr-2021.)
𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥𝑎 ↦ {𝑦𝑏𝑥𝑟𝑦})))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   𝐹 = (𝐴𝑂𝐵)       (𝜑 → (𝐹:𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝐴) ∧ 𝐹 = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})))

Theoremrfovcnvd 37319* Value of the converse of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵. (Contributed by RP, 27-Apr-2021.)
𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥𝑎 ↦ {𝑦𝑏𝑥𝑟𝑦})))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   𝐹 = (𝐴𝑂𝐵)       (𝜑𝐹 = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}))

Theoremrfovf1od 37320* The value of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵, is a bijection. (Contributed by RP, 27-Apr-2021.)
𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥𝑎 ↦ {𝑦𝑏𝑥𝑟𝑦})))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   𝐹 = (𝐴𝑂𝐵)       (𝜑𝐹:𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝐴))

Theoremrfovcnvfvd 37321* Value of the converse of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵, evaluated at function 𝐺. (Contributed by RP, 27-Apr-2021.)
𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥𝑎 ↦ {𝑦𝑏𝑥𝑟𝑦})))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   𝐹 = (𝐴𝑂𝐵)    &   (𝜑𝐺 ∈ (𝒫 𝐵𝑚 𝐴))       (𝜑 → (𝐹𝐺) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐺𝑥))})

Theoremfsovd 37322* Value of the operator, (𝐴𝑂𝐵), which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, 𝐴 and 𝐵. (Contributed by RP, 25-Apr-2021.)
𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → (𝐴𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})))

Theoremfsovrfovd 37323* The operator which gives a 1-to-1 a mapping to a subset and a reverse mapping from elements can be composed from the operator which gives a 1-to-1 mapping between relations and functions to subsets and the converse operator. (Contributed by RP, 15-May-2021.)
𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   𝑅 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑢𝑎 ↦ {𝑣𝑏𝑢𝑟𝑣})))    &   𝐶 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑠 ∈ 𝒫 (𝑎 × 𝑏) ↦ 𝑠))       (𝜑 → (𝐴𝑂𝐵) = ((𝐵𝑅𝐴) ∘ ((𝐴𝐶𝐵) ∘ (𝐴𝑅𝐵))))

Theoremfsovfvd 37324* Value of the operator, (𝐴𝑂𝐵), which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, 𝐴 and 𝐵, when applied to function 𝐹. (Contributed by RP, 25-Apr-2021.)
𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   𝐺 = (𝐴𝑂𝐵)    &   (𝜑𝐹 ∈ (𝒫 𝐵𝑚 𝐴))       (𝜑 → (𝐺𝐹) = (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝐹𝑥)}))

Theoremfsovfvfvd 37325* Value of the operator, (𝐴𝑂𝐵), which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, 𝐴 and 𝐵, when applied to function 𝐹 and element 𝑌. (Contributed by RP, 25-Apr-2021.)
𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   𝐺 = (𝐴𝑂𝐵)    &   (𝜑𝐹 ∈ (𝒫 𝐵𝑚 𝐴))    &   𝐻 = (𝐺𝐹)    &   (𝜑𝑌𝐵)       (𝜑 → (𝐻𝑌) = {𝑥𝐴𝑌 ∈ (𝐹𝑥)})

Theoremfsovfd 37326* The operator, (𝐴𝑂𝐵), which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, 𝐴 and 𝐵, gives a function between two sets of functions. (Contributed by RP, 27-Apr-2021.)
𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   𝐺 = (𝐴𝑂𝐵)       (𝜑𝐺:(𝒫 𝐵𝑚 𝐴)⟶(𝒫 𝐴𝑚 𝐵))

Theoremfsovcnvlem 37327* The 𝑂 operator, which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, gives a family of functions that include their own inverse. (Contributed by RP, 27-Apr-2021.)
𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   𝐺 = (𝐴𝑂𝐵)    &   𝐻 = (𝐵𝑂𝐴)       (𝜑 → (𝐻𝐺) = ( I ↾ (𝒫 𝐵𝑚 𝐴)))

Theoremfsovcnvd 37328* The value of the converse (𝐴𝑂𝐵) is (𝐵𝑂𝐴), where 𝑂 is the operator which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, gives a family of functions that include their own inverse. (Contributed by RP, 27-Apr-2021.)
𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   𝐺 = (𝐴𝑂𝐵)    &   𝐻 = (𝐵𝑂𝐴)       (𝜑𝐺 = 𝐻)

Theoremfsovcnvfvd 37329* The value of the converse of (𝐴𝑂𝐵), where 𝑂 is the operator which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, evaluated at function 𝐹. (Contributed by RP, 27-Apr-2021.)
𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   𝐺 = (𝐴𝑂𝐵)    &   (𝜑𝐹 ∈ (𝒫 𝐴𝑚 𝐵))       (𝜑 → (𝐺𝐹) = (𝑦𝐴 ↦ {𝑥𝐵𝑦 ∈ (𝐹𝑥)}))

Theoremfsovf1od 37330* The value of (𝐴𝑂𝐵) is a bijection, where 𝑂 is the operator which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets. (Contributed by RP, 27-Apr-2021.)
𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   𝐺 = (𝐴𝑂𝐵)       (𝜑𝐺:(𝒫 𝐵𝑚 𝐴)–1-1-onto→(𝒫 𝐴𝑚 𝐵))

Theoremdssmapfvd 37331* Value of the duality operator for self-mappings of subsets of a base set, 𝐵. (Contributed by RP, 19-Apr-2021.)
𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐵𝑉)       (𝜑𝐷 = (𝑓 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))))

Theoremdssmapfv2d 37332* Value of the duality operator for self-mappings of subsets of a base set, 𝐵 when applied to function 𝐹. (Contributed by RP, 19-Apr-2021.)
𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐵𝑉)    &   (𝜑𝐹 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))    &   𝐺 = (𝐷𝐹)       (𝜑𝐺 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝐹‘(𝐵𝑠)))))

Theoremdssmapfv3d 37333* Value of the duality operator for self-mappings of subsets of a base set, 𝐵 when applied to function 𝐹 and subset 𝑆. (Contributed by RP, 19-Apr-2021.)
𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐵𝑉)    &   (𝜑𝐹 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))    &   𝐺 = (𝐷𝐹)    &   (𝜑𝑆 ∈ 𝒫 𝐵)    &   𝑇 = (𝐺𝑆)       (𝜑𝑇 = (𝐵 ∖ (𝐹‘(𝐵𝑆))))

Theoremdssmapnvod 37334* For any base set 𝐵 the duality operator for self-mappings of subsets of that base set is its own inverse, an involution. (Contributed by RP, 20-Apr-2021.)
𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐵𝑉)       (𝜑𝐷 = 𝐷)

Theoremdssmapf1od 37335* For any base set 𝐵 the duality operator for self-mappings of subsets of that base set is one-to-one and onto. (Contributed by RP, 21-Apr-2021.)
𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐵𝑉)       (𝜑𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵))

Theoremdssmap2d 37336* For any base set 𝐵 the duality operator for self-mappings of subsets of that base set when composed with itself is the restricted identity operator. (Contributed by RP, 21-Apr-2021.)
𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐵𝑉)       (𝜑 → (𝐷𝐷) = ( I ↾ (𝒫 𝐵𝑚 𝒫 𝐵)))

21.26.4.2  Generic Pseudoclosure Spaces, Pseudointeror Spaces, and Pseudoneighborhoods

For any base set, 𝐵, an arbitrary mapping of subsets to subsets can be called a pseudoclosure (pseudointerior) function, 𝐾, with its dual of a pseudointerior (pseudoclosure), 𝐼, related by the involution in dssmapfvd 37331. As 𝐾 gains properties of the closure (interior) function of a topology on 𝐵, so does its dual gain corresponding properties of the interior (closure) function of that topology.

As (𝒫 𝐵𝑚 𝒫 𝐵) ≈ (𝒫 𝒫 𝐵𝑚 𝐵) there is also a natural isomorphism which maps from 𝐼 to 𝑁 (and likewise for 𝐾 and 𝑀, introduced below) which identically gains the properties of the neighborhood function of a topology (modified and restricted to operate on single points). A function dual to 𝑁, which Stadler and Stadler refer to as a convergent function, is represented by 𝑀 in this section.

Based on this and the early treatment of topology in Seifert and Threlfall, it seems reasonable to define a pseudotopology as defined in terms of its base set and one of these functions with theorems treating the equivalence of the other definitions and adding topological structure if enough properties hold true.

 Neighborhoods Interior Closure Convergents Theorems Functions 𝑁 ∈ (𝒫 𝒫 𝐵 ↑𝑚 𝐵) 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) 𝐾 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) 𝑀 ∈ (𝒫 𝒫 𝐵 ↑𝑚 𝐵) Correspondences (assuming (𝑋 ∈ 𝐵 ∧ 𝑆 ∈ 𝒫 𝐵)) 𝑆 ∈ (𝑁‘𝑋) ↔ 𝑋 ∈ (𝐼‘𝑆) ↔ ¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆)) ↔ ¬ (𝐵 ∖ 𝑆) ∈ (𝑀‘𝑋) ntrclselnel1 37375, ntrneiel 37399, neicvgel1 37437 ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋) ↔ ¬ 𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑆)) ↔ 𝑋 ∈ (𝐾‘𝑆) ↔ 𝑆 ∈ (𝑀‘𝑋) (𝑁‘𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑠)} = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑠))} = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵 ∖ 𝑠) ∈ (𝑀‘𝑋)} ntrneifv3 37400, clsneifv3 37428, neicvgfv 37439 {𝑥 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑥)} = (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) = {𝑥 ∈ 𝐵 ∣ ¬ (𝐵 ∖ 𝑆) ∈ (𝑀‘𝑥)} ntrneifv4 37403, ntrclsfv 37377, clsneifv4 37429 {𝑥 ∈ 𝐵 ∣ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑥)} = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑆))) = (𝐾‘𝑆) = {𝑥 ∈ 𝐵 ∣ 𝑆 ∈ (𝑀‘𝑥)} clsneifv4 37429, ntrclsfv 37377, ntrneifv4 37403 {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵 ∖ 𝑠) ∈ (𝑁‘𝑋)} = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑠))} = {𝑠 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐾‘𝑠)} = (𝑀‘𝑋) neicvgfv 37439, clsneifv3 37428, ntrneifv3 37400

We have the following table of equivalences to axioms largely established by Kuratowski. In the formulas in this table, to reduce the width of the columns, if any of the variables 𝑥, 𝑠, or 𝑡 are used, then they are implicitly universally quantified and 𝑥 (respectively 𝑠 and 𝑡) ranges over 𝐵 (respectively 𝒫 𝐵 and 𝒫 𝐵).

 Neighborhoods Interior Closure Convergents Equivalence Theorems Assuming a prefix of: ∀𝑥 ∈ 𝐵∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵 (𝑁‘𝑥) ≠ ∅ ∃𝑢 ∈ 𝒫 𝐵𝑥 ∈ (𝐼‘𝑢) ∃𝑢 ∈ 𝒫 𝐵¬ 𝑥 ∈ (𝐾‘𝑢) (𝑀‘𝑥) ≠ 𝒫 𝐵 ntrclsneine0 37383, ntrneineine0 37405, ntrneineine1 37406 (𝑁‘𝑥) ≠ 𝒫 𝐵 ∃𝑢 ∈ 𝒫 𝐵¬ 𝑥 ∈ (𝐼‘𝑢) ∃𝑢 ∈ 𝒫 𝐵𝑥 ∈ (𝐾‘𝑢) (𝑀‘𝑥) ≠ ∅ ntrclsneine0 37383, ntrneineine0 37405, ntrneineine1 37406 𝐵 ∈ (𝑁‘𝑥) (𝐼‘𝐵) = 𝐵 (𝐾‘∅) = ∅ ¬ ∅ ∈ (𝑀‘𝑥) ntrclscls00 37384, ntrneicls00 37407, ntrneicls11 37408 ¬ ∅ ∈ (𝑁‘𝑥) (𝐼‘∅) = ∅ (𝐾‘𝐵) = 𝐵 𝐵 ∈ (𝑀‘𝑥) ntrclscls00 37384, ntrneicls00 37407, ntrneicls11 37408 ((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥)) (𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) — or — ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∪ 𝑡)) — or — (𝐼‘(𝑠 ∩ 𝑡)) ⊆ ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) (𝑠 ⊆ 𝑡 → (𝐾‘𝑠) ⊆ (𝐾‘𝑡)) — or — ((𝐾‘𝑠) ∪ (𝐾‘𝑡)) ⊆ (𝐾‘(𝑠 ∪ 𝑡)) — or — (𝐾‘(𝑠 ∩ 𝑡)) ⊆ ((𝐾‘𝑠) ∩ (𝐾‘𝑡)) ((𝑠 ∈ (𝑀‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑀‘𝑥)) isotone1 37366, isotone2 37367, ntrclsiso 37385, ntrneiiso 37409 (𝑠 ∈ (𝑁‘𝑥) → 𝑥 ∈ 𝑠) (𝐼‘𝑠) ⊆ 𝑠 𝑠 ⊆ (𝐾‘𝑠) (𝑥 ∈ 𝑠 → 𝑠 ∈ (𝑀‘𝑥)) ntrclsk2 37386, ntrneik2 37410, ntrneix2 37411 ((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑡 ∈ (𝑁‘𝑥)) → (𝑠 ∩ 𝑡) ≠ ∅) ((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) ((𝑠 ∪ 𝑡) = 𝐵 → ((𝐾‘𝑠) ∪ (𝐾‘𝑡)) = 𝐵) ((𝑠 ∪ 𝑡) = 𝐵 → (𝑠 ∈ (𝑀‘𝑥) ∨ 𝑡 ∈ (𝑀‘𝑥))) ntrclskb 37387, ntrneikb 37412, ntrneixb 37413 ((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑡 ∈ (𝑁‘𝑥)) → (𝑠 ∩ 𝑡) ∈ (𝑁‘𝑥)) ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∩ 𝑡)) (𝐾‘(𝑠 ∪ 𝑡)) ⊆ ((𝐾‘𝑠) ∪ (𝐾‘𝑡)) ((𝑠 ∪ 𝑡) ∈ (𝑀‘𝑥) → (𝑠 ∈ (𝑀‘𝑥) ∨ 𝑡 ∈ (𝑀‘𝑥))) ntrclsk3 37388, ntrneik3 37414, ntrneix3 37415 ((𝑠 ∩ 𝑡) ∈ (𝑁‘𝑥) ↔ (𝑠 ∈ (𝑁‘𝑥) ∧ 𝑡 ∈ (𝑁‘𝑥))) (𝐼‘(𝑠 ∩ 𝑡)) = ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) (𝐾‘(𝑠 ∪ 𝑡)) = ((𝐾‘𝑠) ∪ (𝐾‘𝑡)) ((𝑠 ∪ 𝑡) ∈ (𝑀‘𝑥) ↔ (𝑠 ∈ (𝑀‘𝑥) ∨ 𝑡 ∈ (𝑀‘𝑥))) ntrclsk13 37389, ntrneik13 37416, ntrneix13 37417 (𝑠 ∈ (𝑁‘𝑥) ↔ ∃𝑢 ∈ (𝑁‘𝑥)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑠 ∈ (𝑁‘𝑦))) (𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) (𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠) (𝑠 ∈ (𝑀‘𝑥) ↔ ∃𝑢 ∈ (𝑀‘𝑥)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑠 ∈ (𝑀‘𝑦))) ntrclsk4 37390, ntrneik4 37419

Using these properties as axiomic constraints on the functions, certain collections of them give rise to named spaces.

Space Foundational Axioms Derived Axioms Theorems
Csázár Generalized Neighborhood Space K2 KA', KA, KB ntrk2imkb 37355, ntrkbimka 37356, neik0imk0p 37354
Min Strong Generalized Neighborhood Space K2, K3 KA', KA, KB ntrk2imkb 37355, ntrkbimka 37356, neik0imk0p 37354
Gniłka Extended Topology K0', K1 K0 neik0pk1imk0 37365
Brissaud Space K0, K2 K0', KA', KA, KB neik0imk0p 37354, ntrk2imkb 37355, ntrkbimka 37356
Neighborhood Space K0', K1, K2 K0, KA', KA, KB neik0pk1imk0 37365, ntrk2imkb 37355, ntrkbimka 37356, neik0imk0p 37354
Davey and Priestley Intersection Structure K1, K4
Moore Closure Space K1, K2, K4 KA', KA, KB ntrk2imkb 37355, ntrkbimka 37356, neik0imk0p 37354
Convex Closure Space K0', K1, K2, K4 K0, KA', KA, KB neik0pk1imk0 37365, ntrk2imkb 37355, ntrkbimka 37356, neik0imk0p 37354
Smyth Neighborhood Space K0', K13 K0, K1, K3 neik0pk1imk0 37365, ntrk1k3eqk13 37368
Čech Closure Space
Pretopological Space
K0', K2, K13 K0, K1, KA', KA, KB, K3 neik0pk1imk0 37365, ntrk2imkb 37355, ntrkbimka 37356, neik0imk0p 37354, ntrk1k3eqk13 37368
Topological Space K0', K2, K13, K4 K0, K1, KA', KA, KB, K3 neik0pk1imk0 37365, ntrk2imkb 37355, ntrkbimka 37356, neik0imk0p 37354, ntrk1k3eqk13 37368
Alexandroff Space K0', K2, K5 K0, K1, KA', KA, KB, K3, K13 neik0pk1imk0 37365, ntrk2imkb 37355, ntrkbimka 37356, neik0imk0p 37354, ntrk1k3eqk13 37368, TBD
Alexandroff Topological Space K0', K2, K4, K5 K0, K1, KA', KA, KB, K3, K13 neik0pk1imk0 37365, ntrk2imkb 37355, ntrkbimka 37356, neik0imk0p 37354, ntrk1k3eqk13 37368, TBD

Theoremsscon34b 37337 Relative complementation reverses inclusion of subclasses. Relativized version of complss 3713. (Contributed by RP, 3-Jun-2021.)
((𝐴𝐶𝐵𝐶) → (𝐴𝐵 ↔ (𝐶𝐵) ⊆ (𝐶𝐴)))

Theoremrcompleq 37338 Two subclasses are equal if and only if their relative complements are equal. Relativized version of compleq 3714. (Contributed by RP, 10-Jun-2021.)
((𝐴𝐶𝐵𝐶) → (𝐴 = 𝐵 ↔ (𝐶𝐴) = (𝐶𝐵)))

Theoremor3or 37339 Decompose disjunction into three cases. (Contributed by RP, 5-Jul-2021.)
((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))

Theoremandi3or 37340 Distribute over triple disjunction. (Contributed by RP, 5-Jul-2021.)
((𝜑 ∧ (𝜓𝜒𝜃)) ↔ ((𝜑𝜓) ∨ (𝜑𝜒) ∨ (𝜑𝜃)))

Theoremuneqsn 37341 If a union of classes is equal to a singleton then at least one class is equal to the singleton while the other may be equal to the empty set. (Contributed by RP, 5-Jul-2021.)
((𝐴𝐵) = {𝐶} ↔ ((𝐴 = {𝐶} ∧ 𝐵 = {𝐶}) ∨ (𝐴 = {𝐶} ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ 𝐵 = {𝐶})))

Theoremdf3o2 37342 Ordinal 3 is the triplet containing ordinals 0, 1 and 2. (Contributed by RP, 8-Jul-2021.)
3𝑜 = {∅, 1𝑜, 2𝑜}

Theoremdf3o3 37343 Ordinal 3 , fully expanded. (Contributed by RP, 8-Jul-2021.)
3𝑜 = {∅, {∅}, {∅, {∅}}}

Theorembrfvimex 37344 If a binary relation holds and the relation is the value of a function, then the argument to that function is a set. (Contributed by RP, 22-May-2021.)
(𝜑𝐴𝑅𝐵)    &   (𝜑𝑅 = (𝐹𝐶))       (𝜑𝐶 ∈ V)

Theorembrovmptimex 37345* If a binary relation holds and the relation is the value of a binary operation built with maps-to, then the arguments to that operation are sets. (Contributed by RP, 22-May-2021.)
𝐹 = (𝑥𝐸, 𝑦𝐺𝐻)    &   (𝜑𝐴𝑅𝐵)    &   (𝜑𝑅 = (𝐶𝐹𝐷))       (𝜑 → (𝐶 ∈ V ∧ 𝐷 ∈ V))

Theorembrovmptimex1 37346* If a binary relation holds and the relation is the value of a binary operation built with maps-to, then the arguments to that operation are sets. (Contributed by RP, 22-May-2021.)
𝐹 = (𝑥𝐸, 𝑦𝐺𝐻)    &   (𝜑𝐴𝑅𝐵)    &   (𝜑𝑅 = (𝐶𝐹𝐷))       (𝜑𝐶 ∈ V)

Theorembrovmptimex2 37347* If a binary relation holds and the relation is the value of a binary operation built with maps-to, then the arguments to that operation are sets. (Contributed by RP, 22-May-2021.)
𝐹 = (𝑥𝐸, 𝑦𝐺𝐻)    &   (𝜑𝐴𝑅𝐵)    &   (𝜑𝑅 = (𝐶𝐹𝐷))       (𝜑𝐷 ∈ V)

Theorembrcoffn 37348 Conditions allowing the decomposition of a binary relation. (Contributed by RP, 7-Jun-2021.)
(𝜑𝐶 Fn 𝑌)    &   (𝜑𝐷:𝑋𝑌)    &   (𝜑𝐴(𝐶𝐷)𝐵)       (𝜑 → (𝐴𝐷(𝐷𝐴) ∧ (𝐷𝐴)𝐶𝐵))

Theorembrcofffn 37349 Conditions allowing the decomposition of a binary relation. (Contributed by RP, 8-Jun-2021.)
(𝜑𝐶 Fn 𝑍)    &   (𝜑𝐷:𝑌𝑍)    &   (𝜑𝐸:𝑋𝑌)    &   (𝜑𝐴(𝐶 ∘ (𝐷𝐸))𝐵)       (𝜑 → (𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵))

Theorembrco2f1o 37350 Conditions allowing the decomposition of a binary relation. (Contributed by RP, 8-Jun-2021.)
(𝜑𝐶:𝑌1-1-onto𝑍)    &   (𝜑𝐷:𝑋1-1-onto𝑌)    &   (𝜑𝐴(𝐶𝐷)𝐵)       (𝜑 → ((𝐶𝐵)𝐶𝐵𝐴𝐷(𝐶𝐵)))

Theorembrco3f1o 37351 Conditions allowing the decomposition of a binary relation. (Contributed by RP, 8-Jun-2021.)
(𝜑𝐶:𝑌1-1-onto𝑍)    &   (𝜑𝐷:𝑋1-1-onto𝑌)    &   (𝜑𝐸:𝑊1-1-onto𝑋)    &   (𝜑𝐴(𝐶 ∘ (𝐷𝐸))𝐵)       (𝜑 → ((𝐶𝐵)𝐶𝐵 ∧ (𝐷‘(𝐶𝐵))𝐷(𝐶𝐵) ∧ 𝐴𝐸(𝐷‘(𝐶𝐵))))

Theoremntrclsbex 37352 If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then the base set exists. (Contributed by RP, 21-May-2021.)
𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)       (𝜑𝐵 ∈ V)

Theoremntrclsrcomplex 37353 The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 25-Jun-2021.)
𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)       (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)

Theoremneik0imk0p 37354 Kuratowski's K0 axiom implies K0'. Neighborhood version. Also a proof the dual KA axiom imples KA' when considering the convergents. (Contributed by RP, 28-Jun-2021.)
(∀𝑥𝐵 𝐵 ∈ (𝑁𝑥) → ∀𝑥𝐵 (𝑁𝑥) ≠ ∅)

Theoremntrk2imkb 37355* 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 37356* 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 37357* 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 37356. (Contributed by RP, 12-Jun-2021.)
((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)) → (𝐼‘∅) = ∅))

Theoremclsk3nimkb 37358* If the base set is not empty, axiom K3 does not imply KB. An 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 37359 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 37360* 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 37361* 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 37362* 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 37363* 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 37364* 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 37365* 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 37366* 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 37367* 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 37368* 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 37369* 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 37370* 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 37371* 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 37372* 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 37373* 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 37374* 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 37375* 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 37376* 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 37377* 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 37378* 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 37379* 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 37380* 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 37381* 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 37382* 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 37383* 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 37384* 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 37385* 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 37386* 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 37387* 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 37388* 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 37389* 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 37390* 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 37391* 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 37392* 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 37393* 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 37394* 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 37395* 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 37396* 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 37397* 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 37398* 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 37399* 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 37400* 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)})

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