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Theorem List for Metamath Proof Explorer - 37201-37300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfrege127 37201 Communte antecedents of frege126 37200. 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 37202 Lemma for frege129 37203. 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 37203 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 37204* Lemma for frege131 37205. 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 37205 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 37206 Lemma for frege133 37207. 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 37207 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))))
 
20.26.4  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).

 
20.26.4.1  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 37208 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 37217 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 37209 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 37210 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 37211 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 37212* 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 37213* 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 37214* 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 37215* 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 37216* 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 37217* 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 37218* 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 37219* 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 37220* 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 37221* 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 37222* 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 37223* 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 37224* 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 37225* 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 37226* 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 37227* 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 37228* 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 37229* 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 37230* 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 37231* 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 37232* 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 37233* 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)))
 
20.26.4.2  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 37228. 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 𝐵𝑚 𝐵) 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) 𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) 𝑀 ∈ (𝒫 𝒫 𝐵𝑚 𝐵)
Correspondences
(assuming (𝑋𝐵𝑆 ∈ 𝒫 𝐵))
𝑆 ∈ (𝑁𝑋) 𝑋 ∈ (𝐼𝑆) ¬ 𝑋 ∈ (𝐾‘(𝐵𝑆)) ¬ (𝐵𝑆) ∈ (𝑀𝑋) ntrclselnel1 37272, ntrneiel 37296, neicvgel1 37334
¬ (𝐵𝑆) ∈ (𝑁𝑋) ¬ 𝑋 ∈ (𝐼‘(𝐵𝑆)) 𝑋 ∈ (𝐾𝑆) 𝑆 ∈ (𝑀𝑋)
Neighborhoods (𝑁𝑋) = {𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠)} = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ (𝐾‘(𝐵𝑠))} = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵𝑠) ∈ (𝑀𝑋)} ntrneifv3 37297, clsneifv3 37325, neicvgfv 37336
Interior {𝑥𝐵𝑆 ∈ (𝑁𝑥)} = (𝐼𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑆))) = {𝑥𝐵 ∣ ¬ (𝐵𝑆) ∈ (𝑀𝑥)} ntrneifv4 37300, ntrclsfv 37274, clsneifv4 37326
Closure {𝑥𝐵 ∣ ¬ (𝐵𝑆) ∈ (𝑁𝑥)} = (𝐵 ∖ (𝐼‘(𝐵𝑆))) = (𝐾𝑆) = {𝑥𝐵𝑆 ∈ (𝑀𝑥)} clsneifv4 37326, ntrclsfv 37274, ntrneifv4 37300
Convergents {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵𝑠) ∈ (𝑁𝑋)} = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ (𝐼‘(𝐵𝑠))} = {𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐾𝑠)} = (𝑀𝑋) neicvgfv 37336, clsneifv3 37325, ntrneifv3 37297

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).

Assuming a prefix of:
𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵
Neighborhoods Interior Closure Convergents Equivalence Theorems
K0'
Neighborhoods are non-empty.
(𝑁𝑥) ≠ ∅ 𝑢 ∈ 𝒫 𝐵𝑥 ∈ (𝐼𝑢) 𝑢 ∈ 𝒫 𝐵¬ 𝑥 ∈ (𝐾𝑢) (𝑀𝑥) ≠ 𝒫 𝐵 ntrclsneine0 37280, ntrneineine0 37302, ntrneineine1 37303
KA'
No neighborhood is equal to the full powerset.
(𝑁𝑥) ≠ 𝒫 𝐵 𝑢 ∈ 𝒫 𝐵¬ 𝑥 ∈ (𝐼𝑢) 𝑢 ∈ 𝒫 𝐵𝑥 ∈ (𝐾𝑢) (𝑀𝑥) ≠ ∅ ntrclsneine0 37280, ntrneineine0 37302, ntrneineine1 37303
K0
Preservation of the Nullary Union of Closures
𝐵 ∈ (𝑁𝑥) (𝐼𝐵) = 𝐵 (𝐾‘∅) = ∅ ¬ ∅ ∈ (𝑀𝑥) ntrclscls00 37281, ntrneicls00 37304, ntrneicls11 37305
KA
Preservation of the Nullary Union of Interiors
¬ ∅ ∈ (𝑁𝑥) (𝐼‘∅) = ∅ (𝐾𝐵) = 𝐵 𝐵 ∈ (𝑀𝑥) ntrclscls00 37281, ntrneicls00 37304, ntrneicls11 37305
K1
Isotonic
Montonic
((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)) (𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡))
— or —
((𝐼𝑠) ∪ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡))
— or —
(𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∩ (𝐼𝑡))
(𝑠𝑡 → (𝐾𝑠) ⊆ (𝐾𝑡))
— or —
((𝐾𝑠) ∪ (𝐾𝑡)) ⊆ (𝐾‘(𝑠𝑡))
— or —
(𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∩ (𝐾𝑡))
((𝑠 ∈ (𝑀𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑀𝑥)) isotone1 37263, isotone2 37264, ntrclsiso 37282, ntrneiiso 37306
K2
Closure is Expansive
(𝑠 ∈ (𝑁𝑥) → 𝑥𝑠) (𝐼𝑠) ⊆ 𝑠 𝑠 ⊆ (𝐾𝑠) (𝑥𝑠𝑠 ∈ (𝑀𝑥)) ntrclsk2 37283, ntrneik2 37307, ntrneix2 37308
KB
Non-disjoint Neighborhoods
((𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)) → (𝑠𝑡) ≠ ∅) ((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) ((𝑠𝑡) = 𝐵 → ((𝐾𝑠) ∪ (𝐾𝑡)) = 𝐵) ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑀𝑥) ∨ 𝑡 ∈ (𝑀𝑥))) ntrclskb 37284, ntrneikb 37309, ntrneixb 37310
K3
Closure is Sub-linear
((𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)) → (𝑠𝑡) ∈ (𝑁𝑥)) ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡)) (𝐾‘(𝑠𝑡)) ⊆ ((𝐾𝑠) ∪ (𝐾𝑡)) ((𝑠𝑡) ∈ (𝑀𝑥) → (𝑠 ∈ (𝑀𝑥) ∨ 𝑡 ∈ (𝑀𝑥))) ntrclsk3 37285, ntrneik3 37311, ntrneix3 37312
K13
Closure is finitely linear
((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥))) (𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) (𝐾‘(𝑠𝑡)) = ((𝐾𝑠) ∪ (𝐾𝑡)) ((𝑠𝑡) ∈ (𝑀𝑥) ↔ (𝑠 ∈ (𝑀𝑥) ∨ 𝑡 ∈ (𝑀𝑥))) ntrclsk13 37286, ntrneik13 37313, ntrneix13 37314
K4
Closure is idempotent
(𝑠 ∈ (𝑁𝑥) ↔ ∃𝑢 ∈ (𝑁𝑥)∀𝑦𝐵 (𝑦𝑢𝑠 ∈ (𝑁𝑦))) (𝐼‘(𝐼𝑠)) = (𝐼𝑠) (𝐾‘(𝐾𝑠)) = (𝐾𝑠) (𝑠 ∈ (𝑀𝑥) ↔ ∃𝑢 ∈ (𝑀𝑥)∀𝑦𝐵 (𝑦𝑢𝑠 ∈ (𝑀𝑦))) ntrclsk4 37287, ntrneik4 37316

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 37252, ntrkbimka 37253, neik0imk0p 37251
Min Strong Generalized Neighborhood Space K2, K3 KA', KA, KB ntrk2imkb 37252, ntrkbimka 37253, neik0imk0p 37251
Gniłka Extended Topology K0', K1 K0 neik0pk1imk0 37262
Brissaud Space K0, K2 K0', KA', KA, KB neik0imk0p 37251, ntrk2imkb 37252, ntrkbimka 37253
Neighborhood Space K0', K1, K2 K0, KA', KA, KB neik0pk1imk0 37262, ntrk2imkb 37252, ntrkbimka 37253, neik0imk0p 37251
Davey and Priestley Intersection Structure K1, K4
Moore Closure Space K1, K2, K4 KA', KA, KB ntrk2imkb 37252, ntrkbimka 37253, neik0imk0p 37251
Convex Closure Space K0', K1, K2, K4 K0, KA', KA, KB neik0pk1imk0 37262, ntrk2imkb 37252, ntrkbimka 37253, neik0imk0p 37251
Smyth Neighborhood Space K0', K13 K0, K1, K3 neik0pk1imk0 37262, ntrk1k3eqk13 37265
Čech Closure Space
Pretopological Space
K0', K2, K13 K0, K1, KA', KA, KB, K3 neik0pk1imk0 37262, ntrk2imkb 37252, ntrkbimka 37253, neik0imk0p 37251, ntrk1k3eqk13 37265
Topological Space K0', K2, K13, K4 K0, K1, KA', KA, KB, K3 neik0pk1imk0 37262, ntrk2imkb 37252, ntrkbimka 37253, neik0imk0p 37251, ntrk1k3eqk13 37265
Alexandroff Space K0', K2, K5 K0, K1, KA', KA, KB, K3, K13 neik0pk1imk0 37262, ntrk2imkb 37252, ntrkbimka 37253, neik0imk0p 37251, ntrk1k3eqk13 37265, TBD
Alexandroff Topological Space K0', K2, K4, K5 K0, K1, KA', KA, KB, K3, K13 neik0pk1imk0 37262, ntrk2imkb 37252, ntrkbimka 37253, neik0imk0p 37251, ntrk1k3eqk13 37265, TBD
 
Theoremsscon34b 37234 Relative complementation reverses inclusion of subclasses. Relativized version of complss 3617. (Contributed by RP, 3-Jun-2021.)
((𝐴𝐶𝐵𝐶) → (𝐴𝐵 ↔ (𝐶𝐵) ⊆ (𝐶𝐴)))
 
Theoremrcompleq 37235 Two subclasses are equal if and only if their relative complements are equal. Relativized version of compleq 3618. (Contributed by RP, 10-Jun-2021.)
((𝐴𝐶𝐵𝐶) → (𝐴 = 𝐵 ↔ (𝐶𝐴) = (𝐶𝐵)))
 
Theoremor3or 37236 Decompose disjunction into three cases. (Contributed by RP, 5-Jul-2021.)
((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))
 
Theoremandi3or 37237 Distribute over triple disjunction. (Contributed by RP, 5-Jul-2021.)
((𝜑 ∧ (𝜓𝜒𝜃)) ↔ ((𝜑𝜓) ∨ (𝜑𝜒) ∨ (𝜑𝜃)))
 
Theoremuneqsn 37238 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 37239 Ordinal 3 is the triplet containing ordinals 0, 1 and 2. (Contributed by RP, 8-Jul-2021.)
3𝑜 = {∅, 1𝑜, 2𝑜}
 
Theoremdf3o3 37240 Ordinal 3 , fully expanded. (Contributed by RP, 8-Jul-2021.)
3𝑜 = {∅, {∅}, {∅, {∅}}}
 
Theorembrfvimex 37241 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 37242* 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 37243* 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 37244* 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 37245 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 37246 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 37247 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 37248 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 37249 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 37250 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 37251 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 37252* 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 37253* 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 37254* 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 37253. (Contributed by RP, 12-Jun-2021.)
((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)) → (𝐼‘∅) = ∅))
 
Theoremclsk3nimkb 37255* 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 37256 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 37257* 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 37258* 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 37259* 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 37260* 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 37261* 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 37262* 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 37263* 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 37264* 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 37265* 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 37266* 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 37267* 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 37268* 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 37269* 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 37270* 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 37271* 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 37272* 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 37273* 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 37274* 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 37275* 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 37276* 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 37277* 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 37278* 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 37279* 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 37280* 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 37281* 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 37282* 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 37283* 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 37284* 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 37285* 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 37286* 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 37287* 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 37288* 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 37289* 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 37290* 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 37291* 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 37292* 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 37293* 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 37294* 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 37295* 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 37296* 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 37297* 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 37298* 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 37299* 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 37300* 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)})
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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42426
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