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Theorem List for Metamath Proof Explorer - 39201-39300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfrege106 39201 Whatever follows 𝑋 in the 𝑅-sequence belongs to the 𝑅 -sequence beginning with 𝑋. Proposition 106 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑍𝑉       (𝑋(t+‘𝑅)𝑍𝑋((t+‘𝑅) ∪ I )𝑍)

Theoremfrege107 39202 Proposition 107 of [Frege1879] p. 74. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑉𝐴       ((𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉𝑍(t+‘𝑅)𝑉)) → (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉𝑍((t+‘𝑅) ∪ I )𝑉)))

Theoremfrege108 39203 If 𝑌 belongs to the 𝑅-sequence beginning with 𝑍, then every result of an application of the procedure 𝑅 to 𝑌 belongs to the 𝑅-sequence beginning with 𝑍. Proposition 108 of [Frege1879] p. 74. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑍𝐴    &   𝑌𝐵    &   𝑉𝐶    &   𝑅𝐷       (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉𝑍((t+‘𝑅) ∪ I )𝑉))

Theoremfrege109 39204 The property of belonging to the 𝑅-sequence beginning with 𝑋 is hereditary in the 𝑅-sequence. Proposition 109 of [Frege1879] p. 74. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑅𝑉       𝑅 hereditary (((t+‘𝑅) ∪ I ) “ {𝑋})

Theoremfrege110 39205* Proposition 110 of [Frege1879] p. 75. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑋𝐴    &   𝑌𝐵    &   𝑀𝐶    &   𝑅𝐷       (∀𝑎(𝑌𝑅𝑎𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀𝑋((t+‘𝑅) ∪ I )𝑀))

Theoremfrege111 39206 If 𝑌 belongs to the 𝑅-sequence beginning with 𝑍, then every result of an application of the procedure 𝑅 to 𝑌 belongs to the 𝑅-sequence beginning with 𝑍 or precedes 𝑍 in the 𝑅-sequence. Proposition 111 of [Frege1879] p. 75. (Contributed by RP, 7-Jul-2020.) (Revised by RP, 8-Jul-2020.) (Proof modification is discouraged.)
𝑍𝐴    &   𝑌𝐵    &   𝑉𝐶    &   𝑅𝐷       (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → (¬ 𝑉(t+‘𝑅)𝑍𝑍((t+‘𝑅) ∪ I )𝑉)))

Theoremfrege112 39207 Identity implies belonging to the 𝑅-sequence beginning with self. Proposition 112 of [Frege1879] p. 76. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑍𝑉       (𝑍 = 𝑋𝑋((t+‘𝑅) ∪ I )𝑍)

Theoremfrege113 39208 Proposition 113 of [Frege1879] p. 76. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑍𝑉       ((𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋𝑍 = 𝑋)) → (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋𝑋((t+‘𝑅) ∪ I )𝑍)))

Theoremfrege114 39209 If 𝑋 belongs to the 𝑅-sequence beginning with 𝑍, then 𝑍 belongs to the 𝑅-sequence beginning with 𝑋 or 𝑋 follows 𝑍 in the 𝑅-sequence. Proposition 114 of [Frege1879] p. 76. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑍𝑉       (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋𝑋((t+‘𝑅) ∪ I )𝑍))

20.28.3.11  _Begriffsschrift_ Chapter III Single-valued procedures

Fun 𝑅 means the relationship content of procedure 𝑅 is single-valued. The double converse allows us to simply apply this syntax in place of Frege's even though the original never explicitly limited discussion of propositional statements which vary on two variables to relations.

dffrege115 39210 through frege133 39228 develop this and how functions relate to transitive and transitive-reflexive closures.

Theoremdffrege115 39210* If from the circumstance that 𝑐 is a result of an application of the procedure 𝑅 to 𝑏, whatever 𝑏 may be, it can be inferred that every result of an application of the procedure 𝑅 to 𝑏 is the same as 𝑐, then we say : "The procedure 𝑅 is single-valued". Definition 115 of [Frege1879] p. 77. (Contributed by RP, 7-Jul-2020.)
(∀𝑐𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑐)) ↔ Fun 𝑅)

Theoremfrege116 39211* One direction of dffrege115 39210. Proposition 116 of [Frege1879] p. 77. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈       (Fun 𝑅 → ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)))

Theoremfrege117 39212* Lemma for frege118 39213. Proposition 117 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈       ((∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎𝑎 = 𝑋)) → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋))) → (Fun 𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋))))

Theoremfrege118 39213* Simplified application of one direction of dffrege115 39210. Proposition 118 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉       (Fun 𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋)))

Theoremfrege119 39214* Lemma for frege120 39215. Proposition 119 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉       ((∀𝑎(𝑌𝑅𝑎𝑎 = 𝑋) → (𝑌𝑅𝐴𝐴 = 𝑋)) → (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴𝐴 = 𝑋))))

Theoremfrege120 39215 Simplified application of one direction of dffrege115 39210. Proposition 120 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝐴𝑊       (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴𝐴 = 𝑋)))

Theoremfrege121 39216 Lemma for frege122 39217. Proposition 121 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝐴𝑊       ((𝐴 = 𝑋𝑋((t+‘𝑅) ∪ I )𝐴) → (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴𝑋((t+‘𝑅) ∪ I )𝐴))))

Theoremfrege122 39217 If 𝑋 is a result of an application of the single-valued procedure 𝑅 to 𝑌, then every result of an application of the procedure 𝑅 to 𝑌 belongs to the 𝑅-sequence beginning with 𝑋. Proposition 122 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝐴𝑊       (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴𝑋((t+‘𝑅) ∪ I )𝐴)))

Theoremfrege123 39218* Lemma for frege124 39219. Proposition 123 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉       ((∀𝑎(𝑌𝑅𝑎𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀𝑋((t+‘𝑅) ∪ I )𝑀)) → (Fun 𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀𝑋((t+‘𝑅) ∪ I )𝑀))))

Theoremfrege124 39219 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 39220 Lemma for frege126 39221. 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 39221 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 39222 Communte antecedents of frege126 39221. 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 39223 Lemma for frege129 39224. 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 39224 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 39225* Lemma for frege131 39226. 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 39226 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 39227 Lemma for frege133 39228. 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 39228 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))))

20.28.4  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).

20.28.4.1  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 39229 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 39238 for a demonstration of a natural one-to-one onto mapping. (Contributed by RP, 27-Apr-2021.)
((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵𝑚 𝐴))

Theoremenrelmapr 39230 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 39231 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 39232 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 39233* 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 39234* 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 39235* 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 39236* 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 39237* 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 39238* 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 39239* 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 39240* 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 39241* 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 39242* 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 39243* 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 39244* 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 39245* 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 39246* 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 39247* 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 39248* 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 39249* 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 39250* 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 39251* 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 39252* 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 39253* 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 39254* 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)))

20.28.4.2  Generic Pseudoclosure Spaces, Pseudointerior 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 39249. 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 39293, ntrneiel 39317, neicvgel1 39355 ¬ (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 39318, clsneifv3 39346, neicvgfv 39357 {𝑥 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑥)} = (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) = {𝑥 ∈ 𝐵 ∣ ¬ (𝐵 ∖ 𝑆) ∈ (𝑀‘𝑥)} ntrneifv4 39321, ntrclsfv 39295, clsneifv4 39347 {𝑥 ∈ 𝐵 ∣ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑥)} = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑆))) = (𝐾‘𝑆) = {𝑥 ∈ 𝐵 ∣ 𝑆 ∈ (𝑀‘𝑥)} clsneifv4 39347, ntrclsfv 39295, ntrneifv4 39321 {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵 ∖ 𝑠) ∈ (𝑁‘𝑋)} = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑠))} = {𝑠 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐾‘𝑠)} = (𝑀‘𝑋) neicvgfv 39357, clsneifv3 39346, ntrneifv3 39318

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 39301, ntrneineine0 39323, ntrneineine1 39324 (𝑁‘𝑥) ≠ 𝒫 𝐵 ∃𝑢 ∈ 𝒫 𝐵¬ 𝑥 ∈ (𝐼‘𝑢) ∃𝑢 ∈ 𝒫 𝐵𝑥 ∈ (𝐾‘𝑢) (𝑀‘𝑥) ≠ ∅ ntrclsneine0 39301, ntrneineine0 39323, ntrneineine1 39324 𝐵 ∈ (𝑁‘𝑥) (𝐼‘𝐵) = 𝐵 (𝐾‘∅) = ∅ ¬ ∅ ∈ (𝑀‘𝑥) ntrclscls00 39302, ntrneicls00 39325, ntrneicls11 39326 ¬ ∅ ∈ (𝑁‘𝑥) (𝐼‘∅) = ∅ (𝐾‘𝐵) = 𝐵 𝐵 ∈ (𝑀‘𝑥) ntrclscls00 39302, ntrneicls00 39325, ntrneicls11 39326 ((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 39284, isotone2 39285, ntrclsiso 39303, ntrneiiso 39327 (𝑠 ∈ (𝑁‘𝑥) → 𝑥 ∈ 𝑠) (𝐼‘𝑠) ⊆ 𝑠 𝑠 ⊆ (𝐾‘𝑠) (𝑥 ∈ 𝑠 → 𝑠 ∈ (𝑀‘𝑥)) ntrclsk2 39304, ntrneik2 39328, ntrneix2 39329 ((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 39305, ntrneikb 39330, ntrneixb 39331 ((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 39306, ntrneik3 39332, ntrneix3 39333 ((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 39307, ntrneik13 39334, ntrneix13 39335 (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 39308, ntrneik4 39337

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 39273, ntrkbimka 39274, neik0imk0p 39272
Min Strong Generalized Neighborhood Space K2, K3 KA', KA, KB ntrk2imkb 39273, ntrkbimka 39274, neik0imk0p 39272
Gniłka Extended Topology K0', K1 K0 neik0pk1imk0 39283
Brissaud Space K0, K2 K0', KA', KA, KB neik0imk0p 39272, ntrk2imkb 39273, ntrkbimka 39274
Neighborhood Space K0', K1, K2 K0, KA', KA, KB neik0pk1imk0 39283, ntrk2imkb 39273, ntrkbimka 39274, neik0imk0p 39272
Davey and Priestley Intersection Structure K1, K4
Moore Closure Space K1, K2, K4 KA', KA, KB ntrk2imkb 39273, ntrkbimka 39274, neik0imk0p 39272
Convex Closure Space K0', K1, K2, K4 K0, KA', KA, KB neik0pk1imk0 39283, ntrk2imkb 39273, ntrkbimka 39274, neik0imk0p 39272
Smyth Neighborhood Space K0', K13 K0, K1, K3 neik0pk1imk0 39283, ntrk1k3eqk13 39286
Čech Closure Space
Pretopological Space
K0', K2, K13 K0, K1, KA', KA, KB, K3 neik0pk1imk0 39283, ntrk2imkb 39273, ntrkbimka 39274, neik0imk0p 39272, ntrk1k3eqk13 39286
Topological Space K0', K2, K13, K4 K0, K1, KA', KA, KB, K3 neik0pk1imk0 39283, ntrk2imkb 39273, ntrkbimka 39274, neik0imk0p 39272, ntrk1k3eqk13 39286
Alexandroff Space K0', K2, K5 K0, K1, KA', KA, KB, K3, K13 neik0pk1imk0 39283, ntrk2imkb 39273, ntrkbimka 39274, neik0imk0p 39272, ntrk1k3eqk13 39286, TBD
Alexandroff Topological Space K0', K2, K4, K5 K0, K1, KA', KA, KB, K3, K13 neik0pk1imk0 39283, ntrk2imkb 39273, ntrkbimka 39274, neik0imk0p 39272, ntrk1k3eqk13 39286, TBD

Theoremsscon34b 39255 Relative complementation reverses inclusion of subclasses. Relativized version of complss 3973. (Contributed by RP, 3-Jun-2021.)
((𝐴𝐶𝐵𝐶) → (𝐴𝐵 ↔ (𝐶𝐵) ⊆ (𝐶𝐴)))

Theoremrcompleq 39256 Two subclasses are equal if and only if their relative complements are equal. Relativized version of compleq 3974. (Contributed by RP, 10-Jun-2021.)
((𝐴𝐶𝐵𝐶) → (𝐴 = 𝐵 ↔ (𝐶𝐴) = (𝐶𝐵)))

Theoremor3or 39257 Decompose disjunction into three cases. (Contributed by RP, 5-Jul-2021.)
((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))

Theoremandi3or 39258 Distribute over triple disjunction. (Contributed by RP, 5-Jul-2021.)
((𝜑 ∧ (𝜓𝜒𝜃)) ↔ ((𝜑𝜓) ∨ (𝜑𝜒) ∨ (𝜑𝜃)))

Theoremuneqsn 39259 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 39260 Ordinal 3 is the triplet containing ordinals 0, 1 and 2. (Contributed by RP, 8-Jul-2021.)
3o = {∅, 1o, 2o}

Theoremdf3o3 39261 Ordinal 3 , fully expanded. (Contributed by RP, 8-Jul-2021.)
3o = {∅, {∅}, {∅, {∅}}}

Theorembrfvimex 39262 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 39263* 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 39264* 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 39265* 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 39266 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 39267 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 39268 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 39269 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 39270 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 39271 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 39272 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 39273* 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 39274* 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 39275* 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 39274. (Contributed by RP, 12-Jun-2021.)
((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)) → (𝐼‘∅) = ∅))

Theoremclsk3nimkb 39276* If the base set is not empty, axiom K3 does not imply KB. A concrete example with a pseudo-closure function of 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏𝑥)) is given. (Contributed by RP, 16-Jun-2021.)
¬ ∀𝑏𝑘 ∈ (𝒫 𝑏𝑚 𝒫 𝑏)(∀𝑠 ∈ 𝒫 𝑏𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡)) → ∀𝑠 ∈ 𝒫 𝑏𝑡 ∈ 𝒫 𝑏((𝑠𝑡) = 𝑏 → ((𝑘𝑠) ∪ (𝑘𝑡)) = 𝑏))

Theoremclsk1indlem0 39277 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 39278* 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 39279* 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 39280* 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 39281* 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 39282* 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 39283* 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 39284* 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 39285* 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 39286* 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 39287* 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 39288* 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 39289* 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 39290* 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 39291* 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 39292* 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 39293* 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 39294* 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 39295* 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 39296* 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 39297* 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 39298* 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 39299* 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 39300* 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)))

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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 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