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Theorem List for Metamath Proof Explorer - 42701-42800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfrege97 42701 The property of following 𝑋 in the 𝑅-sequence is hereditary in the 𝑅-sequence. Proposition 97 of [Frege1879] p. 71.

Here we introduce the image of a singleton under a relation as class which stands for the property of following 𝑋 in the 𝑅 -sequence. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)

𝑋𝑈    &   𝑅𝑊       𝑅 hereditary ((t+‘𝑅) “ {𝑋})
 
Theoremfrege98 42702 If 𝑌 follows 𝑋 and 𝑍 follows 𝑌 in the 𝑅-sequence then 𝑍 follows 𝑋 in the 𝑅-sequence because the transitive closure of a relation has the transitive property. Proposition 98 of [Frege1879] p. 71. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 6-Jul-2020.) (Proof modification is discouraged.)
𝑋𝐴    &   𝑌𝐵    &   𝑍𝐶    &   𝑅𝐷       (𝑋(t+‘𝑅)𝑌 → (𝑌(t+‘𝑅)𝑍𝑋(t+‘𝑅)𝑍))
 
21.34.6.10  _Begriffsschrift_ Chapter III Member of sequence

𝑝((t+‘𝑅) ∪ I )𝑐 means 𝑐 is a member of the 𝑅 -sequence begining with 𝑝 and 𝑝 is a member of the 𝑅 -sequence ending with 𝑐.

dffrege99 42703 through frege114 42718 develop this.

This will be shown to be related to the transitive-reflexive closure of relation 𝑅. But more work needs to be done on transitive closure of relations before this is ready for Metamath.

 
Theoremdffrege99 42703 If 𝑍 is identical with 𝑋 or follows 𝑋 in the 𝑅 -sequence, then we say : "𝑍 belongs to the 𝑅-sequence beginning with 𝑋 " or "𝑋 belongs to the 𝑅-sequence ending with 𝑍". Definition 99 of [Frege1879] p. 71. (Contributed by RP, 2-Jul-2020.)
𝑍𝑈       ((¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍)
 
Theoremfrege100 42704 One direction of dffrege99 42703. Proposition 100 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑍𝑈       (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋))
 
Theoremfrege101 42705 Lemma for frege102 42706. Proposition 101 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑍𝑈       ((𝑍 = 𝑋 → (𝑍𝑅𝑉𝑋(t+‘𝑅)𝑉)) → ((𝑋(t+‘𝑅)𝑍 → (𝑍𝑅𝑉𝑋(t+‘𝑅)𝑉)) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (𝑍𝑅𝑉𝑋(t+‘𝑅)𝑉))))
 
Theoremfrege102 42706 If 𝑍 belongs to the 𝑅-sequence beginning with 𝑋, then every result of an application of the procedure 𝑅 to 𝑍 follows 𝑋 in the 𝑅-sequence. Proposition 102 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑋𝐴    &   𝑍𝐵    &   𝑉𝐶    &   𝑅𝐷       (𝑋((t+‘𝑅) ∪ I )𝑍 → (𝑍𝑅𝑉𝑋(t+‘𝑅)𝑉))
 
Theoremfrege103 42707 Proposition 103 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑍𝑉       ((𝑍 = 𝑋𝑋 = 𝑍) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍𝑋 = 𝑍)))
 
Theoremfrege104 42708 Proposition 104 of [Frege1879] p. 73.

Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the minor clause and result swapped. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

𝑍𝑉       (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍𝑋 = 𝑍))
 
Theoremfrege105 42709 Proposition 105 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑍𝑉       ((¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋) → 𝑋((t+‘𝑅) ∪ I )𝑍)
 
Theoremfrege106 42710 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 42711 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 42712 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 42713 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 42714* 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 42715 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 42716 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 42717 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 42718 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))
 
21.34.6.11  _Begriffsschrift_ Chapter III Single-valued procedures

Fun 𝑅 means the relationship content of procedure 𝑅 is single-valued. The double converse allows 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 42719 through frege133 42737 develop this and how functions relate to transitive and transitive-reflexive closures.

 
Theoremdffrege115 42719* 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 42720* One direction of dffrege115 42719. 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 42721* Lemma for frege118 42722. 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 42722* Simplified application of one direction of dffrege115 42719. 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 42723* Lemma for frege120 42724. 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 42724 Simplified application of one direction of dffrege115 42719. 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 42725 Lemma for frege122 42726. 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 42726 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 42727* Lemma for frege124 42728. 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 42728 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 42729 Lemma for frege126 42730. 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 42730 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 42731 Communte antecedents of frege126 42730. 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 42732 Lemma for frege129 42733. 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 42733 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 42734* Lemma for frege131 42735. 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 42735 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 42736 Lemma for frege133 42737. 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 42737 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))))
 
21.34.7  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).

 
21.34.7.1  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 42738 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 42747 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 42739 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 42740 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 42741 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 42742* 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 42743* 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 42744* 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 42745* 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 42746* 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 42747* 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 42748* 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 42749* 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 42750* 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 42751* 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 42752* 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 42753* 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 42754* 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 42755* 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 42756* 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 42757* 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 42758* 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 42759* 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 42760* 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 42761* 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 42762* 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 42763* 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)))
 
21.34.7.2  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 42758. 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 42798, ntrneiel 42822, neicvgel1 42860
¬ (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 42823, clsneifv3 42851, neicvgfv 42862
Interior {𝑥𝐵𝑆 ∈ (𝑁𝑥)} = (𝐼𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑆))) = {𝑥𝐵 ∣ ¬ (𝐵𝑆) ∈ (𝑀𝑥)} ntrneifv4 42826, ntrclsfv 42800, clsneifv4 42852
Closure {𝑥𝐵 ∣ ¬ (𝐵𝑆) ∈ (𝑁𝑥)} = (𝐵 ∖ (𝐼‘(𝐵𝑆))) = (𝐾𝑆) = {𝑥𝐵𝑆 ∈ (𝑀𝑥)} clsneifv4 42852, ntrclsfv 42800, ntrneifv4 42826
Convergents {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵𝑠) ∈ (𝑁𝑋)} = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ (𝐼‘(𝐵𝑠))} = {𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐾𝑠)} = (𝑀𝑋) neicvgfv 42862, clsneifv3 42851, ntrneifv3 42823

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 nonempty.
(𝑁𝑥) ≠ ∅ 𝑢 ∈ 𝒫 𝐵𝑥 ∈ (𝐼𝑢) 𝑢 ∈ 𝒫 𝐵¬ 𝑥 ∈ (𝐾𝑢) (𝑀𝑥) ≠ 𝒫 𝐵 ntrclsneine0 42806, ntrneineine0 42828, ntrneineine1 42829
KA'
No neighborhood is equal to the full powerset.
(𝑁𝑥) ≠ 𝒫 𝐵 𝑢 ∈ 𝒫 𝐵¬ 𝑥 ∈ (𝐼𝑢) 𝑢 ∈ 𝒫 𝐵𝑥 ∈ (𝐾𝑢) (𝑀𝑥) ≠ ∅ ntrclsneine0 42806, ntrneineine0 42828, ntrneineine1 42829
K0
Preservation of the Nullary Union of Closures
𝐵 ∈ (𝑁𝑥) (𝐼𝐵) = 𝐵 (𝐾‘∅) = ∅ ¬ ∅ ∈ (𝑀𝑥) ntrclscls00 42807, ntrneicls00 42830, ntrneicls11 42831
KA
Preservation of the Nullary Union of Interiors
¬ ∅ ∈ (𝑁𝑥) (𝐼‘∅) = ∅ (𝐾𝐵) = 𝐵 𝐵 ∈ (𝑀𝑥) ntrclscls00 42807, ntrneicls00 42830, ntrneicls11 42831
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 42789, isotone2 42790, ntrclsiso 42808, ntrneiiso 42832
K2
Closure is Expansive
(𝑠 ∈ (𝑁𝑥) → 𝑥𝑠) (𝐼𝑠) ⊆ 𝑠 𝑠 ⊆ (𝐾𝑠) (𝑥𝑠𝑠 ∈ (𝑀𝑥)) ntrclsk2 42809, ntrneik2 42833, ntrneix2 42834
KB
Non-disjoint Neighborhoods
((𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)) → (𝑠𝑡) ≠ ∅) ((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) ((𝑠𝑡) = 𝐵 → ((𝐾𝑠) ∪ (𝐾𝑡)) = 𝐵) ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑀𝑥) ∨ 𝑡 ∈ (𝑀𝑥))) ntrclskb 42810, ntrneikb 42835, ntrneixb 42836
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 42811, ntrneik3 42837, ntrneix3 42838
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 42812, ntrneik13 42839, ntrneix13 42840
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 42813, ntrneik4 42842

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 42778, ntrkbimka 42779, neik0imk0p 42777
Min Strong Generalized Neighborhood Space K2, K3 KA', KA, KB ntrk2imkb 42778, ntrkbimka 42779, neik0imk0p 42777
Gniłka Extended Topology K0', K1 K0 neik0pk1imk0 42788
Brissaud Space K0, K2 K0', KA', KA, KB neik0imk0p 42777, ntrk2imkb 42778, ntrkbimka 42779
Neighborhood Space K0', K1, K2 K0, KA', KA, KB neik0pk1imk0 42788, ntrk2imkb 42778, ntrkbimka 42779, neik0imk0p 42777
Davey and Priestley Intersection Structure K1, K4
Moore Closure Space K1, K2, K4 KA', KA, KB ntrk2imkb 42778, ntrkbimka 42779, neik0imk0p 42777
Convex Closure Space K0', K1, K2, K4 K0, KA', KA, KB neik0pk1imk0 42788, ntrk2imkb 42778, ntrkbimka 42779, neik0imk0p 42777
Smyth Neighborhood Space K0', K13 K0, K1, K3 neik0pk1imk0 42788, ntrk1k3eqk13 42791
Čech Closure Space
Pretopological Space
K0', K2, K13 K0, K1, KA', KA, KB, K3 neik0pk1imk0 42788, ntrk2imkb 42778, ntrkbimka 42779, neik0imk0p 42777, ntrk1k3eqk13 42791
Topological Space K0', K2, K13, K4 K0, K1, KA', KA, KB, K3 neik0pk1imk0 42788, ntrk2imkb 42778, ntrkbimka 42779, neik0imk0p 42777, ntrk1k3eqk13 42791
Alexandroff Space K0', K2, K5 K0, K1, KA', KA, KB, K3, K13 neik0pk1imk0 42788, ntrk2imkb 42778, ntrkbimka 42779, neik0imk0p 42777, ntrk1k3eqk13 42791, TBD
Alexandroff Topological Space K0', K2, K4, K5 K0, K1, KA', KA, KB, K3, K13 neik0pk1imk0 42788, ntrk2imkb 42778, ntrkbimka 42779, neik0imk0p 42777, ntrk1k3eqk13 42791, TBD
 
Theoremor3or 42764 Decompose disjunction into three cases. (Contributed by RP, 5-Jul-2021.)
((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))
 
Theoremandi3or 42765 Distribute over triple disjunction. (Contributed by RP, 5-Jul-2021.)
((𝜑 ∧ (𝜓𝜒𝜃)) ↔ ((𝜑𝜓) ∨ (𝜑𝜒) ∨ (𝜑𝜃)))
 
Theoremuneqsn 42766 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})))
 
Theorembrfvimex 42767 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 42768* 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 42769* 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 42770* 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 42771 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 42772 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 42773 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 42774 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 42775 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 42776 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 42777 Kuratowski's K0 axiom implies K0'. Neighborhood version. Also a proof the dual KA axiom implies KA' when considering the convergents. (Contributed by RP, 28-Jun-2021.)
(∀𝑥𝐵 𝐵 ∈ (𝑁𝑥) → ∀𝑥𝐵 (𝑁𝑥) ≠ ∅)
 
Theoremntrk2imkb 42778* 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 42779* 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 42780* 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 42779. (Contributed by RP, 12-Jun-2021.)
((𝐵𝑉𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) → (((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)) → (𝐼‘∅) = ∅))
 
Theoremclsk3nimkb 42781* 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 42782 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 42783* 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 42784* 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 42785* 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 42786* 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 42787* 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 42788* 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 42789* 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 42790* 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 42791* 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 42792* 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 42793* 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 42794* 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 42795* 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 42796* 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 42797* 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 42798* 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 42799* 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 42800* 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))))
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