P. 441 of Theorem List - Metamath Proof Explorer

Type

Label

Description

Statement

Theorem

frege106 44001

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.)

⊢ 𝑍 ∈ 𝑉    ⇒   ⊢ (𝑋(t+‘𝑅)𝑍 → 𝑋((t+‘𝑅) ∪ I )𝑍)

Theorem

frege107 44002

Proposition 107 of [Frege1879] p. 74. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑉 ∈ 𝐴    ⇒   ⊢ ((𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → 𝑍(t+‘𝑅)𝑉)) → (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → 𝑍((t+‘𝑅) ∪ I )𝑉)))

Theorem

frege108 44003

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.)

⊢ 𝑍 ∈ 𝐴    &   ⊢ 𝑌 ∈ 𝐵    &   ⊢ 𝑉 ∈ 𝐶    &   ⊢ 𝑅 ∈ 𝐷    ⇒   ⊢ (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → 𝑍((t+‘𝑅) ∪ I )𝑉))

Theorem

frege109 44004

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.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑅 ∈ 𝑉    ⇒   ⊢ 𝑅 hereditary (((t+‘𝑅) ∪ I ) “ {𝑋})

Theorem

frege110 44005*

Proposition 110 of [Frege1879] p. 75. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝐴    &   ⊢ 𝑌 ∈ 𝐵    &   ⊢ 𝑀 ∈ 𝐶    &   ⊢ 𝑅 ∈ 𝐷    ⇒   ⊢ (∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀))

Theorem

frege111 44006

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.)

⊢ 𝑍 ∈ 𝐴    &   ⊢ 𝑌 ∈ 𝐵    &   ⊢ 𝑉 ∈ 𝐶    &   ⊢ 𝑅 ∈ 𝐷    ⇒   ⊢ (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → (¬ 𝑉(t+‘𝑅)𝑍 → 𝑍((t+‘𝑅) ∪ I )𝑉)))

Theorem

frege112 44007

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.)

⊢ 𝑍 ∈ 𝑉    ⇒   ⊢ (𝑍 = 𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍)

Theorem

frege113 44008

Proposition 113 of [Frege1879] p. 76. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑍 ∈ 𝑉    ⇒   ⊢ ((𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑍 = 𝑋)) → (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍)))

Theorem

frege114 44009

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.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑍 ∈ 𝑉    ⇒   ⊢ (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍))

21.36.6.11  _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 44010 through frege133 44028 develop this and how functions relate to transitive and transitive-reflexive closures.

Theorem

dffrege115 44010*

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.)

⊢ (∀𝑐∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ Fun ^◡◡𝑅)

Theorem

frege116 44011*

One direction of dffrege115 44010. Proposition 116 of [Frege1879] p. 77. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    ⇒   ⊢ (Fun ^◡◡𝑅 → ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))

Theorem

frege117 44012*

Lemma for frege118 44013. Proposition 117 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    ⇒   ⊢ ((∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋))) → (Fun ^◡◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋))))

Theorem

frege118 44013*

Simplified application of one direction of dffrege115 44010. Proposition 118 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    ⇒   ⊢ (Fun ^◡◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋)))

Theorem

frege119 44014*

Lemma for frege120 44015. Proposition 119 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    ⇒   ⊢ ((∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋) → (𝑌𝑅𝐴 → 𝐴 = 𝑋)) → (Fun ^◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋))))

Theorem

frege120 44015

Simplified application of one direction of dffrege115 44010. Proposition 120 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝐴 ∈ 𝑊    ⇒   ⊢ (Fun ^◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋)))

Theorem

frege121 44016

Lemma for frege122 44017. Proposition 121 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝐴 ∈ 𝑊    ⇒   ⊢ ((𝐴 = 𝑋 → 𝑋((t+‘𝑅) ∪ I )𝐴) → (Fun ^◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝑋((t+‘𝑅) ∪ I )𝐴))))

Theorem

frege122 44017

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.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝐴 ∈ 𝑊    ⇒   ⊢ (Fun ^◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝑋((t+‘𝑅) ∪ I )𝐴)))

Theorem

frege123 44018*

Lemma for frege124 44019. Proposition 123 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    ⇒   ⊢ ((∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀)) → (Fun ^◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀))))

Theorem

frege124 44019

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.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑀 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝑆    ⇒   ⊢ (Fun ^◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀)))

Theorem

frege125 44020

Lemma for frege126 44021. Proposition 125 of [Frege1879] p. 81. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑀 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝑆    ⇒   ⊢ ((𝑋((t+‘𝑅) ∪ I )𝑀 → (¬ 𝑋(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑋)) → (Fun ^◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀 → (¬ 𝑋(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑋)))))

Theorem

frege126 44021

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.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑀 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝑆    ⇒   ⊢ (Fun ^◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀 → (¬ 𝑋(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑋))))

Theorem

frege127 44022

Communte antecedents of frege126 44021. Proposition 127 of [Frege1879] p. 82. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑀 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝑆    ⇒   ⊢ (Fun ^◡◡𝑅 → (𝑌(t+‘𝑅)𝑀 → (𝑌𝑅𝑋 → (¬ 𝑋(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑋))))

Theorem

frege128 44023

Lemma for frege129 44024. Proposition 128 of [Frege1879] p. 83. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑀 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝑆    ⇒   ⊢ ((𝑀((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑋 → (¬ 𝑋(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑋))) → (Fun ^◡◡𝑅 → ((¬ 𝑌(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑌) → (𝑌𝑅𝑋 → (¬ 𝑋(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑋)))))

Theorem

frege129 44024

If the procedure 𝑅 is single-valued and 𝑌 belongs to the 𝑅 -sequence beginning 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 beginning 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.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑀 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝑆    ⇒   ⊢ (Fun ^◡◡𝑅 → ((¬ 𝑌(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑌) → (𝑌𝑅𝑋 → (¬ 𝑋(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑋))))

Theorem

frege130 44025*

Lemma for frege131 44026. Proposition 130 of [Frege1879] p. 84. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑀 ∈ 𝑈    &   ⊢ 𝑅 ∈ 𝑉    ⇒   ⊢ ((∀𝑏((¬ 𝑏(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑏) → ∀𝑎(𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑎))) → 𝑅 hereditary ((^◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))) → (Fun ^◡◡𝑅 → 𝑅 hereditary ((^◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))))

Theorem

frege131 44026

If the procedure 𝑅 is single-valued, then the property of belonging to the 𝑅-sequence beginning 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.)

⊢ 𝑀 ∈ 𝑈    &   ⊢ 𝑅 ∈ 𝑉    ⇒   ⊢ (Fun ^◡◡𝑅 → 𝑅 hereditary ((^◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})))

Theorem

frege132 44027

Lemma for frege133 44028. Proposition 132 of [Frege1879] p. 86. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑀 ∈ 𝑈    &   ⊢ 𝑅 ∈ 𝑉    ⇒   ⊢ ((𝑅 hereditary ((^◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) → (𝑋(t+‘𝑅)𝑀 → (𝑋(t+‘𝑅)𝑌 → (¬ 𝑌(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑌)))) → (Fun ^◡◡𝑅 → (𝑋(t+‘𝑅)𝑀 → (𝑋(t+‘𝑅)𝑌 → (¬ 𝑌(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑌)))))

Theorem

frege133 44028

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.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑀 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝑆    ⇒   ⊢ (Fun ^◡◡𝑅 → (𝑋(t+‘𝑅)𝑀 → (𝑋(t+‘𝑅)𝑌 → (¬ 𝑌(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑌))))

21.36.7  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).

21.36.7.1  Equinumerosity of sets of relations and maps

Because ((2_o ↑_m 𝐵) ↑_m 𝐴) ≈ (2_o ↑_m (𝐴 × 𝐵)) ≈ ((2_o ↑_m 𝐴) ↑_m 𝐵) is an instance of the law of exponents: ((𝐶 ↑_m 𝐵) ↑_m 𝐴) ≈ (𝐶 ↑_m (𝐴 × 𝐵)) ≈ ((𝐶 ↑_m 𝐴) ↑_m 𝐵) we are led to see that (𝒫 𝐵 ↑_m 𝐴) ≈ 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐴 ↑_m 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.

Theorem

enrelmap 44029

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 44038 for a demonstration of a natural one-to-one onto mapping. (Contributed by RP, 27-Apr-2021.)

⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵 ↑_m 𝐴))

Theorem

enrelmapr 44030

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.)

⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐴 ↑_m 𝐵))

Theorem

enmappw 44031

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.)

⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐵 ↑_m 𝐴) ≈ (𝒫 𝐴 ↑_m 𝐵))

Theorem

enmappwid 44032

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.)

⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ↑_m 𝒫 𝐴) ≈ (𝒫 𝒫 𝐴 ↑_m 𝐴))

Theorem

rfovd 44033*

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) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐴𝑂𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})))

Theorem

rfovfvd 44034*

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) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ 𝒫 (𝐴 × 𝐵))    &   ⊢ 𝐹 = (𝐴𝑂𝐵)    ⇒   ⊢ (𝜑 → (𝐹‘𝑅) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦}))

Theorem

rfovfvfvd 44035*

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) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ 𝒫 (𝐴 × 𝐵))    &   ⊢ 𝐹 = (𝐴𝑂𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ 𝐺 = (𝐹‘𝑅)    ⇒   ⊢ (𝜑 → (𝐺‘𝑋) = {𝑦 ∈ 𝐵 ∣ 𝑋𝑅𝑦})

Theorem

rfovcnvf1od 44036*

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) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐹 = (𝐴𝑂𝐵)    ⇒   ⊢ (𝜑 → (𝐹:𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵 ↑_m 𝐴) ∧ ^◡𝐹 = (𝑓 ∈ (𝒫 𝐵 ↑_m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})))

Theorem

rfovcnvd 44037*

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) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐹 = (𝐴𝑂𝐵)    ⇒   ⊢ (𝜑 → ^◡𝐹 = (𝑓 ∈ (𝒫 𝐵 ↑_m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}))

Theorem

rfovf1od 44038*

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) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐹 = (𝐴𝑂𝐵)    ⇒   ⊢ (𝜑 → 𝐹:𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵 ↑_m 𝐴))

Theorem

rfovcnvfvd 44039*

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) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐹 = (𝐴𝑂𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ (𝒫 𝐵 ↑_m 𝐴))    ⇒   ⊢ (𝜑 → (^◡𝐹‘𝐺) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐺‘𝑥))})

Theorem

fsovd 44040*

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 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐴𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵 ↑_m 𝐴) ↦ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})))

Theorem

fsovrfovd 44041*

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 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝑅 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑢 ∈ 𝑎 ↦ {𝑣 ∈ 𝑏 ∣ 𝑢𝑟𝑣})))    &   ⊢ 𝐶 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑠 ∈ 𝒫 (𝑎 × 𝑏) ↦ ^◡𝑠))    ⇒   ⊢ (𝜑 → (𝐴𝑂𝐵) = ((𝐵𝑅𝐴) ∘ ((𝐴𝐶𝐵) ∘ ^◡(𝐴𝑅𝐵))))

Theorem

fsovfvd 44042*

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 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐺 = (𝐴𝑂𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐵 ↑_m 𝐴))    ⇒   ⊢ (𝜑 → (𝐺‘𝐹) = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝐹‘𝑥)}))

Theorem

fsovfvfvd 44043*

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 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐺 = (𝐴𝑂𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐵 ↑_m 𝐴))    &   ⊢ 𝐻 = (𝐺‘𝐹)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐻‘𝑌) = {𝑥 ∈ 𝐴 ∣ 𝑌 ∈ (𝐹‘𝑥)})

Theorem

fsovfd 44044*

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 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐺 = (𝐴𝑂𝐵)    ⇒   ⊢ (𝜑 → 𝐺:(𝒫 𝐵 ↑_m 𝐴)⟶(𝒫 𝐴 ↑_m 𝐵))

Theorem

fsovcnvlem 44045*

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 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐺 = (𝐴𝑂𝐵)    &   ⊢ 𝐻 = (𝐵𝑂𝐴)    ⇒   ⊢ (𝜑 → (𝐻 ∘ 𝐺) = ( I ↾ (𝒫 𝐵 ↑_m 𝐴)))

Theorem

fsovcnvd 44046*

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 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐺 = (𝐴𝑂𝐵)    &   ⊢ 𝐻 = (𝐵𝑂𝐴)    ⇒   ⊢ (𝜑 → ^◡𝐺 = 𝐻)

Theorem

fsovcnvfvd 44047*

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 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐺 = (𝐴𝑂𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐴 ↑_m 𝐵))    ⇒   ⊢ (𝜑 → (^◡𝐺‘𝐹) = (𝑦 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ 𝑦 ∈ (𝐹‘𝑥)}))

Theorem

fsovf1od 44048*

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 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐺 = (𝐴𝑂𝐵)    ⇒   ⊢ (𝜑 → 𝐺:(𝒫 𝐵 ↑_m 𝐴)–1-1-onto→(𝒫 𝐴 ↑_m 𝐵))

Theorem

dssmapfvd 44049*

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 𝑏 ↑_m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐷 = (𝑓 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))))

Theorem

dssmapfv2d 44050*

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 𝑏 ↑_m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))    &   ⊢ 𝐺 = (𝐷‘𝐹)    ⇒   ⊢ (𝜑 → 𝐺 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠)))))

Theorem

dssmapfv3d 44051*

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 𝑏 ↑_m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))    &   ⊢ 𝐺 = (𝐷‘𝐹)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    &   ⊢ 𝑇 = (𝐺‘𝑆)    ⇒   ⊢ (𝜑 → 𝑇 = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆))))

Theorem

dssmapnvod 44052*

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 𝑏 ↑_m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ^◡𝐷 = 𝐷)

Theorem

dssmapf1od 44053*

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 𝑏 ↑_m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑_m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑_m 𝒫 𝐵))

Theorem

dssmap2d 44054*

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 𝑏 ↑_m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐷 ∘ 𝐷) = ( I ↾ (𝒫 𝐵 ↑_m 𝒫 𝐵)))

21.36.7.2  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 44049. 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 𝐵 ↑_m 𝒫 𝐵) ≈ (𝒫 𝒫 𝐵 ↑_m 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 44089, ntrneiel 44113, neicvgel1 44151
	¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋)	↔	¬ 𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑆))	↔	𝑋 ∈ (𝐾‘𝑆)	↔	𝑆 ∈ (𝑀‘𝑋)
Neighborhoods	(𝑁‘𝑋)	=	{𝑠 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑠)}	=	{𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑠))}	=	{𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵 ∖ 𝑠) ∈ (𝑀‘𝑋)}	ntrneifv3 44114, clsneifv3 44142, neicvgfv 44153
Interior	{𝑥 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑥)}	=	(𝐼‘𝑆)	=	(𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆)))	=	{𝑥 ∈ 𝐵 ∣ ¬ (𝐵 ∖ 𝑆) ∈ (𝑀‘𝑥)}	ntrneifv4 44117, ntrclsfv 44091, clsneifv4 44143
Closure	{𝑥 ∈ 𝐵 ∣ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑥)}	=	(𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑆)))	=	(𝐾‘𝑆)	=	{𝑥 ∈ 𝐵 ∣ 𝑆 ∈ (𝑀‘𝑥)}	clsneifv4 44143, ntrclsfv 44091, ntrneifv4 44117
Convergents	{𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵 ∖ 𝑠) ∈ (𝑁‘𝑋)}	=	{𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑠))}	=	{𝑠 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐾‘𝑠)}	=	(𝑀‘𝑋)	neicvgfv 44153, clsneifv3 44142, ntrneifv3 44114

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: ∀𝑥 ∈ 𝐵∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵	Neighborhoods	Interior	Closure	Convergents	Equivalence Theorems
K0' Neighborhoods are nonempty.	(𝑁‘𝑥) ≠ ∅	∃𝑢 ∈ 𝒫 𝐵𝑥 ∈ (𝐼‘𝑢)	∃𝑢 ∈ 𝒫 𝐵¬ 𝑥 ∈ (𝐾‘𝑢)	(𝑀‘𝑥) ≠ 𝒫 𝐵	ntrclsneine0 44097, ntrneineine0 44119, ntrneineine1 44120
KA' No neighborhood is equal to the full powerset.	(𝑁‘𝑥) ≠ 𝒫 𝐵	∃𝑢 ∈ 𝒫 𝐵¬ 𝑥 ∈ (𝐼‘𝑢)	∃𝑢 ∈ 𝒫 𝐵𝑥 ∈ (𝐾‘𝑢)	(𝑀‘𝑥) ≠ ∅	ntrclsneine0 44097, ntrneineine0 44119, ntrneineine1 44120
K0 Preservation of the Nullary Union of Closures	𝐵 ∈ (𝑁‘𝑥)	(𝐼‘𝐵) = 𝐵	(𝐾‘∅) = ∅	¬ ∅ ∈ (𝑀‘𝑥)	ntrclscls00 44098, ntrneicls00 44121, ntrneicls11 44122
KA Preservation of the Nullary Union of Interiors	¬ ∅ ∈ (𝑁‘𝑥)	(𝐼‘∅) = ∅	(𝐾‘𝐵) = 𝐵	𝐵 ∈ (𝑀‘𝑥)	ntrclscls00 44098, ntrneicls00 44121, ntrneicls11 44122
K1 Isotonic Montonic	((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 44080, isotone2 44081, ntrclsiso 44099, ntrneiiso 44123
K2 Closure is Expansive	(𝑠 ∈ (𝑁‘𝑥) → 𝑥 ∈ 𝑠)	(𝐼‘𝑠) ⊆ 𝑠	𝑠 ⊆ (𝐾‘𝑠)	(𝑥 ∈ 𝑠 → 𝑠 ∈ (𝑀‘𝑥))	ntrclsk2 44100, ntrneik2 44124, ntrneix2 44125
KB Non-disjoint Neighborhoods	((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 44101, ntrneikb 44126, ntrneixb 44127
K3 Closure is Sub-linear	((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 44102, ntrneik3 44128, ntrneix3 44129
K13 Closure is finitely linear	((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 44103, ntrneik13 44130, ntrneix13 44131
K4 Closure is idempotent	(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 44104, ntrneik4 44133

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 44069, ntrkbimka 44070, neik0imk0p 44068
Min Strong Generalized Neighborhood Space	K2, K3	KA', KA, KB	ntrk2imkb 44069, ntrkbimka 44070, neik0imk0p 44068
Gniłka Extended Topology	K0', K1	K0	neik0pk1imk0 44079
Brissaud Space	K0, K2	K0', KA', KA, KB	neik0imk0p 44068, ntrk2imkb 44069, ntrkbimka 44070
Neighborhood Space	K0', K1, K2	K0, KA', KA, KB	neik0pk1imk0 44079, ntrk2imkb 44069, ntrkbimka 44070, neik0imk0p 44068
Davey and Priestley Intersection Structure	K1, K4
Moore Closure Space	K1, K2, K4	KA', KA, KB	ntrk2imkb 44069, ntrkbimka 44070, neik0imk0p 44068
Convex Closure Space	K0', K1, K2, K4	K0, KA', KA, KB	neik0pk1imk0 44079, ntrk2imkb 44069, ntrkbimka 44070, neik0imk0p 44068
Smyth Neighborhood Space	K0', K13	K0, K1, K3	neik0pk1imk0 44079, ntrk1k3eqk13 44082
Čech Closure Space Pretopological Space	K0', K2, K13	K0, K1, KA', KA, KB, K3	neik0pk1imk0 44079, ntrk2imkb 44069, ntrkbimka 44070, neik0imk0p 44068, ntrk1k3eqk13 44082
Topological Space	K0', K2, K13, K4	K0, K1, KA', KA, KB, K3	neik0pk1imk0 44079, ntrk2imkb 44069, ntrkbimka 44070, neik0imk0p 44068, ntrk1k3eqk13 44082
Alexandroff Space	K0', K2, K5	K0, K1, KA', KA, KB, K3, K13	neik0pk1imk0 44079, ntrk2imkb 44069, ntrkbimka 44070, neik0imk0p 44068, ntrk1k3eqk13 44082, TBD
Alexandroff Topological Space	K0', K2, K4, K5	K0, K1, KA', KA, KB, K3, K13	neik0pk1imk0 44079, ntrk2imkb 44069, ntrkbimka 44070, neik0imk0p 44068, ntrk1k3eqk13 44082, TBD

Theorem

or3or 44055

Decompose disjunction into three cases. (Contributed by RP, 5-Jul-2021.)

⊢ ((𝜑 ∨ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑 ∧ 𝜓)))

Theorem

andi3or 44056

Distribute over triple disjunction. (Contributed by RP, 5-Jul-2021.)

⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒 ∨ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒) ∨ (𝜑 ∧ 𝜃)))

Theorem

uneqsn 44057

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.)

⊢ ((𝐴 ∪ 𝐵) = {𝐶} ↔ ((𝐴 = {𝐶} ∧ 𝐵 = {𝐶}) ∨ (𝐴 = {𝐶} ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ 𝐵 = {𝐶})))

Theorem

brfvimex 44058

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.)

⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝑅 = (𝐹‘𝐶))    ⇒   ⊢ (𝜑 → 𝐶 ∈ V)

Theorem

brovmptimex 44059*

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 = (𝑥 ∈ 𝐸, 𝑦 ∈ 𝐺 ↦ 𝐻)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝑅 = (𝐶𝐹𝐷))    ⇒   ⊢ (𝜑 → (𝐶 ∈ V ∧ 𝐷 ∈ V))

Theorem

brovmptimex1 44060*

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 = (𝑥 ∈ 𝐸, 𝑦 ∈ 𝐺 ↦ 𝐻)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝑅 = (𝐶𝐹𝐷))    ⇒   ⊢ (𝜑 → 𝐶 ∈ V)

Theorem

brovmptimex2 44061*

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 = (𝑥 ∈ 𝐸, 𝑦 ∈ 𝐺 ↦ 𝐻)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝑅 = (𝐶𝐹𝐷))    ⇒   ⊢ (𝜑 → 𝐷 ∈ V)

Theorem

brcoffn 44062

Conditions allowing the decomposition of a binary relation. (Contributed by RP, 7-Jun-2021.)

⊢ (𝜑 → 𝐶 Fn 𝑌)    &   ⊢ (𝜑 → 𝐷:𝑋⟶𝑌)    &   ⊢ (𝜑 → 𝐴(𝐶 ∘ 𝐷)𝐵)    ⇒   ⊢ (𝜑 → (𝐴𝐷(𝐷‘𝐴) ∧ (𝐷‘𝐴)𝐶𝐵))

Theorem

brcofffn 44063

Conditions allowing the decomposition of a binary relation. (Contributed by RP, 8-Jun-2021.)

⊢ (𝜑 → 𝐶 Fn 𝑍)    &   ⊢ (𝜑 → 𝐷:𝑌⟶𝑍)    &   ⊢ (𝜑 → 𝐸:𝑋⟶𝑌)    &   ⊢ (𝜑 → 𝐴(𝐶 ∘ (𝐷 ∘ 𝐸))𝐵)    ⇒   ⊢ (𝜑 → (𝐴𝐸(𝐸‘𝐴) ∧ (𝐸‘𝐴)𝐷(𝐷‘(𝐸‘𝐴)) ∧ (𝐷‘(𝐸‘𝐴))𝐶𝐵))

Theorem

brco2f1o 44064

Conditions allowing the decomposition of a binary relation. (Contributed by RP, 8-Jun-2021.)

⊢ (𝜑 → 𝐶:𝑌–1-1-onto→𝑍)    &   ⊢ (𝜑 → 𝐷:𝑋–1-1-onto→𝑌)    &   ⊢ (𝜑 → 𝐴(𝐶 ∘ 𝐷)𝐵)    ⇒   ⊢ (𝜑 → ((^◡𝐶‘𝐵)𝐶𝐵 ∧ 𝐴𝐷(^◡𝐶‘𝐵)))

Theorem

brco3f1o 44065

Conditions allowing the decomposition of a binary relation. (Contributed by RP, 8-Jun-2021.)

⊢ (𝜑 → 𝐶:𝑌–1-1-onto→𝑍)    &   ⊢ (𝜑 → 𝐷:𝑋–1-1-onto→𝑌)    &   ⊢ (𝜑 → 𝐸:𝑊–1-1-onto→𝑋)    &   ⊢ (𝜑 → 𝐴(𝐶 ∘ (𝐷 ∘ 𝐸))𝐵)    ⇒   ⊢ (𝜑 → ((^◡𝐶‘𝐵)𝐶𝐵 ∧ (^◡𝐷‘(^◡𝐶‘𝐵))𝐷(^◡𝐶‘𝐵) ∧ 𝐴𝐸(^◡𝐷‘(^◡𝐶‘𝐵))))

Theorem

ntrclsbex 44066

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 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → 𝐵 ∈ V)

Theorem

ntrclsrcomplex 44067

The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 25-Jun-2021.)

⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)

Theorem

neik0imk0p 44068

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.)

⊢ (∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥) → ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) ≠ ∅)

Theorem

ntrk2imkb 44069*

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(𝐼‘𝑠) ⊆ 𝑠 → ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅))

Theorem

ntrkbimka 44070*

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 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) → (𝐼‘∅) = ∅)

Theorem

ntrk0kbimka 44071*

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 44070. (Contributed by RP, 12-Jun-2021.)

⊢ ((𝐵 ∈ 𝑉 ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) → (((𝐼‘𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅)) → (𝐼‘∅) = ∅))

Theorem

clsk3nimkb 44072*

If the base set is not empty, axiom K3 does not imply KB. A concrete example with a pseudo-closure function of 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥)) is given. (Contributed by RP, 16-Jun-2021.)

⊢ ¬ ∀𝑏∀𝑘 ∈ (𝒫 𝑏 ↑_m 𝒫 𝑏)(∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) → ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏))

Theorem

clsk1indlem0 44073

The ansatz closure function (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟)) has the K0 property of preserving the nullary union. (Contributed by RP, 6-Jul-2021.)

⊢ 𝐾 = (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟))    ⇒   ⊢ (𝐾‘∅) = ∅

Theorem

clsk1indlem2 44074*

The ansatz closure function (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟)) has the K2 property of expanding. (Contributed by RP, 6-Jul-2021.)

⊢ 𝐾 = (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟))    ⇒   ⊢ ∀𝑠 ∈ 𝒫 3_o𝑠 ⊆ (𝐾‘𝑠)

Theorem

clsk1indlem3 44075*

The ansatz closure function (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟)) has the K3 property of being sub-linear. (Contributed by RP, 6-Jul-2021.)

⊢ 𝐾 = (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟))    ⇒   ⊢ ∀𝑠 ∈ 𝒫 3_o∀𝑡 ∈ 𝒫 3_o(𝐾‘(𝑠 ∪ 𝑡)) ⊆ ((𝐾‘𝑠) ∪ (𝐾‘𝑡))

Theorem

clsk1indlem4 44076*

The ansatz closure function (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟)) has the K4 property of idempotence. (Contributed by RP, 6-Jul-2021.)

⊢ 𝐾 = (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟))    ⇒   ⊢ ∀𝑠 ∈ 𝒫 3_o(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)

Theorem

clsk1indlem1 44077*

The ansatz closure function (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟)) does not have the K1 property of isotony. (Contributed by RP, 6-Jul-2021.)

⊢ 𝐾 = (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟))    ⇒   ⊢ ∃𝑠 ∈ 𝒫 3_o∃𝑡 ∈ 𝒫 3_o(𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡))

Theorem

clsk1independent 44078*

For generalized closure functions, property K1 (isotony) is independent of the properties K0, K2, K3, K4. This contradicts a claim which appears in preprints of Table 2 in Bärbel M. R. Stadler and Peter F. Stadler. "Generalized Topological Spaces in Evolutionary Theory and Combinatorial Chemistry." J. Chem. Inf. Comput. Sci., 42:577-585, 2002. Proceedings MCC 2001, Dubrovnik. The same table row implying K1 follows from the other four appears in the supplemental materials Bärbel M. R. Stadler and Peter F. Stadler. "Basic Properties of Closure Spaces" 2001 on page 12. (Contributed by RP, 5-Jul-2021.)

⊢ (𝜑 ↔ (𝑘‘∅) = ∅)    &   ⊢ (𝜓 ↔ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡)))    &   ⊢ (𝜒 ↔ ∀𝑠 ∈ 𝒫 𝑏𝑠 ⊆ (𝑘‘𝑠))    &   ⊢ (𝜃 ↔ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)))    &   ⊢ (𝜏 ↔ ∀𝑠 ∈ 𝒫 𝑏(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠))    ⇒   ⊢ ¬ ∀𝑏∀𝑘 ∈ (𝒫 𝑏 ↑_m 𝒫 𝑏)(((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜓)

Theorem

neik0pk1imk0 44079*

Kuratowski's K0' and K1 axioms imply K0. Neighborhood version. (Contributed by RP, 3-Jun-2021.)

⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑_m 𝐵))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) ≠ ∅)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥)))    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥))

Theorem

isotone1 44080*

Two different ways to say subset relation persists across applications of a function. (Contributed by RP, 31-May-2021.)

⊢ (∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝑎 ⊆ 𝑏 → (𝐹‘𝑎) ⊆ (𝐹‘𝑏)) ↔ ∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘(𝑎 ∪ 𝑏)))

Theorem

isotone2 44081*

Two different ways to say subset relation persists across applications of a function. (Contributed by RP, 31-May-2021.)

⊢ (∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝑎 ⊆ 𝑏 → (𝐹‘𝑎) ⊆ (𝐹‘𝑏)) ↔ ∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎 ∩ 𝑏)) ⊆ ((𝐹‘𝑎) ∩ (𝐹‘𝑏)))

Theorem

ntrk1k3eqk13 44082*

An interior function is both monotone and sub-linear if and only if it is finitely linear. (Contributed by RP, 18-Jun-2021.)

⊢ ((∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ∧ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∩ 𝑡))) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠 ∩ 𝑡)) = ((𝐼‘𝑠) ∩ (𝐼‘𝑡)))

Theorem

ntrclsf1o 44083*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator we may characterize the relation as part of a 1-to-1 onto function. (Contributed by RP, 29-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑_m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑_m 𝒫 𝐵))

Theorem

ntrclsnvobr 44084*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then they are related the opposite way. (Contributed by RP, 21-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → 𝐾𝐷𝐼)

Theorem

ntrclsiex 44085*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then those functions are maps of subsets to subsets. (Contributed by RP, 21-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))

Theorem

ntrclskex 44086*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then those functions are maps of subsets to subsets. (Contributed by RP, 21-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))

Theorem

ntrclsfv1 44087*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is a functional relation between them (Contributed by RP, 28-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾)

Theorem

ntrclsfv2 44088*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is a functional relation between them (Contributed by RP, 28-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (𝐷‘𝐾) = 𝐼)

Theorem

ntrclselnel1 44089*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is an equivalence between membership in the interior of a set and non-membership in the closure of the complement of the set. (Contributed by RP, 28-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝐼‘𝑆) ↔ ¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆))))

Theorem

ntrclselnel2 44090*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is an equivalence between membership in interior of the complement of a set and non-membership in the closure of the set. (Contributed by RP, 28-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑆)) ↔ ¬ 𝑋 ∈ (𝐾‘𝑆)))

Theorem

ntrclsfv 44091*

The value of the interior (closure) expressed in terms of the closure (interior). (Contributed by RP, 25-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))))

Theorem

ntrclsfveq1 44092*

If interior and closure functions are related then specific function values are complementary. (Contributed by RP, 27-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → ((𝐼‘𝑆) = 𝐶 ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ 𝐶)))

Theorem

ntrclsfveq2 44093*

If interior and closure functions are related then specific function values are complementary. (Contributed by RP, 27-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → ((𝐼‘(𝐵 ∖ 𝑆)) = 𝐶 ↔ (𝐾‘𝑆) = (𝐵 ∖ 𝐶)))

Theorem

ntrclsfveq 44094*

If interior and closure functions are related then equality of a pair of function values is equivalent to equality of a pair of the other function's values. (Contributed by RP, 27-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    &   ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → ((𝐼‘𝑆) = (𝐼‘𝑇) ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐾‘(𝐵 ∖ 𝑇))))

Theorem

ntrclsss 44095*

If interior and closure functions are related then a subset relation of a pair of function values is equivalent to subset relation of a pair of the other function's values. (Contributed by RP, 27-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    &   ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → ((𝐼‘𝑆) ⊆ (𝐼‘𝑇) ↔ (𝐾‘(𝐵 ∖ 𝑇)) ⊆ (𝐾‘(𝐵 ∖ 𝑆))))

Theorem

ntrclsneine0lem 44096*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that at least one (pseudo-)neighborbood of a particular point exists hold equally. (Contributed by RP, 21-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼‘𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐾‘𝑠)))

Theorem

ntrclsneine0 44097*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that for every point, at least one (pseudo-)neighborbood exists hold equally. (Contributed by RP, 21-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∃𝑠 ∈ 𝒫 𝐵𝑥 ∈ (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑥 ∈ (𝐾‘𝑠)))

Theorem

ntrclscls00 44098*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 1-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅))

Theorem

ntrclsiso 44099*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that either is isotonic hold equally. (Contributed by RP, 3-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐾‘𝑠) ⊆ (𝐾‘𝑡))))

Theorem

ntrclsk2 44100*

An interior function is contracting if and only if the closure function is expansive. (Contributed by RP, 9-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐾‘𝑠)))