P. 233 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 23201-23300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cpmadumatpolylem2 23201*	Lemma 2 for cpmadumatpoly 23202. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 15-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ × = (.r‘𝑌)    &   ⊢ − = (-g‘𝑌)    &   ⊢ 0 = (0g‘𝑌)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))    &   ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑌)    &   ⊢ 1 = (1r‘𝑌)    &   ⊢ 𝑍 = (var1‘𝑅)    &   ⊢ 𝐷 = ((𝑍 · 1 ) − (𝑇‘𝑀))    &   ⊢ 𝐽 = (𝑁 maAdju 𝑃)    &   ⊢ 𝑊 = (Base‘𝑌)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝑋 = (var1‘𝐴)    &   ⊢ ∗ = ( ·𝑠 ‘𝑄)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑄))    &   ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)    ⇒   ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑈 ∘ 𝐺) finSupp (0g‘𝐴))
Theorem	cpmadumatpoly 23202*	The product of the characteristic matrix of a given matrix and its adjunct represented as a polynomial over matrices. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 7-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ × = (.r‘𝑌)    &   ⊢ − = (-g‘𝑌)    &   ⊢ 0 = (0g‘𝑌)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))    &   ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑌)    &   ⊢ 1 = (1r‘𝑌)    &   ⊢ 𝑍 = (var1‘𝑅)    &   ⊢ 𝐷 = ((𝑍 · 1 ) − (𝑇‘𝑀))    &   ⊢ 𝐽 = (𝑁 maAdju 𝑃)    &   ⊢ 𝑊 = (Base‘𝑌)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝑋 = (var1‘𝐴)    &   ⊢ ∗ = ( ·𝑠 ‘𝑄)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑄))    &   ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)    &   ⊢ 𝐼 = (𝑁 pMatToMatPoly 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))(𝐼‘(𝐷 × (𝐽‘𝐷))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛)) ∗ (𝑛 ↑ 𝑋)))))
Theorem	cayhamlem2 23203	Lemma for cayhamlem3 23206. (Contributed by AV, 24-Nov-2019.)
⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 1 = (1r‘𝐴)    &   ⊢ ∗ = ( ·𝑠 ‘𝐴)    &   ⊢ ↑ = (.g‘(mulGrp‘𝐴))    &   ⊢ · = (.r‘𝐴)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑m ℕ0) ∧ 𝐿 ∈ ℕ0)) → ((𝐻‘𝐿) ∗ (𝐿 ↑ 𝑀)) = ((𝐿 ↑ 𝑀) · ((𝐻‘𝐿) ∗ 1 )))
Theorem	chcoeffeqlem 23204*	Lemma for chcoeffeq 23205. (Contributed by AV, 21-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.) (Revised by AV, 15-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ × = (.r‘𝑌)    &   ⊢ − = (-g‘𝑌)    &   ⊢ 0 = (0g‘𝑌)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝐾 = (𝐶‘𝑀)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))    &   ⊢ 𝑊 = (Base‘𝑌)    &   ⊢ 1 = (1r‘𝐴)    &   ⊢ ∗ = ( ·𝑠 ‘𝐴)    &   ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺‘𝑛)) = (((coe1‘𝐾)‘𝑛) ∗ 1 )))
Theorem	chcoeffeq 23205*	The coefficients of the characteristic polynomial multiplied with the identity matrix represented by (transformed) ring elements obtained from the adjunct of the characteristic matrix. (Contributed by AV, 21-Nov-2019.) (Proof shortened by AV, 8-Dec-2019.) (Revised by AV, 15-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ × = (.r‘𝑌)    &   ⊢ − = (-g‘𝑌)    &   ⊢ 0 = (0g‘𝑌)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝐾 = (𝐶‘𝑀)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))    &   ⊢ 𝑊 = (Base‘𝑌)    &   ⊢ 1 = (1r‘𝐴)    &   ⊢ ∗ = ( ·𝑠 ‘𝐴)    &   ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))∀𝑛 ∈ ℕ0 (𝑈‘(𝐺‘𝑛)) = (((coe1‘𝐾)‘𝑛) ∗ 1 ))
Theorem	cayhamlem3 23206*	Lemma for cayhamlem4 23207. (Contributed by AV, 24-Nov-2019.) (Revised by AV, 15-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ × = (.r‘𝑌)    &   ⊢ − = (-g‘𝑌)    &   ⊢ 0 = (0g‘𝑌)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝐾 = (𝐶‘𝑀)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))    &   ⊢ 𝑊 = (Base‘𝑌)    &   ⊢ 1 = (1r‘𝐴)    &   ⊢ ∗ = ( ·𝑠 ‘𝐴)    &   ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)    &   ⊢ ↑ = (.g‘(mulGrp‘𝐴))    &   ⊢ · = (.r‘𝐴)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))(𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑀) · (𝑈‘(𝐺‘𝑛))))))
Theorem	cayhamlem4 23207*	Lemma for cayleyhamilton 23209. (Contributed by AV, 25-Nov-2019.) (Revised by AV, 15-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ × = (.r‘𝑌)    &   ⊢ − = (-g‘𝑌)    &   ⊢ 0 = (0g‘𝑌)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝐾 = (𝐶‘𝑀)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))    &   ⊢ 𝑊 = (Base‘𝑌)    &   ⊢ 1 = (1r‘𝐴)    &   ⊢ ∗ = ( ·𝑠 ‘𝐴)    &   ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)    &   ⊢ ↑ = (.g‘(mulGrp‘𝐴))    &   ⊢ 𝐸 = (.g‘(mulGrp‘𝑌))    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))(𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))))
Theorem	cayleyhamilton0 23208*	The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation". This version of cayleyhamilton 23209 provides definitions not used in the theorem itself, but in its proof to make it clearer, more readable and shorter compared with a proof without them (see cayleyhamiltonALT 23210)! (Contributed by AV, 25-Nov-2019.) (Revised by AV, 15-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 0 = (0g‘𝐴)    &   ⊢ 1 = (1r‘𝐴)    &   ⊢ ∗ = ( ·𝑠 ‘𝐴)    &   ⊢ ↑ = (.g‘(mulGrp‘𝐴))    &   ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝐾 = (coe1‘(𝐶‘𝑀))    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ × = (.r‘𝑌)    &   ⊢ − = (-g‘𝑌)    &   ⊢ 𝑍 = (0g‘𝑌)    &   ⊢ 𝑊 = (Base‘𝑌)    &   ⊢ 𝐸 = (.g‘(mulGrp‘𝑌))    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, (𝑍 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 𝑍, ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))    &   ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 )
Theorem	cayleyhamilton 23209*	The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation", see theorem 7.8 in [Roman] p. 170 (without proof!), or theorem 3.1 in [Lang] p. 561. In other words, a matrix over a commutative ring "inserted" into its characteristic polynomial results in zero. This is Metamath 100 proof #49. (Contributed by Alexander van der Vekens, 25-Nov-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 0 = (0g‘𝐴)    &   ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝐾 = (coe1‘(𝐶‘𝑀))    &   ⊢ ∗ = ( ·𝑠 ‘𝐴)    &   ⊢ ↑ = (.g‘(mulGrp‘𝐴))    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 )
Theorem	cayleyhamiltonALT 23210*	Alternate proof of cayleyhamilton 23209, the Cayley-Hamilton theorem. This proof does not use cayleyhamilton0 23208 directly, but has the same structure as the proof of cayleyhamilton0 23208. In contrast to the proof of cayleyhamilton0 23208, only the definitions required to formulate the theorem itself are used, causing the definitions used in the lemmas being expanded, which makes the proof longer and more difficult to read. (Contributed by AV, 25-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 0 = (0g‘𝐴)    &   ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝐾 = (coe1‘(𝐶‘𝑀))    &   ⊢ ∗ = ( ·𝑠 ‘𝐴)    &   ⊢ ↑ = (.g‘(mulGrp‘𝐴))    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 )
Theorem	cayleyhamilton1 23211*	The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation", or, in other words, a matrix over a commutative ring "inserted" into its characteristic polynomial results in zero. In this variant of cayleyhamilton 23209, the meaning of "inserted" is made more transparent: If the characteristic polynomial is a polynomial with coefficients (𝐹‘𝑛), then a matrix over a commutative ring "inserted" into its characteristic polynomial is the sum of these coefficients multiplied with the corresponding power of the matrix. (Contributed by AV, 25-Nov-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 0 = (0g‘𝐴)    &   ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝐾 = (coe1‘(𝐶‘𝑀))    &   ⊢ ∗ = ( ·𝑠 ‘𝐴)    &   ⊢ ↑ = (.g‘(mulGrp‘𝐴))    &   ⊢ 𝐿 = (Base‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝐸 = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑍 = (0g‘𝑅)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → ((𝐶‘𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 ))
PART 12  BASIC TOPOLOGY
12.1  Topology
12.1.1  Topological spaces A topology on a set is a set of subsets of that set, called open sets, which satisfy certain conditions. One condition is that the whole set be an open set. Therefore, a set is recoverable from a topology on it (as its union, see toponuni 23233), and it may sometimes be more convenient to consider topologies without reference to the underlying set. This is why we define successively the class of topologies (df-top 23213), then the function which associates with a set the set of topologies on it (df-topon 23230), and finally the class of topological spaces, as extensible structures having an underlying set and a topology on it (df-topsp 23252). Of course, a topology is the same thing as a topology on a set (see toprntopon 23244).
12.1.1.1  Topologies
Syntax	ctop 23212	Syntax for the class of topologies.
class Top
Definition	df-top 23213*	Define the class of topologies. It is a proper class (see topnex 23315). See istopg 23214 and istop2g 23215 for the corresponding characterizations, using respectively binary intersections like in this definition and nonempty finite intersections. The final form of the definition is due to Bourbaki (Def. 1 of [BourbakiTop1] p. I.1), while the idea of defining a topology in terms of its open sets is due to Aleksandrov. For the convoluted history of the definitions of these notions, see Gregory H. Moore, The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica 35 (2008) 220--241. (Contributed by NM, 3-Mar-2006.) (Revised by BJ, 20-Oct-2018.)
⊢ Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)}
Theorem	istopg 23214*	Express the predicate "𝐽 is a topology". See istop2g 23215 for another characterization using nonempty finite intersections instead of binary intersections. Note: In the literature, a topology is often represented by a calligraphic letter T, which resembles the letter J. This confusion may have led to J being used by some authors (e.g., K. D. Joshi, Introduction to General Topology (1983), p. 114) and it is convenient for us since we later use 𝑇 to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ (𝐽 ∈ 𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽)))
Theorem	istop2g 23215*	Express the predicate "𝐽 is a topology" using nonempty finite intersections instead of binary intersections as in istopg 23214. (Contributed by NM, 19-Jul-2006.)
⊢ (𝐽 ∈ 𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥((𝑥 ⊆ 𝐽 ∧ 𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → ∩ 𝑥 ∈ 𝐽))))
Theorem	uniopn 23216	The union of a subset of a topology (that is, the union of any family of open sets of a topology) is an open set. (Contributed by Stefan Allan, 27-Feb-2006.)
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → ∪ 𝐴 ∈ 𝐽)
Theorem	iunopn 23217*	The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)
⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)
Theorem	inopn 23218	The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∩ 𝐵) ∈ 𝐽)
Theorem	fitop 23219	A topology is closed under finite intersections. (Contributed by Jeff Hankins, 7-Oct-2009.)
⊢ (𝐽 ∈ Top → (fi‘𝐽) = 𝐽)
Theorem	fiinopn 23220	The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)
⊢ (𝐽 ∈ Top → ((𝐴 ⊆ 𝐽 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → ∩ 𝐴 ∈ 𝐽))
Theorem	iinopn 23221*	The intersection of a nonempty finite family of open sets is open. (Contributed by Mario Carneiro, 14-Sep-2014.)
⊢ ((𝐽 ∈ Top ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)
Theorem	unopn 23222	The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∪ 𝐵) ∈ 𝐽)
Theorem	0opn 23223	The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.)
⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
Theorem	0ntop 23224	The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)
⊢ ¬ ∅ ∈ Top
Theorem	topopn 23225	The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
Theorem	eltopss 23226	A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋)
Theorem	riinopn 23227*	A finite indexed relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → (𝑋 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) ∈ 𝐽)
Theorem	rintopn 23228	A finite relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽 ∧ 𝐴 ∈ Fin) → (𝑋 ∩ ∩ 𝐴) ∈ 𝐽)
12.1.1.2  Topologies on sets
Syntax	ctopon 23229	Syntax for the function of topologies on sets.
class TopOn
Definition	df-topon 23230*	Define the function that associates with a set the set of topologies on it. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗})
Theorem	istopon 23231	Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.)
⊢ (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽))
Theorem	topontop 23232	A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
Theorem	toponuni 23233	The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽)
Theorem	topontopi 23234	A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ 𝐽 ∈ (TopOn‘𝐵)    ⇒   ⊢ 𝐽 ∈ Top
Theorem	toponunii 23235	The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ 𝐽 ∈ (TopOn‘𝐵)    ⇒   ⊢ 𝐵 = ∪ 𝐽
Theorem	toptopon 23236	Alternative definition of Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
Theorem	toptopon2 23237	A topology is the same thing as a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)
⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
Theorem	topontopon 23238	A topology on a set is a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ (TopOn‘∪ 𝐽))
Theorem	funtopon 23239	The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.)
⊢ Fun TopOn
Theorem	toponrestid 23240	Given a topology on a set, restricting it to that same set has no effect. (Contributed by Jim Kingdon, 6-Jul-2022.)
⊢ 𝐴 ∈ (TopOn‘𝐵)    ⇒   ⊢ 𝐴 = (𝐴 ↾t 𝐵)
Theorem	toponsspwpw 23241	The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.)
⊢ (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴
Theorem	dmtopon 23242	The domain of TopOn is the universal class V. (Contributed by BJ, 29-Apr-2021.)
⊢ dom TopOn = V
Theorem	fntopon 23243	The class TopOn is a function with domain the universal class V. Analogue for topologies of fnmre 17762 for Moore collections. (Contributed by BJ, 29-Apr-2021.)
⊢ TopOn Fn V
Theorem	toprntopon 23244	A topology is the same thing as a topology on a set (variable-free version). (Contributed by BJ, 27-Apr-2021.)
⊢ Top = ∪ ran TopOn
Theorem	toponmax 23245	The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ 𝐽)
Theorem	toponss 23246	A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋)
Theorem	toponcom 23247	If 𝐾 is a topology on the base set of topology 𝐽, then 𝐽 is a topology on the base of 𝐾. (Contributed by Mario Carneiro, 22-Aug-2015.)
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘∪ 𝐽)) → 𝐽 ∈ (TopOn‘∪ 𝐾))
Theorem	toponcomb 23248	Biconditional form of toponcom 23247. (Contributed by BJ, 5-Dec-2021.)
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘∪ 𝐾) ↔ 𝐾 ∈ (TopOn‘∪ 𝐽)))
Theorem	topgele 23249	The topologies over the same set have the greatest element (the discrete topology) and the least element (the indiscrete topology). (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 16-Sep-2015.)
⊢ (𝐽 ∈ (TopOn‘𝑋) → ({∅, 𝑋} ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 𝑋))
Theorem	topsn 23250	The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 4859). (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
⊢ (𝐽 ∈ (TopOn‘{𝐴}) → 𝐽 = 𝒫 {𝐴})
12.1.1.3  Topological spaces
Syntax	ctps 23251	Syntax for the class of topological spaces.
class TopSp
Definition	df-topsp 23252	Define the class of topological spaces (as extensible structures). (Contributed by Stefan O'Rear, 13-Aug-2015.)
⊢ TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}
Theorem	istps 23253	Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ 𝐴 = (Base‘𝐾)    &   ⊢ 𝐽 = (TopOpen‘𝐾)    ⇒   ⊢ (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐴))
Theorem	istps2 23254	Express the predicate "is a topological space." (Contributed by NM, 20-Oct-2012.)
⊢ 𝐴 = (Base‘𝐾)    &   ⊢ 𝐽 = (TopOpen‘𝐾)    ⇒   ⊢ (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = ∪ 𝐽))
Theorem	tpsuni 23255	The base set of a topological space. (Contributed by FL, 27-Jun-2014.)
⊢ 𝐴 = (Base‘𝐾)    &   ⊢ 𝐽 = (TopOpen‘𝐾)    ⇒   ⊢ (𝐾 ∈ TopSp → 𝐴 = ∪ 𝐽)
Theorem	tpstop 23256	The topology extractor on a topological space is a topology. (Contributed by FL, 27-Jun-2014.)
⊢ 𝐽 = (TopOpen‘𝐾)    ⇒   ⊢ (𝐾 ∈ TopSp → 𝐽 ∈ Top)
Theorem	tpspropd 23257	A topological space depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)
⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))    ⇒   ⊢ (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp))
Theorem	tpsprop2d 23258	A topological space depends only on the base and topology components. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   ⊢ (𝜑 → (TopSet‘𝐾) = (TopSet‘𝐿))    ⇒   ⊢ (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp))
Theorem	topontopn 23259	Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ 𝐴 = (Base‘𝐾)    &   ⊢ 𝐽 = (TopSet‘𝐾)    ⇒   ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopOpen‘𝐾))
Theorem	tsettps 23260	If the topology component is already correctly truncated, then it forms a topological space (with the topology extractor function coming out the same as the component). (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ 𝐴 = (Base‘𝐾)    &   ⊢ 𝐽 = (TopSet‘𝐾)    ⇒   ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp)
Theorem	istpsi 23261	Properties that determine a topological space. (Contributed by NM, 20-Oct-2012.)
⊢ (Base‘𝐾) = 𝐴    &   ⊢ (TopOpen‘𝐾) = 𝐽    &   ⊢ 𝐴 = ∪ 𝐽    &   ⊢ 𝐽 ∈ Top    ⇒   ⊢ 𝐾 ∈ TopSp
Theorem	eltpsg 23262	Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by AV, 31-Oct-2024.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), 𝐽⟩}    ⇒   ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp)
Theorem	eltpsi 23263	Properties that determine a topological space from a construction (using no explicit indices). (Contributed by NM, 20-Oct-2012.) (Revised by Mario Carneiro, 13-Aug-2015.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), 𝐽⟩}    &   ⊢ 𝐴 = ∪ 𝐽    &   ⊢ 𝐽 ∈ Top    ⇒   ⊢ 𝐾 ∈ TopSp
12.1.2  Topological bases
Syntax	ctb 23264	Syntax for the class of topological bases.
class TopBases
Definition	df-bases 23265*	Define the class of topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 23267). Note that "bases" is the plural of "basis". (Contributed by NM, 17-Jul-2006.)
⊢ TopBases = {𝑥 ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ⊆ ∪ (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧))}
Theorem	isbasisg 23266*	Express the predicate "the set 𝐵 is a basis for a topology". (Contributed by NM, 17-Jul-2006.)
⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
Theorem	isbasis2g 23267*	Express the predicate "the set 𝐵 is a basis for a topology". (Contributed by NM, 17-Jul-2006.)
⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
Theorem	isbasis3g 23268*	Express the predicate "the set 𝐵 is a basis for a topology". Definition of basis in [Munkres] p. 78. (Contributed by NM, 17-Jul-2006.)
⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ (∀𝑥 ∈ 𝐵 𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ ∪ 𝐵∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))))
Theorem	basis1 23269	Property of a basis. (Contributed by NM, 16-Jul-2006.)
⊢ ((𝐵 ∈ TopBases ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐶 ∩ 𝐷) ⊆ ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝐷)))
Theorem	basis2 23270*	Property of a basis. (Contributed by NM, 17-Jul-2006.)
⊢ (((𝐵 ∈ TopBases ∧ 𝐶 ∈ 𝐵) ∧ (𝐷 ∈ 𝐵 ∧ 𝐴 ∈ (𝐶 ∩ 𝐷))) → ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷)))
Theorem	fiinbas 23271*	If a set is closed under finite intersection, then it is a basis for a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝐵 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases)
Theorem	basdif0 23272	A basis is not affected by the addition or removal of the empty set. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ ((𝐵 ∖ {∅}) ∈ TopBases ↔ 𝐵 ∈ TopBases)
Theorem	baspartn 23273*	A disjoint system of sets is a basis for a topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ ((𝑃 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) → 𝑃 ∈ TopBases)
Theorem	tgval 23274*	The topology generated by a basis. See also tgval2 23275 and tgval3 23282. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})
Theorem	tgval2 23275*	Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 23288) that (topGen‘𝐵) is indeed a topology (on ∪ 𝐵, see unitg 23286). See also tgval 23274 and tgval3 23282. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))})
Theorem	eltg 23276	Membership in a topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴)))
Theorem	eltg2 23277*	Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ (𝐴 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))))
Theorem	eltg2b 23278*	Membership in a topology generated by a basis. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 10-Jan-2015.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))
Theorem	eltg4i 23279	An open set in a topology generated by a basis is the union of all basic open sets contained in it. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐴 = ∪ (𝐵 ∩ 𝒫 𝐴))
Theorem	eltg3i 23280	The union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 30-Aug-2015.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → ∪ 𝐴 ∈ (topGen‘𝐵))
Theorem	eltg3 23281*	Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Mario Carneiro, 30-Aug-2015.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 = ∪ 𝑥)))
Theorem	tgval3 23282*	Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80. See also tgval 23274 and tgval2 23275. (Contributed by NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.)
⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)})
Theorem	tg1 23283	Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐴 ⊆ ∪ 𝐵)
Theorem	tg2 23284*	Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
⊢ ((𝐴 ∈ (topGen‘𝐵) ∧ 𝐶 ∈ 𝐴) → ∃𝑥 ∈ 𝐵 (𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴))
Theorem	bastg 23285	A member of a basis is a subset of the topology it generates. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
⊢ (𝐵 ∈ 𝑉 → 𝐵 ⊆ (topGen‘𝐵))
Theorem	unitg 23286	The topology generated by a basis 𝐵 is a topology on ∪ 𝐵. Importantly, this theorem means that we don't have to specify separately the base set for the topological space generated by a basis. In other words, any member of the class TopBases completely specifies the basis it corresponds to. (Contributed by NM, 16-Jul-2006.) (Proof shortened by OpenAI, 30-Mar-2020.)
⊢ (𝐵 ∈ 𝑉 → ∪ (topGen‘𝐵) = ∪ 𝐵)
Theorem	tgss 23287	Subset relation for generated topologies. (Contributed by NM, 7-May-2007.)
⊢ ((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶))
Theorem	tgcl 23288	Show that a basis generates a topology. Remark in [Munkres] p. 79. (Contributed by NM, 17-Jul-2006.)
⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top)
Theorem	tgclb 23289	The property tgcl 23288 can be reversed: if the topology generated by 𝐵 is actually a topology, then 𝐵 must be a topological basis. This yields an alternative definition of TopBases. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ (𝐵 ∈ TopBases ↔ (topGen‘𝐵) ∈ Top)
Theorem	tgtopon 23290	A basis generates a topology on ∪ 𝐵. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ (TopOn‘∪ 𝐵))
Theorem	topbas 23291	A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
⊢ (𝐽 ∈ Top → 𝐽 ∈ TopBases)
Theorem	tgtop 23292	A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
Theorem	eltop 23293	Membership in a topology, expressed without quantifiers. (Contributed by NM, 19-Jul-2006.)
⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ 𝐴 ⊆ ∪ (𝐽 ∩ 𝒫 𝐴)))
Theorem	eltop2 23294*	Membership in a topology. (Contributed by NM, 19-Jul-2006.)
⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))
Theorem	eltop3 23295*	Membership in a topology. (Contributed by NM, 19-Jul-2006.)
⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∃𝑥(𝑥 ⊆ 𝐽 ∧ 𝐴 = ∪ 𝑥)))
Theorem	fibas 23296	A collection of finite intersections is a basis. The initial set is a subbasis for the topology. (Contributed by Jeff Hankins, 25-Aug-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
⊢ (fi‘𝐴) ∈ TopBases
Theorem	tgdom 23297	A space has no more open sets than subsets of a basis. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 9-Apr-2015.)
⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) ≼ 𝒫 𝐵)
Theorem	tgiun 23298*	The indexed union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐶 ∈ (topGen‘𝐵))
Theorem	tgidm 23299	The topology generator function is idempotent. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.)
⊢ (𝐵 ∈ 𝑉 → (topGen‘(topGen‘𝐵)) = (topGen‘𝐵))
Theorem	bastop 23300	Two ways to express that a basis is a topology. (Contributed by NM, 18-Jul-2006.)
⊢ (𝐵 ∈ TopBases → (𝐵 ∈ Top ↔ (topGen‘𝐵) = 𝐵))