P. 229 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 22801-22900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	chcoeffeq 22801*	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 22802*	Lemma for cayhamlem4 22803. (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 22803*	Lemma for cayleyhamilton 22805. (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 22804*	The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation". This version of cayleyhamilton 22805 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 22806)! (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 22805*	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 22806*	Alternate proof of cayleyhamilton 22805, the Cayley-Hamilton theorem. This proof does not use cayleyhamilton0 22804 directly, but has the same structure as the proof of cayleyhamilton0 22804. In contrast to the proof of cayleyhamilton0 22804, 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 22807*	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 22805, 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 22829), 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 22809), then the function which associates with a set the set of topologies on it (df-topon 22826), and finally the class of topological spaces, as extensible structures having an underlying set and a topology on it (df-topsp 22848). Of course, a topology is the same thing as a topology on a set (see toprntopon 22840).
12.1.1.1  Topologies
Syntax	ctop 22808	Syntax for the class of topologies.
class Top
Definition	df-top 22809*	Define the class of topologies. It is a proper class (see topnex 22912). See istopg 22810 and istop2g 22811 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 22810*	Express the predicate "𝐽 is a topology". See istop2g 22811 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 22811*	Express the predicate "𝐽 is a topology" using nonempty finite intersections instead of binary intersections as in istopg 22810. (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 22812	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 22813*	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 22814	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 22815	A topology is closed under finite intersections. (Contributed by Jeff Hankins, 7-Oct-2009.)
⊢ (𝐽 ∈ Top → (fi‘𝐽) = 𝐽)
Theorem	fiinopn 22816	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 22817*	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 22818	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 22819	The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.)
⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
Theorem	0ntop 22820	The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)
⊢ ¬ ∅ ∈ Top
Theorem	topopn 22821	The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
Theorem	eltopss 22822	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 22823*	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 22824	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 22825	Syntax for the function of topologies on sets.
class TopOn
Definition	df-topon 22826*	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 22827	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 22828	A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
Theorem	toponuni 22829	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 22830	A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ 𝐽 ∈ (TopOn‘𝐵)    ⇒   ⊢ 𝐽 ∈ Top
Theorem	toponunii 22831	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 22832	Alternative definition of Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
Theorem	toptopon2 22833	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 22834	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 22835	The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.)
⊢ Fun TopOn
Theorem	toponrestid 22836	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 22837	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 22838	The domain of TopOn is the universal class V. (Contributed by BJ, 29-Apr-2021.)
⊢ dom TopOn = V
Theorem	fntopon 22839	The class TopOn is a function with domain the universal class V. Analogue for topologies of fnmre 17565 for Moore collections. (Contributed by BJ, 29-Apr-2021.)
⊢ TopOn Fn V
Theorem	toprntopon 22840	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 22841	The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ 𝐽)
Theorem	toponss 22842	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 22843	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 22844	Biconditional form of toponcom 22843. (Contributed by BJ, 5-Dec-2021.)
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘∪ 𝐾) ↔ 𝐾 ∈ (TopOn‘∪ 𝐽)))
Theorem	topgele 22845	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 22846	The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 4897). (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 22847	Syntax for the class of topological spaces.
class TopSp
Definition	df-topsp 22848	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 22849	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 22850	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 22851	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 22852	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 22853	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 22854	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 22855	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 22856	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 22857	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 22858	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	eltpsgOLD 22859	Obsolete version of eltpsg 22858 as of 31-Oct-2024. Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), 𝐽⟩}    ⇒   ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp)
Theorem	eltpsi 22860	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 22861	Syntax for the class of topological bases.
class TopBases
Definition	df-bases 22862*	Define the class of topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 22864). 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 22863*	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 22864*	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 22865*	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 22866	Property of a basis. (Contributed by NM, 16-Jul-2006.)
⊢ ((𝐵 ∈ TopBases ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐶 ∩ 𝐷) ⊆ ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝐷)))
Theorem	basis2 22867*	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 22868*	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 22869	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 22870*	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 22871*	The topology generated by a basis. See also tgval2 22872 and tgval3 22879. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})
Theorem	tgval2 22872*	Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 22885) that (topGen‘𝐵) is indeed a topology (on ∪ 𝐵, see unitg 22883). See also tgval 22871 and tgval3 22879. (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 22873	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 22874*	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 22875*	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 22876	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 22877	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 22878*	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 22879*	Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80. See also tgval 22871 and tgval2 22872. (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 22880	Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐴 ⊆ ∪ 𝐵)
Theorem	tg2 22881*	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 22882	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 22883	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 22884	Subset relation for generated topologies. (Contributed by NM, 7-May-2007.)
⊢ ((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶))
Theorem	tgcl 22885	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 22886	The property tgcl 22885 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 22887	A basis generates a topology on ∪ 𝐵. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ (TopOn‘∪ 𝐵))
Theorem	topbas 22888	A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
⊢ (𝐽 ∈ Top → 𝐽 ∈ TopBases)
Theorem	tgtop 22889	A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
Theorem	eltop 22890	Membership in a topology, expressed without quantifiers. (Contributed by NM, 19-Jul-2006.)
⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ 𝐴 ⊆ ∪ (𝐽 ∩ 𝒫 𝐴)))
Theorem	eltop2 22891*	Membership in a topology. (Contributed by NM, 19-Jul-2006.)
⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))
Theorem	eltop3 22892*	Membership in a topology. (Contributed by NM, 19-Jul-2006.)
⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∃𝑥(𝑥 ⊆ 𝐽 ∧ 𝐴 = ∪ 𝑥)))
Theorem	fibas 22893	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 22894	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 22895*	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 22896	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 22897	Two ways to express that a basis is a topology. (Contributed by NM, 18-Jul-2006.)
⊢ (𝐵 ∈ TopBases → (𝐵 ∈ Top ↔ (topGen‘𝐵) = 𝐵))
Theorem	tgtop11 22898	The topology generation function is one-to-one when applied to completed topologies. (Contributed by NM, 18-Jul-2006.)
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ (topGen‘𝐽) = (topGen‘𝐾)) → 𝐽 = 𝐾)
Theorem	0top 22899	The singleton of the empty set is the only topology possible for an empty underlying set. (Contributed by NM, 9-Sep-2006.)
⊢ (𝐽 ∈ Top → (∪ 𝐽 = ∅ ↔ 𝐽 = {∅}))
Theorem	en1top 22900	{∅} is the only topology with one element. (Contributed by FL, 18-Aug-2008.)
⊢ (𝐽 ∈ Top → (𝐽 ≈ 1o ↔ 𝐽 = {∅}))