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	eltopss 22801	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 22802*	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 22803	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 22804	Syntax for the function of topologies on sets.
class TopOn
Definition	df-topon 22805*	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 22806	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 22807	A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
Theorem	toponuni 22808	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 22809	A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ 𝐽 ∈ (TopOn‘𝐵)    ⇒   ⊢ 𝐽 ∈ Top
Theorem	toponunii 22810	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 22811	Alternative definition of Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
Theorem	toptopon2 22812	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 22813	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 22814	The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.)
⊢ Fun TopOn
Theorem	toponrestid 22815	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 22816	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 22817	The domain of TopOn is the universal class V. (Contributed by BJ, 29-Apr-2021.)
⊢ dom TopOn = V
Theorem	fntopon 22818	The class TopOn is a function with domain the universal class V. Analogue for topologies of fnmre 17559 for Moore collections. (Contributed by BJ, 29-Apr-2021.)
⊢ TopOn Fn V
Theorem	toprntopon 22819	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 22820	The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ 𝐽)
Theorem	toponss 22821	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 22822	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 22823	Biconditional form of toponcom 22822. (Contributed by BJ, 5-Dec-2021.)
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘∪ 𝐾) ↔ 𝐾 ∈ (TopOn‘∪ 𝐽)))
Theorem	topgele 22824	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 22825	The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 4867). (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 22826	Syntax for the class of topological spaces.
class TopSp
Definition	df-topsp 22827	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 22828	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 22829	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 22830	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 22831	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 22832	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 22833	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 22834	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 22835	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 22836	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 22837	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 22838	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 22839	Syntax for the class of topological bases.
class TopBases
Definition	df-bases 22840*	Define the class of topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 22842). 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 22841*	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 22842*	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 22843*	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 22844	Property of a basis. (Contributed by NM, 16-Jul-2006.)
⊢ ((𝐵 ∈ TopBases ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐶 ∩ 𝐷) ⊆ ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝐷)))
Theorem	basis2 22845*	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 22846*	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 22847	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 22848*	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 22849*	The topology generated by a basis. See also tgval2 22850 and tgval3 22857. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})
Theorem	tgval2 22850*	Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 22863) that (topGen‘𝐵) is indeed a topology (on ∪ 𝐵, see unitg 22861). See also tgval 22849 and tgval3 22857. (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 22851	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 22852*	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 22853*	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 22854	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 22855	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 22856*	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 22857*	Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80. See also tgval 22849 and tgval2 22850. (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 22858	Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐴 ⊆ ∪ 𝐵)
Theorem	tg2 22859*	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 22860	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 22861	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 22862	Subset relation for generated topologies. (Contributed by NM, 7-May-2007.)
⊢ ((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶))
Theorem	tgcl 22863	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 22864	The property tgcl 22863 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 22865	A basis generates a topology on ∪ 𝐵. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ (TopOn‘∪ 𝐵))
Theorem	topbas 22866	A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
⊢ (𝐽 ∈ Top → 𝐽 ∈ TopBases)
Theorem	tgtop 22867	A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
Theorem	eltop 22868	Membership in a topology, expressed without quantifiers. (Contributed by NM, 19-Jul-2006.)
⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ 𝐴 ⊆ ∪ (𝐽 ∩ 𝒫 𝐴)))
Theorem	eltop2 22869*	Membership in a topology. (Contributed by NM, 19-Jul-2006.)
⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))
Theorem	eltop3 22870*	Membership in a topology. (Contributed by NM, 19-Jul-2006.)
⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∃𝑥(𝑥 ⊆ 𝐽 ∧ 𝐴 = ∪ 𝑥)))
Theorem	fibas 22871	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 22872	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 22873*	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 22874	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 22875	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 22876	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 22877	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 22878	{∅} is the only topology with one element. (Contributed by FL, 18-Aug-2008.)
⊢ (𝐽 ∈ Top → (𝐽 ≈ 1o ↔ 𝐽 = {∅}))
Theorem	en2top 22879	If a topology has two elements, it is the indiscrete topology. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ≈ 2o ↔ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)))
Theorem	tgss3 22880	A criterion for determining whether one topology is finer than another. Lemma 2.2 of [Munkres] p. 80 using abbreviations. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶)))
Theorem	tgss2 22881*	A criterion for determining whether one topology is finer than another, based on a comparison of their bases. Lemma 2.2 of [Munkres] p. 80. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ ∀𝑥 ∈ ∪ 𝐵∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
Theorem	basgen 22882	Given a topology 𝐽, show that a subset 𝐵 satisfying the third antecedent is a basis for it. Lemma 2.3 of [Munkres] p. 81 using abbreviations. (Contributed by NM, 22-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.)
⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽 ∧ 𝐽 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = 𝐽)
Theorem	basgen2 22883*	Given a topology 𝐽, show that a subset 𝐵 satisfying the third antecedent is a basis for it. Lemma 2.3 of [Munkres] p. 81. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽 ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)) → (topGen‘𝐵) = 𝐽)
Theorem	2basgen 22884	Conditions that determine the equality of two generated topologies. (Contributed by NM, 8-May-2007.) (Revised by Mario Carneiro, 2-Sep-2015.)
⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = (topGen‘𝐶))
Theorem	tgfiss 22885	If a subbase is included into a topology, so is the generated topology. (Contributed by FL, 20-Apr-2012.) (Proof shortened by Mario Carneiro, 10-Jan-2015.)
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → (topGen‘(fi‘𝐴)) ⊆ 𝐽)
Theorem	tgdif0 22886	A generated topology is not affected by the addition or removal of the empty set from the base. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵)
Theorem	bastop1 22887*	A subset of a topology is a basis for the topology iff every member of the topology is a union of members of the basis. We use the idiom "(topGen‘𝐵) = 𝐽 " to express "𝐵 is a basis for topology 𝐽 " since we do not have a separate notation for this. Definition 15.35 of [Schechter] p. 428. (Contributed by NM, 2-Feb-2008.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥 ∈ 𝐽 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
Theorem	bastop2 22888*	A version of bastop1 22887 that doesn't have 𝐵 ⊆ 𝐽 in the antecedent. (Contributed by NM, 3-Feb-2008.)
⊢ (𝐽 ∈ Top → ((topGen‘𝐵) = 𝐽 ↔ (𝐵 ⊆ 𝐽 ∧ ∀𝑥 ∈ 𝐽 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦))))
12.1.3  Examples of topologies
Theorem	distop 22889	The discrete topology on a set 𝐴. Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 17-Jul-2006.) (Revised by Mario Carneiro, 19-Mar-2015.)
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)
Theorem	topnex 22890	The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 7738; an alternate proof uses indiscrete topologies (see indistop 22896) and the analogue of pwnex 7738 with pairs {∅, 𝑥} instead of power sets 𝒫 𝑥 (that analogue is also a consequence of abnex 7736). (Contributed by BJ, 2-May-2021.)
⊢ Top ∉ V
Theorem	distopon 22891	The discrete topology on a set 𝐴, with base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
Theorem	sn0topon 22892	The singleton of the empty set is a topology on the empty set. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ {∅} ∈ (TopOn‘∅)
Theorem	sn0top 22893	The singleton of the empty set is a topology. (Contributed by Stefan Allan, 3-Mar-2006.) (Proof shortened by Mario Carneiro, 13-Aug-2015.)
⊢ {∅} ∈ Top
Theorem	indislem 22894	A lemma to eliminate some sethood hypotheses when dealing with the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴}
Theorem	indistopon 22895	The indiscrete topology on a set 𝐴. Part of Example 2 in [Munkres] p. 77. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ (𝐴 ∈ 𝑉 → {∅, 𝐴} ∈ (TopOn‘𝐴))
Theorem	indistop 22896	The indiscrete topology on a set 𝐴. Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 16-Jul-2006.) (Revised by Stefan Allan, 6-Nov-2008.) (Revised by Mario Carneiro, 13-Aug-2015.)
⊢ {∅, 𝐴} ∈ Top
Theorem	indisuni 22897	The base set of the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ ( I ‘𝐴) = ∪ {∅, 𝐴}
Theorem	fctop 22898*	The finite complement topology on a set 𝐴. Example 3 in [Munkres] p. 77. (Contributed by FL, 15-Aug-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ∈ Fin ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))
Theorem	fctop2 22899*	The finite complement topology on a set 𝐴. Example 3 in [Munkres] p. 77. (This version of fctop 22898 requires the Axiom of Infinity.) (Contributed by FL, 20-Aug-2006.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≺ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))
Theorem	cctop 22900*	The countable complement topology on a set 𝐴. Example 4 in [Munkres] p. 77. (Contributed by FL, 23-Aug-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))