P. 129 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12801-12900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eltopss 12801	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)
8.1.1.2  Topologies on sets
Syntax	ctopon 12802	Syntax for the function of topologies on sets.
class TopOn
Definition	df-topon 12803*	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	funtopon 12804	The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.)
⊢ Fun TopOn
Theorem	istopon 12805	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 12806	A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
Theorem	toponuni 12807	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 12808	A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ 𝐽 ∈ (TopOn‘𝐵)    ⇒   ⊢ 𝐽 ∈ Top
Theorem	toponunii 12809	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 12810	Alternative definition of Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
Theorem	toptopon2 12811	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 12812	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	toponrestid 12813	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	toponsspwpwg 12814	The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.) (Revised by Jim Kingdon, 16-Jan-2023.)
⊢ (𝐴 ∈ 𝑉 → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)
Theorem	dmtopon 12815	The domain of TopOn is V. (Contributed by BJ, 29-Apr-2021.)
⊢ dom TopOn = V
Theorem	fntopon 12816	The class TopOn is a function with domain V. (Contributed by BJ, 29-Apr-2021.)
⊢ TopOn Fn V
Theorem	toponmax 12817	The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ 𝐽)
Theorem	toponss 12818	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 12819	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 12820	Biconditional form of toponcom 12819. (Contributed by BJ, 5-Dec-2021.)
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘∪ 𝐾) ↔ 𝐾 ∈ (TopOn‘∪ 𝐽)))
Theorem	topgele 12821	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))
8.1.1.3  Topological spaces
Syntax	ctps 12822	Syntax for the class of topological spaces.
class TopSp
Definition	df-topsp 12823	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 12824	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 12825	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 12826	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 12827	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 12828	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	topontopn 12829	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 12830	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 12831	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 12832	Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), 𝐽⟩}    ⇒   ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp)
Theorem	eltpsi 12833	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
8.1.2  Topological bases
Syntax	ctb 12834	Syntax for the class of topological bases.
class TopBases
Definition	df-bases 12835*	Define the class of topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 12837). 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 12836*	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 12837*	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 12838*	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 12839	Property of a basis. (Contributed by NM, 16-Jul-2006.)
⊢ ((𝐵 ∈ TopBases ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐶 ∩ 𝐷) ⊆ ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝐷)))
Theorem	basis2 12840*	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 12841*	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	baspartn 12842*	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 12843*	The topology generated by a basis. See also tgval2 12845 and tgval3 12852. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})
Theorem	tgvalex 12844	The topology generated by a basis is a set. (Contributed by Jim Kingdon, 4-Mar-2023.)
⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) ∈ V)
Theorem	tgval2 12845*	Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 12858) that (topGen‘𝐵) is indeed a topology (on ∪ 𝐵, see unitg 12856). See also tgval 12843 and tgval3 12852. (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 12846	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 12847*	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 12848*	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 12849	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 12850	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 12851*	Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Jim Kingdon, 4-Mar-2023.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 = ∪ 𝑥)))
Theorem	tgval3 12852*	Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80. See also tgval 12843 and tgval2 12845. (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 12853	Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐴 ⊆ ∪ 𝐵)
Theorem	tg2 12854*	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 12855	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 12856	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 12857	Subset relation for generated topologies. (Contributed by NM, 7-May-2007.)
⊢ ((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶))
Theorem	tgcl 12858	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 12859	The property tgcl 12858 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 12860	A basis generates a topology on ∪ 𝐵. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ (TopOn‘∪ 𝐵))
Theorem	topbas 12861	A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
⊢ (𝐽 ∈ Top → 𝐽 ∈ TopBases)
Theorem	tgtop 12862	A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
Theorem	eltop 12863	Membership in a topology, expressed without quantifiers. (Contributed by NM, 19-Jul-2006.)
⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ 𝐴 ⊆ ∪ (𝐽 ∩ 𝒫 𝐴)))
Theorem	eltop2 12864*	Membership in a topology. (Contributed by NM, 19-Jul-2006.)
⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))
Theorem	eltop3 12865*	Membership in a topology. (Contributed by NM, 19-Jul-2006.)
⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∃𝑥(𝑥 ⊆ 𝐽 ∧ 𝐴 = ∪ 𝑥)))
Theorem	tgdom 12866	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 12867*	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 12868	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 12869	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 12870	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	en1top 12871	{∅} is the only topology with one element. (Contributed by FL, 18-Aug-2008.)
⊢ (𝐽 ∈ Top → (𝐽 ≈ 1o ↔ 𝐽 = {∅}))
Theorem	tgss3 12872	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 12873*	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 12874	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 12875*	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	2basgeng 12876	Conditions that determine the equality of two generated topologies. (Contributed by NM, 8-May-2007.) (Revised by Jim Kingdon, 5-Mar-2023.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = (topGen‘𝐶))
Theorem	bastop1 12877*	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 12878*	A version of bastop1 12877 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))))
8.1.3  Examples of topologies
Theorem	distop 12879	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 12880	The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 4434. (Contributed by BJ, 2-May-2021.)
⊢ Top ∉ V
Theorem	distopon 12881	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 12882	The singleton of the empty set is a topology on the empty set. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ {∅} ∈ (TopOn‘∅)
Theorem	sn0top 12883	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	epttop 12884*	The excluded point topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ∈ (TopOn‘𝐴))
Theorem	distps 12885	The discrete topology on a set 𝐴 expressed as a topological space. (Contributed by FL, 20-Aug-2006.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), 𝒫 𝐴⟩}    ⇒   ⊢ 𝐾 ∈ TopSp
8.1.4  Closure and interior
Syntax	ccld 12886	Extend class notation with the set of closed sets of a topology.
class Clsd
Syntax	cnt 12887	Extend class notation with interior of a subset of a topology base set.
class int
Syntax	ccl 12888	Extend class notation with closure of a subset of a topology base set.
class cls
Definition	df-cld 12889*	Define a function on topologies whose value is the set of closed sets of the topology. (Contributed by NM, 2-Oct-2006.)
⊢ Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗})
Definition	df-ntr 12890*	Define a function on topologies whose value is the interior function on the subsets of the base set. See ntrval 12904. (Contributed by NM, 10-Sep-2006.)
⊢ int = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∪ (𝑗 ∩ 𝒫 𝑥)))
Definition	df-cls 12891*	Define a function on topologies whose value is the closure function on the subsets of the base set. See clsval 12905. (Contributed by NM, 3-Oct-2006.)
⊢ cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∩ {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥 ⊆ 𝑦}))
Theorem	fncld 12892	The closed-set generator is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ Clsd Fn Top
Theorem	cldval 12893*	The set of closed sets of a topology. (Note that the set of open sets is just the topology itself, so we don't have a separate definition.) (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽})
Theorem	ntrfval 12894*	The interior function on the subsets of a topology's base set. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥)))
Theorem	clsfval 12895*	The closure function on the subsets of a topology's base set. (Contributed by NM, 3-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top → (cls‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦}))
Theorem	cldrcl 12896	Reverse closure of the closed set operation. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
Theorem	iscld 12897	The predicate "the class 𝑆 is a closed set". (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽)))
Theorem	iscld2 12898	A subset of the underlying set of a topology is closed iff its complement is open. (Contributed by NM, 4-Oct-2006.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ 𝑆) ∈ 𝐽))
Theorem	cldss 12899	A closed set is a subset of the underlying set of a topology. (Contributed by NM, 5-Oct-2006.) (Revised by Stefan O'Rear, 22-Feb-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ 𝑋)
Theorem	cldss2 12900	The set of closed sets is contained in the powerset of the base. (Contributed by Mario Carneiro, 6-Jan-2014.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋