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Theorem List for Metamath Proof Explorer - 21501-21600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiunopn 21501* The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)
((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵𝐽) → 𝑥𝐴 𝐵𝐽)
 
Theoreminopn 21502 The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)
 
Theoremfitop 21503 A topology is closed under finite intersections. (Contributed by Jeff Hankins, 7-Oct-2009.)
(𝐽 ∈ Top → (fi‘𝐽) = 𝐽)
 
Theoremfiinopn 21504 The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)
(𝐽 ∈ Top → ((𝐴𝐽𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → 𝐴𝐽))
 
Theoremiinopn 21505* The intersection of a nonempty finite family of open sets is open. (Contributed by Mario Carneiro, 14-Sep-2014.)
((𝐽 ∈ Top ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐽)) → 𝑥𝐴 𝐵𝐽)
 
Theoremunopn 21506 The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)
 
Theorem0opn 21507 The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.)
(𝐽 ∈ Top → ∅ ∈ 𝐽)
 
Theorem0ntop 21508 The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)
¬ ∅ ∈ Top
 
Theoremtopopn 21509 The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
𝑋 = 𝐽       (𝐽 ∈ Top → 𝑋𝐽)
 
Theoremeltopss 21510 A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴𝑋)
 
Theoremriinopn 21511* A finite indexed relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵𝐽) → (𝑋 𝑥𝐴 𝐵) ∈ 𝐽)
 
Theoremrintopn 21512 A finite relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝐽𝐴 ∈ Fin) → (𝑋 𝐴) ∈ 𝐽)
 
12.1.1.2  Topologies on sets
 
Syntaxctopon 21513 Syntax for the function of topologies on sets.
class TopOn
 
Definitiondf-topon 21514* 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})
 
Theoremistopon 21515 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))
 
Theoremtopontop 21516 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
 
Theoremtoponuni 21517 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOn‘𝐵) → 𝐵 = 𝐽)
 
Theoremtopontopi 21518 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐽 ∈ (TopOn‘𝐵)       𝐽 ∈ Top
 
Theoremtoponunii 21519 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐽 ∈ (TopOn‘𝐵)       𝐵 = 𝐽
 
Theoremtoptopon 21520 Alternative definition of Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝑋 = 𝐽       (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
 
Theoremtoptopon2 21521 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))
 
Theoremtopontopon 21522 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))
 
Theoremfuntopon 21523 The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.)
Fun TopOn
 
Theoremtoponrestid 21524 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 = (𝐴t 𝐵)
 
Theoremtoponsspwpw 21525 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
 
Theoremdmtopon 21526 The domain of TopOn is the universal class V. (Contributed by BJ, 29-Apr-2021.)
dom TopOn = V
 
Theoremfntopon 21527 The class TopOn is a function with domain the universal class V. Analogue for topologies of fnmre 16860 for Moore collections. (Contributed by BJ, 29-Apr-2021.)
TopOn Fn V
 
Theoremtoprntopon 21528 A topology is the same thing as a topology on a set (variable-free version). (Contributed by BJ, 27-Apr-2021.)
Top = ran TopOn
 
Theoremtoponmax 21529 The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOn‘𝐵) → 𝐵𝐽)
 
Theoremtoponss 21530 A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝐴𝑋)
 
Theoremtoponcom 21531 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))
 
Theoremtoponcomb 21532 Biconditional form of toponcom 21531. (Contributed by BJ, 5-Dec-2021.)
((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘ 𝐾) ↔ 𝐾 ∈ (TopOn‘ 𝐽)))
 
Theoremtopgele 21533 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} ⊆ 𝐽𝐽 ⊆ 𝒫 𝑋))
 
Theoremtopsn 21534 The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 4817). (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
 
Syntaxctps 21535 Syntax for the class of topological spaces.
class TopSp
 
Definitiondf-topsp 21536 Define the class of topological spaces (as extensible structures). (Contributed by Stefan O'Rear, 13-Aug-2015.)
TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}
 
Theoremistps 21537 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))
 
Theoremistps2 21538 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))
 
Theoremtpsuni 21539 The base set of a topological space. (Contributed by FL, 27-Jun-2014.)
𝐴 = (Base‘𝐾)    &   𝐽 = (TopOpen‘𝐾)       (𝐾 ∈ TopSp → 𝐴 = 𝐽)
 
Theoremtpstop 21540 The topology extractor on a topological space is a topology. (Contributed by FL, 27-Jun-2014.)
𝐽 = (TopOpen‘𝐾)       (𝐾 ∈ TopSp → 𝐽 ∈ Top)
 
Theoremtpspropd 21541 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))
 
Theoremtpsprop2d 21542 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))
 
Theoremtopontopn 21543 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))
 
Theoremtsettps 21544 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)
 
Theoremistpsi 21545 Properties that determine a topological space. (Contributed by NM, 20-Oct-2012.)
(Base‘𝐾) = 𝐴    &   (TopOpen‘𝐾) = 𝐽    &   𝐴 = 𝐽    &   𝐽 ∈ Top       𝐾 ∈ TopSp
 
Theoremeltpsg 21546 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)
 
Theoremeltpsi 21547 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
 
Syntaxctb 21548 Syntax for the class of topological bases.
class TopBases
 
Definitiondf-bases 21549* Define the class of topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 21551). Note that "bases" is the plural of "basis". (Contributed by NM, 17-Jul-2006.)
TopBases = {𝑥 ∣ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ⊆ (𝑥 ∩ 𝒫 (𝑦𝑧))}
 
Theoremisbasisg 21550* Express the predicate "the set 𝐵 is a basis for a topology". (Contributed by NM, 17-Jul-2006.)
(𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
 
Theoremisbasis2g 21551* Express the predicate "the set 𝐵 is a basis for a topology". (Contributed by NM, 17-Jul-2006.)
(𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
 
Theoremisbasis3g 21552* 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.)
(𝐵𝐶 → (𝐵 ∈ TopBases ↔ (∀𝑥𝐵 𝑥 𝐵 ∧ ∀𝑥 𝐵𝑦𝐵 𝑥𝑦 ∧ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))))
 
Theorembasis1 21553 Property of a basis. (Contributed by NM, 16-Jul-2006.)
((𝐵 ∈ TopBases ∧ 𝐶𝐵𝐷𝐵) → (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷)))
 
Theorembasis2 21554* Property of a basis. (Contributed by NM, 17-Jul-2006.)
(((𝐵 ∈ TopBases ∧ 𝐶𝐵) ∧ (𝐷𝐵𝐴 ∈ (𝐶𝐷))) → ∃𝑥𝐵 (𝐴𝑥𝑥 ⊆ (𝐶𝐷)))
 
Theoremfiinbas 21555* If a set is closed under finite intersection, then it is a basis for a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases)
 
Theorembasdif0 21556 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)
 
Theorembaspartn 21557* A disjoint system of sets is a basis for a topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝑃𝑉 ∧ ∀𝑥𝑃𝑦𝑃 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)) → 𝑃 ∈ TopBases)
 
Theoremtgval 21558* The topology generated by a basis. See also tgval2 21559 and tgval3 21566. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
 
Theoremtgval2 21559* Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 21572) that (topGen‘𝐵) is indeed a topology (on 𝐵, see unitg 21570). See also tgval 21558 and tgval3 21566. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (topGen‘𝐵) = {𝑥 ∣ (𝑥 𝐵 ∧ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))})
 
Theoremeltg 21560 Membership in a topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 (𝐵 ∩ 𝒫 𝐴)))
 
Theoremeltg2 21561* Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ (𝐴 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝑦𝐴))))
 
Theoremeltg2b 21562* Membership in a topology generated by a basis. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝑦𝐴)))
 
Theoremeltg4i 21563 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))
 
Theoremeltg3i 21564 The union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 30-Aug-2015.)
((𝐵𝑉𝐴𝐵) → 𝐴 ∈ (topGen‘𝐵))
 
Theoremeltg3 21565* Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Mario Carneiro, 30-Aug-2015.)
(𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∃𝑥(𝑥𝐵𝐴 = 𝑥)))
 
Theoremtgval3 21566* Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80. See also tgval 21558 and tgval2 21559. (Contributed by NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.)
(𝐵𝑉 → (topGen‘𝐵) = {𝑥 ∣ ∃𝑦(𝑦𝐵𝑥 = 𝑦)})
 
Theoremtg1 21567 Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
(𝐴 ∈ (topGen‘𝐵) → 𝐴 𝐵)
 
Theoremtg2 21568* Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
((𝐴 ∈ (topGen‘𝐵) ∧ 𝐶𝐴) → ∃𝑥𝐵 (𝐶𝑥𝑥𝐴))
 
Theorembastg 21569 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.)
(𝐵𝑉𝐵 ⊆ (topGen‘𝐵))
 
Theoremunitg 21570 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.)
(𝐵𝑉 (topGen‘𝐵) = 𝐵)
 
Theoremtgss 21571 Subset relation for generated topologies. (Contributed by NM, 7-May-2007.)
((𝐶𝑉𝐵𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶))
 
Theoremtgcl 21572 Show that a basis generates a topology. Remark in [Munkres] p. 79. (Contributed by NM, 17-Jul-2006.)
(𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top)
 
Theoremtgclb 21573 The property tgcl 21572 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)
 
Theoremtgtopon 21574 A basis generates a topology on 𝐵. (Contributed by Mario Carneiro, 14-Aug-2015.)
(𝐵 ∈ TopBases → (topGen‘𝐵) ∈ (TopOn‘ 𝐵))
 
Theoremtopbas 21575 A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
(𝐽 ∈ Top → 𝐽 ∈ TopBases)
 
Theoremtgtop 21576 A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
(𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
 
Theoremeltop 21577 Membership in a topology, expressed without quantifiers. (Contributed by NM, 19-Jul-2006.)
(𝐽 ∈ Top → (𝐴𝐽𝐴 (𝐽 ∩ 𝒫 𝐴)))
 
Theoremeltop2 21578* Membership in a topology. (Contributed by NM, 19-Jul-2006.)
(𝐽 ∈ Top → (𝐴𝐽 ↔ ∀𝑥𝐴𝑦𝐽 (𝑥𝑦𝑦𝐴)))
 
Theoremeltop3 21579* Membership in a topology. (Contributed by NM, 19-Jul-2006.)
(𝐽 ∈ Top → (𝐴𝐽 ↔ ∃𝑥(𝑥𝐽𝐴 = 𝑥)))
 
Theoremfibas 21580 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
 
Theoremtgdom 21581 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.)
(𝐵𝑉 → (topGen‘𝐵) ≼ 𝒫 𝐵)
 
Theoremtgiun 21582* The indexed union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
((𝐵𝑉 ∧ ∀𝑥𝐴 𝐶𝐵) → 𝑥𝐴 𝐶 ∈ (topGen‘𝐵))
 
Theoremtgidm 21583 The topology generator function is idempotent. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.)
(𝐵𝑉 → (topGen‘(topGen‘𝐵)) = (topGen‘𝐵))
 
Theorembastop 21584 Two ways to express that a basis is a topology. (Contributed by NM, 18-Jul-2006.)
(𝐵 ∈ TopBases → (𝐵 ∈ Top ↔ (topGen‘𝐵) = 𝐵))
 
Theoremtgtop11 21585 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)
 
Theorem0top 21586 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 = {∅}))
 
Theoremen1top 21587 {∅} is the only topology with one element. (Contributed by FL, 18-Aug-2008.)
(𝐽 ∈ Top → (𝐽 ≈ 1o𝐽 = {∅}))
 
Theoremen2top 21588 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 ≠ ∅)))
 
Theoremtgss3 21589 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.)
((𝐵𝑉𝐶𝑊) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶)))
 
Theoremtgss2 21590* 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.)
((𝐵𝑉 𝐵 = 𝐶) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ ∀𝑥 𝐵𝑦𝐵 (𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))
 
Theorembasgen 21591 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 ∧ 𝐵𝐽𝐽 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = 𝐽)
 
Theorembasgen2 21592* 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 ∧ 𝐵𝐽 ∧ ∀𝑥𝐽𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)) → (topGen‘𝐵) = 𝐽)
 
Theorem2basgen 21593 Conditions that determine the equality of two generated topologies. (Contributed by NM, 8-May-2007.) (Revised by Mario Carneiro, 2-Sep-2015.)
((𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = (topGen‘𝐶))
 
Theoremtgfiss 21594 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 ∧ 𝐴𝐽) → (topGen‘(fi‘𝐴)) ⊆ 𝐽)
 
Theoremtgdif0 21595 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)
 
Theorembastop1 21596* 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 ∧ 𝐵𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥𝐽𝑦(𝑦𝐵𝑥 = 𝑦)))
 
Theorembastop2 21597* A version of bastop1 21596 that doesn't have 𝐵𝐽 in the antecedent. (Contributed by NM, 3-Feb-2008.)
(𝐽 ∈ Top → ((topGen‘𝐵) = 𝐽 ↔ (𝐵𝐽 ∧ ∀𝑥𝐽𝑦(𝑦𝐵𝑥 = 𝑦))))
 
12.1.3  Examples of topologies
 
Theoremdistop 21598 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.)
(𝐴𝑉 → 𝒫 𝐴 ∈ Top)
 
Theoremtopnex 21599 The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 7472; an alternate proof uses indiscrete topologies (see indistop 21605) and the analogue of pwnex 7472 with pairs {∅, 𝑥} instead of power sets 𝒫 𝑥 (that analogue is also a consequence of abnex 7470). (Contributed by BJ, 2-May-2021.)
Top ∉ V
 
Theoremdistopon 21600 The discrete topology on a set 𝐴, with base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐴𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
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