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Theorem List for Metamath Proof Explorer - 21701-21800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremneindisj 21701 Any neighborhood of an element in the closure of a subset intersects the subset. Part of proof of Theorem 6.6 of [Munkres] p. 97. (Contributed by NM, 26-Feb-2007.)
𝑋 = 𝐽       (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑃 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃}))) → (𝑁𝑆) ≠ ∅)

Theoremopnneiss 21702 An open set is a neighborhood of any of its subsets. (Contributed by NM, 13-Feb-2007.)
((𝐽 ∈ Top ∧ 𝑁𝐽𝑆𝑁) → 𝑁 ∈ ((nei‘𝐽)‘𝑆))

Theoremopnneip 21703 An open set is a neighborhood of any of its members. (Contributed by NM, 8-Mar-2007.)
((𝐽 ∈ Top ∧ 𝑁𝐽𝑃𝑁) → 𝑁 ∈ ((nei‘𝐽)‘{𝑃}))

Theoremopnnei 21704* A set is open iff it is a neighborhood of all of its points. (Contributed by Jeff Hankins, 15-Sep-2009.)
(𝐽 ∈ Top → (𝑆𝐽 ↔ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))

Theoremtpnei 21705 The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 21702. (Contributed by FL, 2-Oct-2006.)
𝑋 = 𝐽       (𝐽 ∈ Top → (𝑆𝑋𝑋 ∈ ((nei‘𝐽)‘𝑆)))

Theoremneiuni 21706 The union of the neighborhoods of a set equals the topology's underlying set. (Contributed by FL, 18-Sep-2007.) (Revised by Mario Carneiro, 9-Apr-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑋 = ((nei‘𝐽)‘𝑆))

Theoremneindisj2 21707* A point 𝑃 belongs to the closure of a set 𝑆 iff every neighborhood of 𝑃 meets 𝑆. (Contributed by FL, 15-Sep-2013.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑃})(𝑛𝑆) ≠ ∅))

Theoremtopssnei 21708 A finer topology has more neighborhoods. (Contributed by Mario Carneiro, 9-Apr-2015.)
𝑋 = 𝐽    &   𝑌 = 𝐾       (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ 𝐽𝐾) → ((nei‘𝐽)‘𝑆) ⊆ ((nei‘𝐾)‘𝑆))

Theoreminnei 21709 The intersection of two neighborhoods of a set is also a neighborhood of the set. Generalization to subsets of Property Vii of [BourbakiTop1] p. I.3 for binary intersections. (Contributed by FL, 28-Sep-2006.)
((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁𝑀) ∈ ((nei‘𝐽)‘𝑆))

Theoremopnneiid 21710 Only an open set is a neighborhood of itself. (Contributed by FL, 2-Oct-2006.)
(𝐽 ∈ Top → (𝑁 ∈ ((nei‘𝐽)‘𝑁) ↔ 𝑁𝐽))

Theoremneissex 21711* For any neighborhood 𝑁 of 𝑆, there is a neighborhood 𝑥 of 𝑆 such that 𝑁 is a neighborhood of all subsets of 𝑥. Generalization to subsets of Property Viv of [BourbakiTop1] p. I.3. (Contributed by FL, 2-Oct-2006.)
((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑥 ∈ ((nei‘𝐽)‘𝑆)∀𝑦(𝑦𝑥𝑁 ∈ ((nei‘𝐽)‘𝑦)))

Theorem0nei 21712 The empty set is a neighborhood of itself. (Contributed by FL, 10-Dec-2006.)
(𝐽 ∈ Top → ∅ ∈ ((nei‘𝐽)‘∅))

Theoremneipeltop 21713* Lemma for neiptopreu 21717. (Contributed by Thierry Arnoux, 6-Jan-2018.)
𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}       (𝐶𝐽 ↔ (𝐶𝑋 ∧ ∀𝑝𝐶 𝐶 ∈ (𝑁𝑝)))

Theoremneiptopuni 21714* Lemma for neiptopreu 21717. (Contributed by Thierry Arnoux, 6-Jan-2018.)
𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}    &   (𝜑𝑁:𝑋⟶𝒫 𝒫 𝑋)    &   ((((𝜑𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑏 ∈ (𝑁𝑝))    &   ((𝜑𝑝𝑋) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))    &   (((𝜑𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)    &   (((𝜑𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑏 ∈ (𝑁𝑝)∀𝑞𝑏 𝑎 ∈ (𝑁𝑞))    &   ((𝜑𝑝𝑋) → 𝑋 ∈ (𝑁𝑝))       (𝜑𝑋 = 𝐽)

Theoremneiptoptop 21715* Lemma for neiptopreu 21717. (Contributed by Thierry Arnoux, 7-Jan-2018.)
𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}    &   (𝜑𝑁:𝑋⟶𝒫 𝒫 𝑋)    &   ((((𝜑𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑏 ∈ (𝑁𝑝))    &   ((𝜑𝑝𝑋) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))    &   (((𝜑𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)    &   (((𝜑𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑏 ∈ (𝑁𝑝)∀𝑞𝑏 𝑎 ∈ (𝑁𝑞))    &   ((𝜑𝑝𝑋) → 𝑋 ∈ (𝑁𝑝))       (𝜑𝐽 ∈ Top)

Theoremneiptopnei 21716* Lemma for neiptopreu 21717. (Contributed by Thierry Arnoux, 7-Jan-2018.)
𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}    &   (𝜑𝑁:𝑋⟶𝒫 𝒫 𝑋)    &   ((((𝜑𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑏 ∈ (𝑁𝑝))    &   ((𝜑𝑝𝑋) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))    &   (((𝜑𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)    &   (((𝜑𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑏 ∈ (𝑁𝑝)∀𝑞𝑏 𝑎 ∈ (𝑁𝑞))    &   ((𝜑𝑝𝑋) → 𝑋 ∈ (𝑁𝑝))       (𝜑𝑁 = (𝑝𝑋 ↦ ((nei‘𝐽)‘{𝑝})))

Theoremneiptopreu 21717* If, to each element 𝑃 of a set 𝑋, we associate a set (𝑁𝑃) fulfilling Properties Vi, Vii, Viii and Property Viv of [BourbakiTop1] p. I.2. , corresponding to ssnei 21694, innei 21709, elnei 21695 and neissex 21711, then there is a unique topology 𝑗 such that for any point 𝑝, (𝑁𝑝) is the set of neighborhoods of 𝑝. Proposition 2 of [BourbakiTop1] p. I.3. This can be used to build a topology from a set of neighborhoods. Note that innei 21709 uses binary intersections whereas Property Vii mentions finite intersections (which includes the empty intersection of subsets of 𝑋, which is equal to 𝑋), so we add the hypothesis that 𝑋 is a neighborhood of all points. TODO: when df-fi 8853 includes the empty intersection, remove that extra hypothesis. (Contributed by Thierry Arnoux, 6-Jan-2018.)
𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}    &   (𝜑𝑁:𝑋⟶𝒫 𝒫 𝑋)    &   ((((𝜑𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑏 ∈ (𝑁𝑝))    &   ((𝜑𝑝𝑋) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))    &   (((𝜑𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)    &   (((𝜑𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑏 ∈ (𝑁𝑝)∀𝑞𝑏 𝑎 ∈ (𝑁𝑞))    &   ((𝜑𝑝𝑋) → 𝑋 ∈ (𝑁𝑝))       (𝜑 → ∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})))

12.1.6  Limit points and perfect sets

Syntaxclp 21718 Extend class notation with the limit point function for topologies.
class limPt

Syntaxcperf 21719 Extend class notation with the class of all perfect spaces.
class Perf

Definitiondf-lp 21720* Define a function on topologies whose value is the set of limit points of the subsets of the base set. See lpval 21723. (Contributed by NM, 10-Feb-2007.)
limPt = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 ↦ {𝑦𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))}))

Definitiondf-perf 21721 Define the class of all perfect spaces. A perfect space is one for which every point in the set is a limit point of the whole space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Perf = {𝑗 ∈ Top ∣ ((limPt‘𝑗)‘ 𝑗) = 𝑗}

Theoremlpfval 21722* The limit point function on the subsets of a topology's base set. (Contributed by NM, 10-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
𝑋 = 𝐽       (𝐽 ∈ Top → (limPt‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑦𝑦 ∈ ((cls‘𝐽)‘(𝑥 ∖ {𝑦}))}))

Theoremlpval 21723* The set of limit points of a subset of the base set of a topology. Alternate definition of limit point in [Munkres] p. 97. (Contributed by NM, 10-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((limPt‘𝐽)‘𝑆) = {𝑥𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))})

Theoremislp 21724 The predicate "the class 𝑃 is a limit point of 𝑆". (Contributed by NM, 10-Feb-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃}))))

Theoremlpsscls 21725 The limit points of a subset are included in the subset's closure. (Contributed by NM, 26-Feb-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((limPt‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝑆))

Theoremlpss 21726 The limit points of a subset are included in the base set. (Contributed by NM, 9-Nov-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((limPt‘𝐽)‘𝑆) ⊆ 𝑋)

Theoremlpdifsn 21727 𝑃 is a limit point of 𝑆 iff it is a limit point of 𝑆 ∖ {𝑃}. (Contributed by Mario Carneiro, 25-Dec-2016.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑃 ∈ ((limPt‘𝐽)‘(𝑆 ∖ {𝑃}))))

Theoremlpss3 21728 Subset relationship for limit points. (Contributed by Mario Carneiro, 25-Dec-2016.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((limPt‘𝐽)‘𝑇) ⊆ ((limPt‘𝐽)‘𝑆))

Theoremislp2 21729* The predicate "𝑃 is a limit point of 𝑆 " in terms of neighborhoods. Definition of limit point in [Munkres] p. 97. Although Munkres uses open neighborhoods, it also works for our more general neighborhoods. (Contributed by NM, 26-Feb-2007.) (Proof shortened by Mario Carneiro, 25-Dec-2016.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅))

Theoremislp3 21730* The predicate "𝑃 is a limit point of 𝑆 " in terms of open sets. see islp2 21729, elcls 21657, islp 21724. (Contributed by FL, 31-Jul-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑥𝐽 (𝑃𝑥 → (𝑥 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)))

Theoremmaxlp 21731 A point is a limit point of the whole space iff the singleton of the point is not open. (Contributed by Mario Carneiro, 24-Dec-2016.)
𝑋 = 𝐽       (𝐽 ∈ Top → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ (𝑃𝑋 ∧ ¬ {𝑃} ∈ 𝐽)))

Theoremclslp 21732 The closure of a subset of a topological space is the subset together with its limit points. Theorem 6.6 of [Munkres] p. 97. (Contributed by NM, 26-Feb-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))

Theoremislpi 21733 A point belonging to a set's closure but not the set itself is a limit point. (Contributed by NM, 8-Nov-2007.)
𝑋 = 𝐽       (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑃 ∈ ((cls‘𝐽)‘𝑆) ∧ ¬ 𝑃𝑆)) → 𝑃 ∈ ((limPt‘𝐽)‘𝑆))

Theoremcldlp 21734 A subset of a topological space is closed iff it contains all its limit points. Corollary 6.7 of [Munkres] p. 97. (Contributed by NM, 26-Feb-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((limPt‘𝐽)‘𝑆) ⊆ 𝑆))

Theoremisperf 21735 Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
𝑋 = 𝐽       (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋))

Theoremisperf2 21736 Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
𝑋 = 𝐽       (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋)))

Theoremisperf3 21737* A perfect space is a topology which has no open singletons. (Contributed by Mario Carneiro, 24-Dec-2016.)
𝑋 = 𝐽       (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))

Theoremperflp 21738 The limit points of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
𝑋 = 𝐽       (𝐽 ∈ Perf → ((limPt‘𝐽)‘𝑋) = 𝑋)

Theoremperfi 21739 Property of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
𝑋 = 𝐽       ((𝐽 ∈ Perf ∧ 𝑃𝑋) → ¬ {𝑃} ∈ 𝐽)

Theoremperftop 21740 A perfect space is a topology. (Contributed by Mario Carneiro, 25-Dec-2016.)
(𝐽 ∈ Perf → 𝐽 ∈ Top)

12.1.7  Subspace topologies

Theoremrestrcl 21741 Reverse closure for the subspace topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 1-May-2015.)
((𝐽t 𝐴) ∈ Top → (𝐽 ∈ V ∧ 𝐴 ∈ V))

Theoremrestbas 21742 A subspace topology basis is a basis. (Contributed by Mario Carneiro, 19-Mar-2015.)
(𝐵 ∈ TopBases → (𝐵t 𝐴) ∈ TopBases)

Theoremtgrest 21743 A subspace can be generated by restricted sets from a basis for the original topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Proof shortened by Mario Carneiro, 30-Aug-2015.)
((𝐵𝑉𝐴𝑊) → (topGen‘(𝐵t 𝐴)) = ((topGen‘𝐵) ↾t 𝐴))

Theoremresttop 21744 A subspace topology is a topology. Definition of subspace topology in [Munkres] p. 89. 𝐴 is normally a subset of the base set of 𝐽. (Contributed by FL, 15-Apr-2007.) (Revised by Mario Carneiro, 1-May-2015.)
((𝐽 ∈ Top ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Top)

Theoremresttopon 21745 A subspace topology is a topology on the base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))

Theoremrestuni 21746 The underlying set of a subspace topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋) → 𝐴 = (𝐽t 𝐴))

Theoremstoig 21747 The topological space built with a subspace topology. (Contributed by FL, 5-Jan-2009.) (Proof shortened by Mario Carneiro, 1-May-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋) → {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), (𝐽t 𝐴)⟩} ∈ TopSp)

Theoremrestco 21748 Composition of subspaces. (Contributed by Mario Carneiro, 15-Dec-2013.) (Revised by Mario Carneiro, 1-May-2015.)
((𝐽𝑉𝐴𝑊𝐵𝑋) → ((𝐽t 𝐴) ↾t 𝐵) = (𝐽t (𝐴𝐵)))

Theoremrestabs 21749 Equivalence of being a subspace of a subspace and being a subspace of the original. (Contributed by Jeff Hankins, 11-Jul-2009.) (Proof shortened by Mario Carneiro, 1-May-2015.)
((𝐽𝑉𝑆𝑇𝑇𝑊) → ((𝐽t 𝑇) ↾t 𝑆) = (𝐽t 𝑆))

Theoremrestin 21750 When the subspace region is not a subset of the base of the topology, the resulting set is the same as the subspace restricted to the base. (Contributed by Mario Carneiro, 15-Dec-2013.)
𝑋 = 𝐽       ((𝐽𝑉𝐴𝑊) → (𝐽t 𝐴) = (𝐽t (𝐴𝑋)))

Theoremrestuni2 21751 The underlying set of a subspace topology. (Contributed by Mario Carneiro, 21-Mar-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑉) → (𝐴𝑋) = (𝐽t 𝐴))

Theoremresttopon2 21752 The underlying set of a subspace topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ (TopOn‘(𝐴𝑋)))

Theoremrest0 21753 The subspace topology induced by the topology 𝐽 on the empty set. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 1-May-2015.)
(𝐽 ∈ Top → (𝐽t ∅) = {∅})

Theoremrestsn 21754 The only subspace topology induced by the topology {∅}. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
(𝐴𝑉 → ({∅} ↾t 𝐴) = {∅})

Theoremrestsn2 21755 The subspace topology induced by a singleton. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t {𝐴}) = 𝒫 {𝐴})

Theoremrestcld 21756* A closed set of a subspace topology is a closed set of the original topology intersected with the subset. (Contributed by FL, 11-Jul-2009.) (Proof shortened by Mario Carneiro, 15-Dec-2013.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐴 ∈ (Clsd‘(𝐽t 𝑆)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))

Theoremrestcldi 21757 A closed set is closed in the subspace topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝑋 = 𝐽       ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐵 ∈ (Clsd‘(𝐽t 𝐴)))

Theoremrestcldr 21758 A set which is closed in the subspace topology induced by a closed set is closed in the original topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘(𝐽t 𝐴))) → 𝐵 ∈ (Clsd‘𝐽))

Theoremrestopnb 21759 If 𝐵 is an open subset of the subspace base set 𝐴, then any subset of 𝐵 is open iff it is open in 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)
(((𝐽 ∈ Top ∧ 𝐴𝑉) ∧ (𝐵𝐽𝐵𝐴𝐶𝐵)) → (𝐶𝐽𝐶 ∈ (𝐽t 𝐴)))

Theoremssrest 21760 If 𝐾 is a finer topology than 𝐽, then the subspace topologies induced by 𝐴 maintain this relationship. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 1-May-2015.)
((𝐾𝑉𝐽𝐾) → (𝐽t 𝐴) ⊆ (𝐾t 𝐴))

Theoremrestopn2 21761 If 𝐴 is open, then 𝐵 is open in 𝐴 iff it is an open subset of 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)
((𝐽 ∈ Top ∧ 𝐴𝐽) → (𝐵 ∈ (𝐽t 𝐴) ↔ (𝐵𝐽𝐵𝐴)))

Theoremrestdis 21762 A subspace of a discrete topology is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.)
((𝐴𝑉𝐵𝐴) → (𝒫 𝐴t 𝐵) = 𝒫 𝐵)

Theoremrestfpw 21763 The restriction of the set of finite subsets of 𝐴 is the set of finite subsets of 𝐵. (Contributed by Mario Carneiro, 18-Sep-2015.)
((𝐴𝑉𝐵𝐴) → ((𝒫 𝐴 ∩ Fin) ↾t 𝐵) = (𝒫 𝐵 ∩ Fin))

Theoremneitr 21764 The neighborhood of a trace is the trace of the neighborhood. (Contributed by Thierry Arnoux, 17-Jan-2018.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → ((nei‘(𝐽t 𝐴))‘𝐵) = (((nei‘𝐽)‘𝐵) ↾t 𝐴))

Theoremrestcls 21765 A closure in a subspace topology. (Contributed by Jeff Hankins, 22-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)
𝑋 = 𝐽    &   𝐾 = (𝐽t 𝑌)       ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘𝐾)‘𝑆) = (((cls‘𝐽)‘𝑆) ∩ 𝑌))

Theoremrestntr 21766 An interior in a subspace topology. Willard in General Topology says that there is no analogue of restcls 21765 for interiors. In some sense, that is true. (Contributed by Jeff Hankins, 23-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)
𝑋 = 𝐽    &   𝐾 = (𝐽t 𝑌)       ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((int‘𝐾)‘𝑆) = (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌))

Theoremrestlp 21767 The limit points of a subset restrict naturally in a subspace. (Contributed by Mario Carneiro, 25-Dec-2016.)
𝑋 = 𝐽    &   𝐾 = (𝐽t 𝑌)       ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((limPt‘𝐾)‘𝑆) = (((limPt‘𝐽)‘𝑆) ∩ 𝑌))

Theoremrestperf 21768 Perfection of a subspace. Note that the term "perfect set" is reserved for closed sets which are perfect in the subspace topology. (Contributed by Mario Carneiro, 25-Dec-2016.)
𝑋 = 𝐽    &   𝐾 = (𝐽t 𝑌)       ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝐾 ∈ Perf ↔ 𝑌 ⊆ ((limPt‘𝐽)‘𝑌)))

Theoremperfopn 21769 An open subset of a perfect space is perfect. (Contributed by Mario Carneiro, 25-Dec-2016.)
𝑋 = 𝐽    &   𝐾 = (𝐽t 𝑌)       ((𝐽 ∈ Perf ∧ 𝑌𝐽) → 𝐾 ∈ Perf)

Theoremresstopn 21770 The topology of a restricted structure. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝐻 = (𝐾s 𝐴)    &   𝐽 = (TopOpen‘𝐾)       (𝐽t 𝐴) = (TopOpen‘𝐻)

Theoremresstps 21771 A restricted topological space is a topological space. Note that this theorem would not be true if TopSp was defined directly in terms of the TopSet slot instead of the TopOpen derived function. (Contributed by Mario Carneiro, 13-Aug-2015.)
((𝐾 ∈ TopSp ∧ 𝐴𝑉) → (𝐾s 𝐴) ∈ TopSp)

12.1.8  Order topology

Theoremordtbaslem 21772* Lemma for ordtbas 21776. In a total order, unbounded-above intervals are closed under intersection. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅    &   𝐴 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})       (𝑅 ∈ TosetRel → (fi‘𝐴) = 𝐴)

Theoremordtval 21773* Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅    &   𝐴 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})    &   𝐵 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})       (𝑅𝑉 → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (𝐴𝐵)))))

Theoremordtuni 21774* Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅    &   𝐴 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})    &   𝐵 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})       (𝑅𝑉𝑋 = ({𝑋} ∪ (𝐴𝐵)))

Theoremordtbas2 21775* Lemma for ordtbas 21776. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅    &   𝐴 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})    &   𝐵 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})    &   𝐶 = ran (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})       (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) = ((𝐴𝐵) ∪ 𝐶))

Theoremordtbas 21776* In a total order, the finite intersections of the open rays generates the set of open intervals, but no more - these four collections form a subbasis for the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅    &   𝐴 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})    &   𝐵 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})    &   𝐶 = ran (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})       (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴𝐵))) = (({𝑋} ∪ (𝐴𝐵)) ∪ 𝐶))

Theoremordttopon 21777 Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       (𝑅𝑉 → (ordTop‘𝑅) ∈ (TopOn‘𝑋))

Theoremordtopn1 21778* An upward ray (𝑃, +∞) is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ (ordTop‘𝑅))

Theoremordtopn2 21779* A downward ray (-∞, 𝑃) is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ (ordTop‘𝑅))

Theoremordtopn3 21780* An open interval (𝐴, 𝐵) is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅𝑉𝐴𝑋𝐵𝑋) → {𝑥𝑋 ∣ (¬ 𝑥𝑅𝐴 ∧ ¬ 𝐵𝑅𝑥)} ∈ (ordTop‘𝑅))

Theoremordtcld1 21781* A downward ray (-∞, 𝑃] is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅𝑉𝑃𝑋) → {𝑥𝑋𝑥𝑅𝑃} ∈ (Clsd‘(ordTop‘𝑅)))

Theoremordtcld2 21782* An upward ray [𝑃, +∞) is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅𝑉𝑃𝑋) → {𝑥𝑋𝑃𝑅𝑥} ∈ (Clsd‘(ordTop‘𝑅)))

Theoremordtcld3 21783* A closed interval [𝐴, 𝐵] is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅𝑉𝐴𝑋𝐵𝑋) → {𝑥𝑋 ∣ (𝐴𝑅𝑥𝑥𝑅𝐵)} ∈ (Clsd‘(ordTop‘𝑅)))

Theoremordttop 21784 The order topology is a topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑅𝑉 → (ordTop‘𝑅) ∈ Top)

Theoremordtcnv 21785 The order dual generates the same topology as the original order. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑅 ∈ PosetRel → (ordTop‘𝑅) = (ordTop‘𝑅))

Theoremordtrest 21786 The subspace topology of an order topology is in general finer than the topology generated by the restricted order, but we do have inclusion in one direction. (Contributed by Mario Carneiro, 9-Sep-2015.)
((𝑅 ∈ PosetRel ∧ 𝐴𝑉) → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘𝑅) ↾t 𝐴))

Theoremordtrest2lem 21787* Lemma for ordtrest2 21788. (Contributed by Mario Carneiro, 9-Sep-2015.)
𝑋 = dom 𝑅    &   (𝜑𝑅 ∈ TosetRel )    &   (𝜑𝐴𝑋)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝑋 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)} ⊆ 𝐴)       (𝜑 → ∀𝑣 ∈ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧})(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))

Theoremordtrest2 21788* An interval-closed set 𝐴 in a total order has the same subspace topology as the restricted order topology. (An interval-closed set is the same thing as an open or half-open or closed interval in , but in other sets like there are interval-closed sets like (π, +∞) ∩ ℚ that are not intervals.) (Contributed by Mario Carneiro, 9-Sep-2015.)
𝑋 = dom 𝑅    &   (𝜑𝑅 ∈ TosetRel )    &   (𝜑𝐴𝑋)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝑋 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)} ⊆ 𝐴)       (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = ((ordTop‘𝑅) ↾t 𝐴))

Theoremletopon 21789 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
(ordTop‘ ≤ ) ∈ (TopOn‘ℝ*)

Theoremletop 21790 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
(ordTop‘ ≤ ) ∈ Top

Theoremletopuni 21791 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
* = (ordTop‘ ≤ )

Theoremxrstopn 21792 The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
(ordTop‘ ≤ ) = (TopOpen‘ℝ*𝑠)

Theoremxrstps 21793 The extended real number structure is a topological space. (Contributed by Mario Carneiro, 21-Aug-2015.)
*𝑠 ∈ TopSp

Theoremleordtvallem1 21794* Lemma for leordtval 21797. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝐴 = ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞))       𝐴 = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑦𝑥})

Theoremleordtvallem2 21795* Lemma for leordtval 21797. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝐴 = ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞))    &   𝐵 = ran (𝑥 ∈ ℝ* ↦ (-∞[,)𝑥))       𝐵 = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑥𝑦})

Theoremleordtval2 21796 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝐴 = ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞))    &   𝐵 = ran (𝑥 ∈ ℝ* ↦ (-∞[,)𝑥))       (ordTop‘ ≤ ) = (topGen‘(fi‘(𝐴𝐵)))

Theoremleordtval 21797 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝐴 = ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞))    &   𝐵 = ran (𝑥 ∈ ℝ* ↦ (-∞[,)𝑥))    &   𝐶 = ran (,)       (ordTop‘ ≤ ) = (topGen‘((𝐴𝐵) ∪ 𝐶))

Theoremiccordt 21798 A closed interval is closed in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝐴[,]𝐵) ∈ (Clsd‘(ordTop‘ ≤ ))

Theoremiocpnfordt 21799 An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝐴(,]+∞) ∈ (ordTop‘ ≤ )

Theoremicomnfordt 21800 An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
(-∞[,)𝐴) ∈ (ordTop‘ ≤ )

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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