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Theorem List for Metamath Proof Explorer - 20901-21000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisnei 20901* The predicate "𝑁 is a neighborhood of 𝑆." (Contributed by FL, 25-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁𝑋 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔𝑁))))

Theoremneiint 20902 An intuitive definition of a neighborhood in terms of interior. (Contributed by Szymon Jaroszewicz, 18-Dec-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋𝑁𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑁)))

Theoremisneip 20903* The predicate "𝑁 is a neighborhood of point 𝑃." (Contributed by NM, 26-Feb-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑁 ∈ ((nei‘𝐽)‘{𝑃}) ↔ (𝑁𝑋 ∧ ∃𝑔𝐽 (𝑃𝑔𝑔𝑁))))

Theoremneii1 20904 A neighborhood is included in the topology's base set. (Contributed by NM, 12-Feb-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁𝑋)

Theoremneisspw 20905 The neighborhoods of any set are subsets of the base set. (Contributed by Stefan O'Rear, 6-Aug-2015.)
𝑋 = 𝐽       (𝐽 ∈ Top → ((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋)

Theoremneii2 20906* Property of a neighborhood. (Contributed by NM, 12-Feb-2007.)
((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑁))

Theoremneiss 20907 Any neighborhood of a set 𝑆 is also a neighborhood of any subset 𝑅𝑆. Theorem of [BourbakiTop1] p. I.2. (Contributed by FL, 25-Sep-2006.)
((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅𝑆) → 𝑁 ∈ ((nei‘𝐽)‘𝑅))

Theoremssnei 20908 A set is included in its neighborhoods. Proposition Viii of [BourbakiTop1] p. I.3 . (Contributed by FL, 16-Nov-2006.)
((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆𝑁)

Theoremelnei 20909 A point belongs to any of its neighborhoods. Proposition Viii of [BourbakiTop1] p. I.3. (Contributed by FL, 28-Sep-2006.)
((𝐽 ∈ Top ∧ 𝑃𝐴𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → 𝑃𝑁)

Theorem0nnei 20910 The empty set is not a neighborhood of a nonempty set. (Contributed by FL, 18-Sep-2007.)
((𝐽 ∈ Top ∧ 𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))

Theoremneips 20911* A neighborhood of a set is a neighborhood of every point in the set. Proposition of [BourbakiTop1] p. I.2. (Contributed by FL, 16-Nov-2006.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ ∀𝑝𝑆 𝑁 ∈ ((nei‘𝐽)‘{𝑝})))

Theoremopnneissb 20912 An open set is a neighborhood of any of its subsets. (Contributed by FL, 2-Oct-2006.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑁𝐽𝑆𝑋) → (𝑆𝑁𝑁 ∈ ((nei‘𝐽)‘𝑆)))

Theoremopnssneib 20913 Any superset of an open set is a neighborhood of it. (Contributed by NM, 14-Feb-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝐽𝑁𝑋) → (𝑆𝑁𝑁 ∈ ((nei‘𝐽)‘𝑆)))

Theoremssnei2 20914 Any subset of 𝑋 containing a neighborhood of a set is a neighborhood of this set. Proposition Vi of [BourbakiTop1] p. I.3. (Contributed by FL, 2-Oct-2006.)
𝑋 = 𝐽       (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑁𝑀𝑀𝑋)) → 𝑀 ∈ ((nei‘𝐽)‘𝑆))

Theoremneindisj 20915 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 20916 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 20917 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 20918* 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 20919 The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 20916. (Contributed by FL, 2-Oct-2006.)
𝑋 = 𝐽       (𝐽 ∈ Top → (𝑆𝑋𝑋 ∈ ((nei‘𝐽)‘𝑆)))

Theoremneiuni 20920 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 20921* 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 20922 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 20923 The intersection of two neighborhoods of a set is also a neighborhood of the set. Proposition Vii of [BourbakiTop1] p. I.3 . (Contributed by FL, 28-Sep-2006.)
((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁𝑀) ∈ ((nei‘𝐽)‘𝑆))

Theoremopnneiid 20924 Only an open set is a neighborhood of itself. (Contributed by FL, 2-Oct-2006.)
(𝐽 ∈ Top → (𝑁 ∈ ((nei‘𝐽)‘𝑁) ↔ 𝑁𝐽))

Theoremneissex 20925* For any neighborhood 𝑁 of 𝑆, there is a neighborhood 𝑥 of 𝑆 such that 𝑁 is a neighborhood of all subsets of 𝑥. Proposition 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 20926 The empty set is a neighborhood of itself. (Contributed by FL, 10-Dec-2006.)
(𝐽 ∈ Top → ∅ ∈ ((nei‘𝐽)‘∅))

Theoremneipeltop 20927* Lemma for neiptopreu 20931. (Contributed by Thierry Arnoux, 6-Jan-2018.)
𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}       (𝐶𝐽 ↔ (𝐶𝑋 ∧ ∀𝑝𝐶 𝐶 ∈ (𝑁𝑝)))

Theoremneiptopuni 20928* Lemma for neiptopreu 20931. (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 20929* Lemma for neiptopreu 20931. (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 20930* Lemma for neiptopreu 20931. (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 20931* If, to each element 𝑃 of a set 𝑋, we associate a set (𝑁𝑃) fulfilling the properties Vi, Vii, Viii and property Viv of [BourbakiTop1] p. I.2. , corresponding to ssnei 20908, innei 20923, elnei 20909 and neissex 20925, 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 the additional condition that 𝑋 is a neighborhood of all points was added. (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 20932 Extend class notation with the limit point function for topologies.
class limPt

Syntaxcperf 20933 Extend class notation with the class of all perfect spaces.
class Perf

Definitiondf-lp 20934* Define a function on topologies whose value is the set of limit points of the subsets of the base set. See lpval 20937. (Contributed by NM, 10-Feb-2007.)
limPt = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 ↦ {𝑦𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))}))

Definitiondf-perf 20935 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 20936* 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 20937* 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 20938 The predicate "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 20939 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 20940 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 20941 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 20942 Subset relationship for limit points. (Contributed by Mario Carneiro, 25-Dec-2016.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((limPt‘𝐽)‘𝑇) ⊆ ((limPt‘𝐽)‘𝑆))

Theoremislp2 20943* 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 20944* The predicate "𝑃 is a limit point of 𝑆 " in terms of open sets. see islp2 20943, elcls 20871, islp 20938. (Contributed by FL, 31-Jul-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑥𝐽 (𝑃𝑥 → (𝑥 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)))

Theoremmaxlp 20945 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 20946 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 20947 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 20948 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 20949 Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
𝑋 = 𝐽       (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋))

Theoremisperf2 20950 Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
𝑋 = 𝐽       (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋)))

Theoremisperf3 20951* 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 20952 The limit points of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
𝑋 = 𝐽       (𝐽 ∈ Perf → ((limPt‘𝐽)‘𝑋) = 𝑋)

Theoremperfi 20953 Property of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
𝑋 = 𝐽       ((𝐽 ∈ Perf ∧ 𝑃𝑋) → ¬ {𝑃} ∈ 𝐽)

Theoremperftop 20954 A perfect space is a topology. (Contributed by Mario Carneiro, 25-Dec-2016.)
(𝐽 ∈ Perf → 𝐽 ∈ Top)

12.1.7  Subspace topologies

Theoremrestrcl 20955 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 20956 A subspace topology basis is a basis. 𝑌 is normally a subset of the base set of 𝐽. (Contributed by Mario Carneiro, 19-Mar-2015.)
(𝐵 ∈ TopBases → (𝐵t 𝐴) ∈ TopBases)

Theoremtgrest 20957 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 20958 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 20959 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 20960 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 20961 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 20962 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 20963 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 20964 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 20965 The underlying set of a subspace topology. (Contributed by Mario Carneiro, 21-Mar-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑉) → (𝐴𝑋) = (𝐽t 𝐴))

Theoremresttopon2 20966 The underlying set of a subspace topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ (TopOn‘(𝐴𝑋)))

Theoremrest0 20967 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 20968 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 20969 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 20970* 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 20971 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 20972 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 20973 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 20974 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 20975 The 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 20976 A subspace of a discrete topology is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.)
((𝐴𝑉𝐵𝐴) → (𝒫 𝐴t 𝐵) = 𝒫 𝐵)

Theoremrestfpw 20977 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 20978 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 20979 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 20980 An interior in a subspace topology. Willard in General Topology says that there is no analogue of restcls 20979 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 20981 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 20982 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 20983 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 20984 The topology of a restricted structure. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝐻 = (𝐾s 𝐴)    &   𝐽 = (TopOpen‘𝐾)       (𝐽t 𝐴) = (TopOpen‘𝐻)

Theoremresstps 20985 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 20986* Lemma for ordtbas 20990. 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 20987* 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 20988* 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 20989* Lemma for ordtbas 20990. (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 20990* 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 20991 Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       (𝑅𝑉 → (ordTop‘𝑅) ∈ (TopOn‘𝑋))

Theoremordtopn1 20992* An upward ray (𝑃, +∞) is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ (ordTop‘𝑅))

Theoremordtopn2 20993* A downward ray (-∞, 𝑃) is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ (ordTop‘𝑅))

Theoremordtopn3 20994* An open interval (𝐴, 𝐵) is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅𝑉𝐴𝑋𝐵𝑋) → {𝑥𝑋 ∣ (¬ 𝑥𝑅𝐴 ∧ ¬ 𝐵𝑅𝑥)} ∈ (ordTop‘𝑅))

Theoremordtcld1 20995* A downward ray (-∞, 𝑃] is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅𝑉𝑃𝑋) → {𝑥𝑋𝑥𝑅𝑃} ∈ (Clsd‘(ordTop‘𝑅)))

Theoremordtcld2 20996* An upward ray [𝑃, +∞) is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅𝑉𝑃𝑋) → {𝑥𝑋𝑃𝑅𝑥} ∈ (Clsd‘(ordTop‘𝑅)))

Theoremordtcld3 20997* A closed interval [𝐴, 𝐵] is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅𝑉𝐴𝑋𝐵𝑋) → {𝑥𝑋 ∣ (𝐴𝑅𝑥𝑥𝑅𝐵)} ∈ (Clsd‘(ordTop‘𝑅)))

Theoremordttop 20998 The order topology is a topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑅𝑉 → (ordTop‘𝑅) ∈ Top)

Theoremordtcnv 20999 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 21000 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))

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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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