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Theorem List for Metamath Proof Explorer - 20701-20800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisclo 20701* A set 𝐴 is clopen iff for every point 𝑥 in the space there is a neighborhood 𝑦 such that all the points in 𝑦 are in 𝐴 iff 𝑥 is. (Contributed by Mario Carneiro, 10-Mar-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑥𝐴𝑧𝐴))))

Theoremisclo2 20702* A set 𝐴 is clopen iff for every point 𝑥 in the space there is a neighborhood 𝑦 of 𝑥 which is either disjoint from 𝐴 or contained in 𝐴. (Contributed by Mario Carneiro, 7-Jul-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∀𝑧𝑦 (𝑧𝐴𝑦𝐴))))

Theoremdiscld 20703 The open sets of a discrete topology are closed and its closed sets are open. (Contributed by FL, 7-Jun-2007.) (Revised by Mario Carneiro, 7-Apr-2015.)
(𝐴𝑉 → (Clsd‘𝒫 𝐴) = 𝒫 𝐴)

Theoremsn0cld 20704 The closed sets of the topology {∅}. (Contributed by FL, 5-Jan-2009.)
(Clsd‘{∅}) = {∅}

Theoremindiscld 20705 The closed sets of an indiscrete topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 14-Aug-2015.)
(Clsd‘{∅, 𝐴}) = {∅, 𝐴}

Theoremmretopd 20706* A Moore collection which is closed under finite unions called topological; such a collection is the closed sets of a canonically associated topology. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝜑𝑀 ∈ (Moore‘𝐵))    &   (𝜑 → ∅ ∈ 𝑀)    &   ((𝜑𝑥𝑀𝑦𝑀) → (𝑥𝑦) ∈ 𝑀)    &   𝐽 = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ 𝑀}       (𝜑 → (𝐽 ∈ (TopOn‘𝐵) ∧ 𝑀 = (Clsd‘𝐽)))

Theoremtoponmre 20707 The topologies over a given base set form a Moore collection: the intersection of any family of them is a topology, including the empty (relative) intersection which gives the discrete topology distop 20610. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 5-May-2015.)
(𝐵𝑉 → (TopOn‘𝐵) ∈ (Moore‘𝒫 𝐵))

Theoremcldmreon 20708 The closed sets of a topology over a set are a Moore collection over the same set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
(𝐽 ∈ (TopOn‘𝐵) → (Clsd‘𝐽) ∈ (Moore‘𝐵))

Theoremiscldtop 20709* A family is the closed sets of a topology iff it is a Moore collection and closed under finite union. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐾 ∈ (Clsd “ (TopOn‘𝐵)) ↔ (𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾))

TheoremmreclatdemoBAD 20710 The closed subspaces of a topology-bearing module form a complete lattice. Demonstration for mreclatBAD 17010. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 6511 update): This proof uses the old df-clat 16931 and references the required instance of mreclatBAD 17010 as a hypothesis. When mreclatBAD 17010 is corrected to become mreclat, delete this theorem and uncomment the mreclatdemo below.
(((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊))) ∈ (Moore‘ (TopOpen‘𝑊)) → (toInc‘((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊)))) ∈ CLat)       (𝑊 ∈ (TopSp ∩ LMod) → (toInc‘((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊)))) ∈ CLat)

12.1.5  Neighborhoods

Syntaxcnei 20711 Extend class notation with neighborhood relation for topologies.
class nei

Definitiondf-nei 20712* Define a function on topologies whose value is a map from a subset to its neighborhoods. (Contributed by NM, 11-Feb-2007.)
nei = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 ↦ {𝑦 ∈ 𝒫 𝑗 ∣ ∃𝑔𝑗 (𝑥𝑔𝑔𝑦)}))

Theoremneifval 20713* The neighborhood function on the subsets of a topology's base set. (Contributed by NM, 11-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
𝑋 = 𝐽       (𝐽 ∈ Top → (nei‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)}))

Theoremneif 20714 The neighborhood function is a function of the subsets of a topology's base set. (Contributed by NM, 12-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
𝑋 = 𝐽       (𝐽 ∈ Top → (nei‘𝐽) Fn 𝒫 𝑋)

Theoremneiss2 20715 A set with a neighborhood is a subset of the topology's base set. (This theorem depends on a function's value being empty outside of its domain, but it will make later theorems simpler to state.) (Contributed by NM, 12-Feb-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆𝑋)

Theoremneival 20716* The set of neighborhoods of a subset of the base set of a topology. (Contributed by NM, 11-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((nei‘𝐽)‘𝑆) = {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑆𝑔𝑔𝑣)})

Theoremisnei 20717* 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 20718 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 20719* 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 20720 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 20721 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 20722* Property of a neighborhood. (Contributed by NM, 12-Feb-2007.)
((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑁))

Theoremneiss 20723 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 20724 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 20725 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 20726 The empty set is not a neighborhood of a nonempty set. (Contributed by FL, 18-Sep-2007.)
((𝐽 ∈ Top ∧ 𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))

Theoremneips 20727* 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 20728 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 20729 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 20730 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 20731 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 20732 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 20733 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 20734* 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 20735 The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 20732. (Contributed by FL, 2-Oct-2006.)
𝑋 = 𝐽       (𝐽 ∈ Top → (𝑆𝑋𝑋 ∈ ((nei‘𝐽)‘𝑆)))

Theoremneiuni 20736 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 20737* 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 20738 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 20739 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 20740 Only an open set is a neighborhood of itself. (Contributed by FL, 2-Oct-2006.)
(𝐽 ∈ Top → (𝑁 ∈ ((nei‘𝐽)‘𝑁) ↔ 𝑁𝐽))

Theoremneissex 20741* 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 20742 The empty set is a neighborhood of itself. (Contributed by FL, 10-Dec-2006.)
(𝐽 ∈ Top → ∅ ∈ ((nei‘𝐽)‘∅))

Theoremneipeltop 20743* Lemma for neiptopreu 20747. (Contributed by Thierry Arnoux, 6-Jan-2018.)
𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}       (𝐶𝐽 ↔ (𝐶𝑋 ∧ ∀𝑝𝐶 𝐶 ∈ (𝑁𝑝)))

Theoremneiptopuni 20744* Lemma for neiptopreu 20747. (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 20745* Lemma for neiptopreu 20747. (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 20746* Lemma for neiptopreu 20747. (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 20747* 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 20724, innei 20739, elnei 20725 and neissex 20741, 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 20748 Extend class notation with the limit point function for topologies.
class limPt

Syntaxcperf 20749 Extend class notation with the class of all perfect spaces.
class Perf

Definitiondf-lp 20750* Define a function on topologies whose value is the set of limit points of the subsets of the base set. See lpval 20753. (Contributed by NM, 10-Feb-2007.)
limPt = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 ↦ {𝑦𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))}))

Definitiondf-perf 20751 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 20752* 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 20753* 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 20754 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 20755 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 20756 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 20757 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 20758 Subset relationship for limit points. (Contributed by Mario Carneiro, 25-Dec-2016.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((limPt‘𝐽)‘𝑇) ⊆ ((limPt‘𝐽)‘𝑆))

Theoremislp2 20759* 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 20760* The predicate "𝑃 is a limit point of 𝑆 " in terms of open sets. see islp2 20759, elcls 20687, islp 20754. (Contributed by FL, 31-Jul-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑥𝐽 (𝑃𝑥 → (𝑥 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)))

Theoremmaxlp 20761 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 20762 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 20763 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 20764 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 20765 Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
𝑋 = 𝐽       (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋))

Theoremisperf2 20766 Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
𝑋 = 𝐽       (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋)))

Theoremisperf3 20767* 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 20768 The limit points of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
𝑋 = 𝐽       (𝐽 ∈ Perf → ((limPt‘𝐽)‘𝑋) = 𝑋)

Theoremperfi 20769 Property of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
𝑋 = 𝐽       ((𝐽 ∈ Perf ∧ 𝑃𝑋) → ¬ {𝑃} ∈ 𝐽)

Theoremperftop 20770 A perfect space is a topology. (Contributed by Mario Carneiro, 25-Dec-2016.)
(𝐽 ∈ Perf → 𝐽 ∈ Top)

12.1.7  Subspace topologies

Theoremrestrcl 20771 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 20772 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 20773 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 20774 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 20775 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 20776 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 20777 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 20778 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 20779 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 20780 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 20781 The underlying set of a subspace topology. (Contributed by Mario Carneiro, 21-Mar-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑉) → (𝐴𝑋) = (𝐽t 𝐴))

Theoremresttopon2 20782 The underlying set of a subspace topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ (TopOn‘(𝐴𝑋)))

Theoremrest0 20783 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 20784 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 20785 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 20786* 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 20787 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 20788 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 20789 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 20790 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 20791 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 20792 A subspace of a discrete topology is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.)
((𝐴𝑉𝐵𝐴) → (𝒫 𝐴t 𝐵) = 𝒫 𝐵)

Theoremrestfpw 20793 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 20794 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 20795 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 20796 An interior in a subspace topology. Willard in General Topology says that there is no analogue of restcls 20795 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 20797 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 20798 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 20799 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 20800 The topology of a restricted structure. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝐻 = (𝐾s 𝐴)    &   𝐽 = (TopOpen‘𝐾)       (𝐽t 𝐴) = (TopOpen‘𝐻)

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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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