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Theorem List for Metamath Proof Explorer - 20801-20900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
12.1.4  Closure and interior
 
Syntaxccld 20801 Extend class notation with the set of closed sets of a topology.
class Clsd
 
Syntaxcnt 20802 Extend class notation with interior of a subset of a topology base set.
class int
 
Syntaxccl 20803 Extend class notation with closure of a subset of a topology base set.
class cls
 
Definitiondf-cld 20804* 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 ∣ ( 𝑗𝑥) ∈ 𝑗})
 
Definitiondf-ntr 20805* Define a function on topologies whose value is the interior function on the subsets of the base set. See ntrval 20821. (Contributed by NM, 10-Sep-2006.)
int = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 (𝑗 ∩ 𝒫 𝑥)))
 
Definitiondf-cls 20806* Define a function on topologies whose value is the closure function on the subsets of the base set. See clsval 20822. (Contributed by NM, 3-Oct-2006.)
cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦}))
 
Theoremfncld 20807 The closed-set generator is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Clsd Fn Top
 
Theoremcldval 20808* 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 ∣ (𝑋𝑥) ∈ 𝐽})
 
Theoremntrfval 20809* 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)))
 
Theoremclsfval 20810* 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) ∣ 𝑥𝑦}))
 
Theoremcldrcl 20811 Reverse closure of the closed set operation. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
 
Theoremiscld 20812 The predicate "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) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
 
Theoremiscld2 20813 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 ∧ 𝑆𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑋𝑆) ∈ 𝐽))
 
Theoremcldss 20814 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) → 𝑆𝑋)
 
Theoremcldss2 20815 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
 
Theoremcldopn 20816 The complement of a closed set is open. (Contributed by NM, 5-Oct-2006.) (Revised by Stefan O'Rear, 22-Feb-2015.)
𝑋 = 𝐽       (𝑆 ∈ (Clsd‘𝐽) → (𝑋𝑆) ∈ 𝐽)
 
Theoremisopn2 20817 A subset of the underlying set of a topology is open iff its complement is closed. (Contributed by NM, 4-Oct-2006.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆𝐽 ↔ (𝑋𝑆) ∈ (Clsd‘𝐽)))
 
Theoremopncld 20818 The complement of an open set is closed. (Contributed by NM, 6-Oct-2006.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝐽) → (𝑋𝑆) ∈ (Clsd‘𝐽))
 
Theoremdifopn 20819 The difference of a closed set with an open set is open. (Contributed by Mario Carneiro, 6-Jan-2014.)
𝑋 = 𝐽       ((𝐴𝐽𝐵 ∈ (Clsd‘𝐽)) → (𝐴𝐵) ∈ 𝐽)
 
Theoremtopcld 20820 The underlying set of a topology is closed. Part of Theorem 6.1(1) of [Munkres] p. 93. (Contributed by NM, 3-Oct-2006.)
𝑋 = 𝐽       (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
 
Theoremntrval 20821 The interior of a subset of a topology's base set is the union of all the open sets it includes. Definition of interior of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) = (𝐽 ∩ 𝒫 𝑆))
 
Theoremclsval 20822* The closure of a subset of a topology's base set is the intersection of all the closed sets that include it. Definition of closure of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
 
Theorem0cld 20823 The empty set is closed. Part of Theorem 6.1(1) of [Munkres] p. 93. (Contributed by NM, 4-Oct-2006.)
(𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽))
 
Theoremiincld 20824* The indexed intersection of a collection 𝐵(𝑥) of closed sets is closed. Theorem 6.1(2) of [Munkres] p. 93. (Contributed by NM, 5-Oct-2006.) (Revised by Mario Carneiro, 3-Sep-2015.)
((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
 
Theoremintcld 20825 The intersection of a set of closed sets is closed. (Contributed by NM, 5-Oct-2006.)
((𝐴 ≠ ∅ ∧ 𝐴 ⊆ (Clsd‘𝐽)) → 𝐴 ∈ (Clsd‘𝐽))
 
Theoremuncld 20826 The union of two closed sets is closed. Equivalent to Theorem 6.1(3) of [Munkres] p. 93. (Contributed by NM, 5-Oct-2006.)
((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴𝐵) ∈ (Clsd‘𝐽))
 
Theoremcldcls 20827 A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.)
(𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆)
 
Theoremincld 20828 The intersection of two closed sets is closed. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴𝐵) ∈ (Clsd‘𝐽))
 
Theoremriincld 20829* An indexed relative intersection of closed sets is closed. (Contributed by Stefan O'Rear, 22-Feb-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 𝑥𝐴 𝐵) ∈ (Clsd‘𝐽))
 
Theoremiuncld 20830* A finite indexed union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
 
Theoremunicld 20831 A finite union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ (Clsd‘𝐽)) → 𝐴 ∈ (Clsd‘𝐽))
 
Theoremclscld 20832 The closure of a subset of a topology's underlying set is closed. (Contributed by NM, 4-Oct-2006.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
 
Theoremclsf 20833 The closure function is a function from subsets of the base to closed sets. (Contributed by Mario Carneiro, 11-Apr-2015.)
𝑋 = 𝐽       (𝐽 ∈ Top → (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽))
 
Theoremntropn 20834 The interior of a subset of a topology's underlying set is open. (Contributed by NM, 11-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
 
Theoremclsval2 20835 Express closure in terms of interior. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆))))
 
Theoremntrval2 20836 Interior expressed in terms of closure. (Contributed by NM, 1-Oct-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑆))))
 
Theoremntrdif 20837 An interior of a complement is the complement of the closure. This set is also known as the exterior of 𝐴. (Contributed by Jeff Hankins, 31-Aug-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘(𝑋𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘𝐴)))
 
Theoremclsdif 20838 A closure of a complement is the complement of the interior. (Contributed by Jeff Hankins, 31-Aug-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘(𝑋𝐴)) = (𝑋 ∖ ((int‘𝐽)‘𝐴)))
 
Theoremclsss 20839 Subset relationship for closure. (Contributed by NM, 10-Feb-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((cls‘𝐽)‘𝑇) ⊆ ((cls‘𝐽)‘𝑆))
 
Theoremntrss 20840 Subset relationship for interior. (Contributed by NM, 3-Oct-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((int‘𝐽)‘𝑇) ⊆ ((int‘𝐽)‘𝑆))
 
Theoremsscls 20841 A subset of a topology's underlying set is included in its closure. (Contributed by NM, 22-Feb-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
 
Theoremntrss2 20842 A subset includes its interior. (Contributed by NM, 3-Oct-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
 
Theoremssntr 20843 An open subset of a set is a subset of the set's interior. (Contributed by Jeff Hankins, 31-Aug-2009.) (Revised by Mario Carneiro, 11-Nov-2013.)
𝑋 = 𝐽       (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑂𝐽𝑂𝑆)) → 𝑂 ⊆ ((int‘𝐽)‘𝑆))
 
Theoremclsss3 20844 The closure of a subset of a topological space is included in the space. (Contributed by NM, 26-Feb-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
 
Theoremntrss3 20845 The interior of a subset of a topological space is included in the space. (Contributed by NM, 1-Oct-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) ⊆ 𝑋)
 
Theoremntrin 20846 A pairwise intersection of interiors is the interior of the intersection. This does not always hold for arbitrary intersections. (Contributed by Jeff Hankins, 31-Aug-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) = (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)))
 
Theoremcmclsopn 20847 The complement of a closure is open. (Contributed by NM, 11-Sep-2006.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
 
Theoremcmntrcld 20848 The complement of an interior is closed. (Contributed by NM, 1-Oct-2007.) (Proof shortened by OpenAI, 3-Jul-2020.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((int‘𝐽)‘𝑆)) ∈ (Clsd‘𝐽))
 
Theoremiscld3 20849 A subset is closed iff it equals its own closure. (Contributed by NM, 2-Oct-2006.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝑆) = 𝑆))
 
Theoremiscld4 20850 A subset is closed iff it contains its own closure. (Contributed by NM, 31-Jan-2008.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝑆) ⊆ 𝑆))
 
Theoremisopn3 20851 A subset is open iff it equals its own interior. (Contributed by NM, 9-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆))
 
Theoremclsidm 20852 The closure operation is idempotent. (Contributed by NM, 2-Oct-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘((cls‘𝐽)‘𝑆)) = ((cls‘𝐽)‘𝑆))
 
Theoremntridm 20853 The interior operation is idempotent. (Contributed by NM, 2-Oct-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘((int‘𝐽)‘𝑆)) = ((int‘𝐽)‘𝑆))
 
Theoremclstop 20854 The closure of a topology's underlying set is entire set. (Contributed by NM, 5-Oct-2007.)
𝑋 = 𝐽       (𝐽 ∈ Top → ((cls‘𝐽)‘𝑋) = 𝑋)
 
Theoremntrtop 20855 The interior of a topology's underlying set is entire set. (Contributed by NM, 12-Sep-2006.)
𝑋 = 𝐽       (𝐽 ∈ Top → ((int‘𝐽)‘𝑋) = 𝑋)
 
Theorem0ntr 20856 A subset with an empty interior cannot cover a whole (nonempty) topology. (Contributed by NM, 12-Sep-2006.)
𝑋 = 𝐽       (((𝐽 ∈ Top ∧ 𝑋 ≠ ∅) ∧ (𝑆𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → (𝑋𝑆) ≠ ∅)
 
Theoremclsss2 20857 If a subset is included in a closed set, so is the subset's closure. (Contributed by NM, 22-Feb-2007.)
𝑋 = 𝐽       ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆𝐶) → ((cls‘𝐽)‘𝑆) ⊆ 𝐶)
 
Theoremelcls 20858* Membership in a closure. Theorem 6.5(a) of [Munkres] p. 95. (Contributed by NM, 22-Feb-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋𝑃𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑥𝐽 (𝑃𝑥 → (𝑥𝑆) ≠ ∅)))
 
Theoremelcls2 20859* Membership in a closure. (Contributed by NM, 5-Mar-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ (𝑃𝑋 ∧ ∀𝑥𝐽 (𝑃𝑥 → (𝑥𝑆) ≠ ∅))))
 
Theoremclsndisj 20860 Any open set containing a point that belongs to the closure of a subset intersects the subset. One direction of Theorem 6.5(a) of [Munkres] p. 95. (Contributed by NM, 26-Feb-2007.)
𝑋 = 𝐽       (((𝐽 ∈ Top ∧ 𝑆𝑋𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑈𝐽𝑃𝑈)) → (𝑈𝑆) ≠ ∅)
 
Theoremntrcls0 20861 A subset whose closure has an empty interior also has an empty interior. (Contributed by NM, 4-Oct-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋 ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) = ∅) → ((int‘𝐽)‘𝑆) = ∅)
 
Theoremntreq0 20862* Two ways to say that a subset has an empty interior. (Contributed by NM, 3-Oct-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → (((int‘𝐽)‘𝑆) = ∅ ↔ ∀𝑥𝐽 (𝑥𝑆𝑥 = ∅)))
 
Theoremcldmre 20863 The closed sets of a topology comprise a Moore system on the points of the topology. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝑋 = 𝐽       (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘𝑋))
 
Theoremmrccls 20864 Moore closure generalizes closure in a topology. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘(Clsd‘𝐽))       (𝐽 ∈ Top → (cls‘𝐽) = 𝐹)
 
Theoremcls0 20865 The closure of the empty set. (Contributed by NM, 2-Oct-2007.)
(𝐽 ∈ Top → ((cls‘𝐽)‘∅) = ∅)
 
Theoremntr0 20866 The interior of the empty set. (Contributed by NM, 2-Oct-2007.)
(𝐽 ∈ Top → ((int‘𝐽)‘∅) = ∅)
 
Theoremisopn3i 20867 An open subset equals its own interior. (Contributed by Mario Carneiro, 30-Dec-2016.)
((𝐽 ∈ Top ∧ 𝑆𝐽) → ((int‘𝐽)‘𝑆) = 𝑆)
 
Theoremelcls3 20868* Membership in a closure in terms of the members of a basis. Theorem 6.5(b) of [Munkres] p. 95. (Contributed by NM, 26-Feb-2007.) (Revised by Mario Carneiro, 3-Sep-2015.)
(𝜑𝐽 = (topGen‘𝐵))    &   (𝜑𝑋 = 𝐽)    &   (𝜑𝐵 ∈ TopBases)    &   (𝜑𝑆𝑋)    &   (𝜑𝑃𝑋)       (𝜑 → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑥𝐵 (𝑃𝑥 → (𝑥𝑆) ≠ ∅)))
 
Theoremopncldf1 20869* A bijection useful for converting statements about open sets to statements about closed sets and vice versa. (Contributed by Jeff Hankins, 27-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
𝑋 = 𝐽    &   𝐹 = (𝑢𝐽 ↦ (𝑋𝑢))       (𝐽 ∈ Top → (𝐹:𝐽1-1-onto→(Clsd‘𝐽) ∧ 𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥))))
 
Theoremopncldf2 20870* The values of the open-closed bijection. (Contributed by Jeff Hankins, 27-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
𝑋 = 𝐽    &   𝐹 = (𝑢𝐽 ↦ (𝑋𝑢))       ((𝐽 ∈ Top ∧ 𝐴𝐽) → (𝐹𝐴) = (𝑋𝐴))
 
Theoremopncldf3 20871* The values of the converse/inverse of the open-closed bijection. (Contributed by Jeff Hankins, 27-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
𝑋 = 𝐽    &   𝐹 = (𝑢𝐽 ↦ (𝑋𝑢))       (𝐵 ∈ (Clsd‘𝐽) → (𝐹𝐵) = (𝑋𝐵))
 
Theoremisclo 20872* 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 20873* 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 20874 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 20875 The closed sets of the topology {∅}. (Contributed by FL, 5-Jan-2009.)
(Clsd‘{∅}) = {∅}
 
Theoremindiscld 20876 The closed sets of an indiscrete topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 14-Aug-2015.)
(Clsd‘{∅, 𝐴}) = {∅, 𝐴}
 
Theoremmretopd 20877* 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 20878 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 20780. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 5-May-2015.)
(𝐵𝑉 → (TopOn‘𝐵) ∈ (Moore‘𝒫 𝐵))
 
Theoremcldmreon 20879 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 20880* 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 20881 The closed subspaces of a topology-bearing module form a complete lattice. Demonstration for mreclatBAD 17168. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 6596 update): This proof uses the old df-clat 17089 and references the required instance of mreclatBAD 17168 as a hypothesis. When mreclatBAD 17168 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 20882 Extend class notation with neighborhood relation for topologies.
class nei
 
Definitiondf-nei 20883* 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 20884* 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 20885 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 20886 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 20887* 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 20888* 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 20889 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 20890* 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 20891 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 20892 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 20893* Property of a neighborhood. (Contributed by NM, 12-Feb-2007.)
((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑁))
 
Theoremneiss 20894 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 20895 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 20896 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 20897 The empty set is not a neighborhood of a nonempty set. (Contributed by FL, 18-Sep-2007.)
((𝐽 ∈ Top ∧ 𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
 
Theoremneips 20898* 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 20899 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 20900 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)))
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