Home Metamath Proof ExplorerTheorem List (p. 209 of 424) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27745) Hilbert Space Explorer (27746-29270) Users' Mathboxes (29271-42316)

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)))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42316
 Copyright terms: Public domain < Previous  Next >