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Theorem List for Intuitionistic Logic Explorer - 11801-11900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfiinbas 11801* If a set is closed under finite intersection, then it is a basis for a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases)

Theorembaspartn 11802* A disjoint system of sets is a basis for a topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝑃𝑉 ∧ ∀𝑥𝑃𝑦𝑃 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)) → 𝑃 ∈ TopBases)

Theoremtgval 11803* The topology generated by a basis. See also tgval2 11805 and tgval3 11812. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})

Theoremtgvalex 11804 The topology generated by a basis is a set. (Contributed by Jim Kingdon, 4-Mar-2023.)
(𝐵𝑉 → (topGen‘𝐵) ∈ V)

Theoremtgval2 11805* Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 11818) that (topGen‘𝐵) is indeed a topology (on 𝐵, see unitg 11816). See also tgval 11803 and tgval3 11812. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (topGen‘𝐵) = {𝑥 ∣ (𝑥 𝐵 ∧ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))})

Theoremeltg 11806 Membership in a topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 (𝐵 ∩ 𝒫 𝐴)))

Theoremeltg2 11807* Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ (𝐴 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝑦𝐴))))

Theoremeltg2b 11808* Membership in a topology generated by a basis. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝑦𝐴)))

Theoremeltg4i 11809 An open set in a topology generated by a basis is the union of all basic open sets contained in it. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝐴 ∈ (topGen‘𝐵) → 𝐴 = (𝐵 ∩ 𝒫 𝐴))

Theoremeltg3i 11810 The union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 30-Aug-2015.)
((𝐵𝑉𝐴𝐵) → 𝐴 ∈ (topGen‘𝐵))

Theoremeltg3 11811* Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Jim Kingdon, 4-Mar-2023.)
(𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∃𝑥(𝑥𝐵𝐴 = 𝑥)))

Theoremtgval3 11812* Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80. See also tgval 11803 and tgval2 11805. (Contributed by NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.)
(𝐵𝑉 → (topGen‘𝐵) = {𝑥 ∣ ∃𝑦(𝑦𝐵𝑥 = 𝑦)})

Theoremtg1 11813 Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
(𝐴 ∈ (topGen‘𝐵) → 𝐴 𝐵)

Theoremtg2 11814* Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
((𝐴 ∈ (topGen‘𝐵) ∧ 𝐶𝐴) → ∃𝑥𝐵 (𝐶𝑥𝑥𝐴))

Theorembastg 11815 A member of a basis is a subset of the topology it generates. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉𝐵 ⊆ (topGen‘𝐵))

Theoremunitg 11816 The topology generated by a basis 𝐵 is a topology on 𝐵. Importantly, this theorem means that we don't have to specify separately the base set for the topological space generated by a basis. In other words, any member of the class TopBases completely specifies the basis it corresponds to. (Contributed by NM, 16-Jul-2006.) (Proof shortened by OpenAI, 30-Mar-2020.)
(𝐵𝑉 (topGen‘𝐵) = 𝐵)

Theoremtgss 11817 Subset relation for generated topologies. (Contributed by NM, 7-May-2007.)
((𝐶𝑉𝐵𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶))

Theoremtgcl 11818 Show that a basis generates a topology. Remark in [Munkres] p. 79. (Contributed by NM, 17-Jul-2006.)
(𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top)

Theoremtgclb 11819 The property tgcl 11818 can be reversed: if the topology generated by 𝐵 is actually a topology, then 𝐵 must be a topological basis. This yields an alternative definition of TopBases. (Contributed by Mario Carneiro, 2-Sep-2015.)
(𝐵 ∈ TopBases ↔ (topGen‘𝐵) ∈ Top)

Theoremtgtopon 11820 A basis generates a topology on 𝐵. (Contributed by Mario Carneiro, 14-Aug-2015.)
(𝐵 ∈ TopBases → (topGen‘𝐵) ∈ (TopOn‘ 𝐵))

Theoremtopbas 11821 A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
(𝐽 ∈ Top → 𝐽 ∈ TopBases)

Theoremtgtop 11822 A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
(𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)

Theoremeltop 11823 Membership in a topology, expressed without quantifiers. (Contributed by NM, 19-Jul-2006.)
(𝐽 ∈ Top → (𝐴𝐽𝐴 (𝐽 ∩ 𝒫 𝐴)))

Theoremeltop2 11824* Membership in a topology. (Contributed by NM, 19-Jul-2006.)
(𝐽 ∈ Top → (𝐴𝐽 ↔ ∀𝑥𝐴𝑦𝐽 (𝑥𝑦𝑦𝐴)))

Theoremeltop3 11825* Membership in a topology. (Contributed by NM, 19-Jul-2006.)
(𝐽 ∈ Top → (𝐴𝐽 ↔ ∃𝑥(𝑥𝐽𝐴 = 𝑥)))

Theoremtgdom 11826 A space has no more open sets than subsets of a basis. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 9-Apr-2015.)
(𝐵𝑉 → (topGen‘𝐵) ≼ 𝒫 𝐵)

Theoremtgiun 11827* The indexed union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
((𝐵𝑉 ∧ ∀𝑥𝐴 𝐶𝐵) → 𝑥𝐴 𝐶 ∈ (topGen‘𝐵))

Theoremtgidm 11828 The topology generator function is idempotent. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.)
(𝐵𝑉 → (topGen‘(topGen‘𝐵)) = (topGen‘𝐵))

Theorembastop 11829 Two ways to express that a basis is a topology. (Contributed by NM, 18-Jul-2006.)
(𝐵 ∈ TopBases → (𝐵 ∈ Top ↔ (topGen‘𝐵) = 𝐵))

Theoremtgtop11 11830 The topology generation function is one-to-one when applied to completed topologies. (Contributed by NM, 18-Jul-2006.)
((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ (topGen‘𝐽) = (topGen‘𝐾)) → 𝐽 = 𝐾)

Theoremen1top 11831 {∅} is the only topology with one element. (Contributed by FL, 18-Aug-2008.)
(𝐽 ∈ Top → (𝐽 ≈ 1o𝐽 = {∅}))

Theoremtgss3 11832 A criterion for determining whether one topology is finer than another. Lemma 2.2 of [Munkres] p. 80 using abbreviations. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
((𝐵𝑉𝐶𝑊) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶)))

Theoremtgss2 11833* A criterion for determining whether one topology is finer than another, based on a comparison of their bases. Lemma 2.2 of [Munkres] p. 80. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
((𝐵𝑉 𝐵 = 𝐶) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ ∀𝑥 𝐵𝑦𝐵 (𝑥𝑦 → ∃𝑧𝐶 (𝑥𝑧𝑧𝑦))))

Theorembasgen 11834 Given a topology 𝐽, show that a subset 𝐵 satisfying the third antecedent is a basis for it. Lemma 2.3 of [Munkres] p. 81 using abbreviations. (Contributed by NM, 22-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.)
((𝐽 ∈ Top ∧ 𝐵𝐽𝐽 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = 𝐽)

Theorembasgen2 11835* Given a topology 𝐽, show that a subset 𝐵 satisfying the third antecedent is a basis for it. Lemma 2.3 of [Munkres] p. 81. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
((𝐽 ∈ Top ∧ 𝐵𝐽 ∧ ∀𝑥𝐽𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)) → (topGen‘𝐵) = 𝐽)

Theorem2basgeng 11836 Conditions that determine the equality of two generated topologies. (Contributed by NM, 8-May-2007.) (Revised by Jim Kingdon, 5-Mar-2023.)
((𝐵𝑉𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = (topGen‘𝐶))

Theorembastop1 11837* A subset of a topology is a basis for the topology iff every member of the topology is a union of members of the basis. We use the idiom "(topGen‘𝐵) = 𝐽 " to express "𝐵 is a basis for topology 𝐽 " since we do not have a separate notation for this. Definition 15.35 of [Schechter] p. 428. (Contributed by NM, 2-Feb-2008.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
((𝐽 ∈ Top ∧ 𝐵𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥𝐽𝑦(𝑦𝐵𝑥 = 𝑦)))

Theorembastop2 11838* A version of bastop1 11837 that doesn't have 𝐵𝐽 in the antecedent. (Contributed by NM, 3-Feb-2008.)
(𝐽 ∈ Top → ((topGen‘𝐵) = 𝐽 ↔ (𝐵𝐽 ∧ ∀𝑥𝐽𝑦(𝑦𝐵𝑥 = 𝑦))))

6.1.3  Examples of topologies

Theoremdistop 11839 The discrete topology on a set 𝐴. Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 17-Jul-2006.) (Revised by Mario Carneiro, 19-Mar-2015.)
(𝐴𝑉 → 𝒫 𝐴 ∈ Top)

Theoremtopnex 11840 The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 4284. (Contributed by BJ, 2-May-2021.)
Top ∉ V

Theoremdistopon 11841 The discrete topology on a set 𝐴, with base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐴𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))

Theoremsn0topon 11842 The singleton of the empty set is a topology on the empty set. (Contributed by Mario Carneiro, 13-Aug-2015.)
{∅} ∈ (TopOn‘∅)

Theoremsn0top 11843 The singleton of the empty set is a topology. (Contributed by Stefan Allan, 3-Mar-2006.) (Proof shortened by Mario Carneiro, 13-Aug-2015.)
{∅} ∈ Top

Theoremepttop 11844* The excluded point topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ (TopOn‘𝐴))

Theoremdistps 11845 The discrete topology on a set 𝐴 expressed as a topological space. (Contributed by FL, 20-Aug-2006.)
𝐴 ∈ V    &   𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), 𝒫 𝐴⟩}       𝐾 ∈ TopSp

6.1.4  Closure and interior

Syntaxccld 11846 Extend class notation with the set of closed sets of a topology.
class Clsd

Syntaxcnt 11847 Extend class notation with interior of a subset of a topology base set.
class int

Syntaxccl 11848 Extend class notation with closure of a subset of a topology base set.
class cls

Definitiondf-cld 11849* 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 11850* Define a function on topologies whose value is the interior function on the subsets of the base set. See ntrval 11864. (Contributed by NM, 10-Sep-2006.)
int = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 (𝑗 ∩ 𝒫 𝑥)))

Definitiondf-cls 11851* Define a function on topologies whose value is the closure function on the subsets of the base set. See clsval 11865. (Contributed by NM, 3-Oct-2006.)
cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦}))

Theoremfncld 11852 The closed-set generator is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Clsd Fn Top

Theoremcldval 11853* 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 11854* 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 11855* 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 11856 Reverse closure of the closed set operation. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)

Theoremiscld 11857 The predicate "the class 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 11858 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 11859 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 11860 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 11861 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)

Theoremdifopn 11862 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 11863 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 11864 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 11865* 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 11866 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))

Theoremuncld 11867 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 11868 A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.)
(𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆)

Theoremiuncld 11869* A finite indexed union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.) (Revised by Jim Kingdon, 10-Mar-2023.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))

Theoremunicld 11870 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))

Theoremntropn 11871 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)

Theoremclsss 11872 Subset relationship for closure. (Contributed by NM, 10-Feb-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((cls‘𝐽)‘𝑇) ⊆ ((cls‘𝐽)‘𝑆))

Theoremntrss 11873 Subset relationship for interior. (Contributed by NM, 3-Oct-2007.) (Revised by Jim Kingdon, 11-Mar-2023.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((int‘𝐽)‘𝑇) ⊆ ((int‘𝐽)‘𝑆))

Theoremsscls 11874 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 11875 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 11876 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))

Theoremntrss3 11877 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 11878 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)))

Theoremisopn3 11879 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))

Theoremntridm 11880 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 11881 The closure of a topology's underlying set is the entire set. (Contributed by NM, 5-Oct-2007.) (Proof shortened by Jim Kingdon, 11-Mar-2023.)
𝑋 = 𝐽       (𝐽 ∈ Top → ((cls‘𝐽)‘𝑋) = 𝑋)

Theoremntrtop 11882 The interior of a topology's underlying set is the entire set. (Contributed by NM, 12-Sep-2006.)
𝑋 = 𝐽       (𝐽 ∈ Top → ((int‘𝐽)‘𝑋) = 𝑋)

Theoremclsss2 11883 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)

Theoremclsss3 11884 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)

Theoremntrcls0 11885 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 11886* Two ways to say that a subset has an empty interior. (Contributed by NM, 3-Oct-2007.) (Revised by Jim Kingdon, 11-Mar-2023.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → (((int‘𝐽)‘𝑆) = ∅ ↔ ∀𝑥𝐽 (𝑥𝑆𝑥 = ∅)))

Theoremcls0 11887 The closure of the empty set. (Contributed by NM, 2-Oct-2007.) (Proof shortened by Jim Kingdon, 12-Mar-2023.)
(𝐽 ∈ Top → ((cls‘𝐽)‘∅) = ∅)

Theoremntr0 11888 The interior of the empty set. (Contributed by NM, 2-Oct-2007.)
(𝐽 ∈ Top → ((int‘𝐽)‘∅) = ∅)

Theoremisopn3i 11889 An open subset equals its own interior. (Contributed by Mario Carneiro, 30-Dec-2016.)
((𝐽 ∈ Top ∧ 𝑆𝐽) → ((int‘𝐽)‘𝑆) = 𝑆)

Theoremdiscld 11890 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 11891 The closed sets of the topology {∅}. (Contributed by FL, 5-Jan-2009.)
(Clsd‘{∅}) = {∅}

6.2  Metric spaces

6.2.1  Topological definitions using the reals

Syntaxccncf 11892 Extend class notation to include the operation which returns a class of continuous complex functions.
class cn

Definitiondf-cncf 11893* Define the operation whose value is a class of continuous complex functions. (Contributed by Paul Chapman, 11-Oct-2007.)
cn→ = (𝑎 ∈ 𝒫 ℂ, 𝑏 ∈ 𝒫 ℂ ↦ {𝑓 ∈ (𝑏𝑚 𝑎) ∣ ∀𝑥𝑎𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑦𝑎 ((abs‘(𝑥𝑦)) < 𝑑 → (abs‘((𝑓𝑥) − (𝑓𝑦))) < 𝑒)})

Theoremcncfval 11894* The value of the continuous complex function operation is the set of continuous functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴cn𝐵) = {𝑓 ∈ (𝐵𝑚 𝐴) ∣ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)})

Theoremelcncf 11895* Membership in the set of continuous complex functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐴cn𝐵) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦))))

Theoremelcncf2 11896* Version of elcncf 11895 with arguments commuted. (Contributed by Mario Carneiro, 28-Apr-2014.)
((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐴cn𝐵) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑤𝑥)) < 𝑧 → (abs‘((𝐹𝑤) − (𝐹𝑥))) < 𝑦))))

Theoremcncfrss 11897 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐴 ⊆ ℂ)

Theoremcncfrss2 11898 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐵 ⊆ ℂ)

Theoremcncff 11899 A continuous complex function's domain and codomain. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐹:𝐴𝐵)

Theoremcncfi 11900* Defining property of a continuous function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 25-Aug-2014.)
((𝐹 ∈ (𝐴cn𝐵) ∧ 𝐶𝐴𝑅 ∈ ℝ+) → ∃𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑤𝐶)) < 𝑧 → (abs‘((𝐹𝑤) − (𝐹𝐶))) < 𝑅))

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