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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | isopn2 21901 | 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 ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐽 ↔ (𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽))) | ||
Theorem | opncld 21902 | The complement of an open set is closed. (Contributed by NM, 6-Oct-2006.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → (𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽)) | ||
Theorem | difopn 21903 | The difference of a closed set with an open set is open. (Contributed by Mario Carneiro, 6-Jan-2014.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∖ 𝐵) ∈ 𝐽) | ||
Theorem | topcld 21904 | 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)) | ||
Theorem | ntrval 21905 | 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 ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = ∪ (𝐽 ∩ 𝒫 𝑆)) | ||
Theorem | clsval 21906* | 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 ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥}) | ||
Theorem | 0cld 21907 | 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)) | ||
Theorem | iincld 21908* | 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 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) | ||
Theorem | intcld 21909 | The intersection of a set of closed sets is closed. (Contributed by NM, 5-Oct-2006.) |
⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ (Clsd‘𝐽)) → ∩ 𝐴 ∈ (Clsd‘𝐽)) | ||
Theorem | uncld 21910 | 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)) → (𝐴 ∪ 𝐵) ∈ (Clsd‘𝐽)) | ||
Theorem | cldcls 21911 | A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.) |
⊢ (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆) | ||
Theorem | incld 21912 | 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)) → (𝐴 ∩ 𝐵) ∈ (Clsd‘𝐽)) | ||
Theorem | riincld 21913* | An indexed relative intersection of closed sets is closed. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) ∈ (Clsd‘𝐽)) | ||
Theorem | iuncld 21914* | A finite indexed union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) | ||
Theorem | unicld 21915 | 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)) | ||
Theorem | clscld 21916 | The closure of a subset of a topology's underlying set is closed. (Contributed by NM, 4-Oct-2006.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽)) | ||
Theorem | clsf 21917 | 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)) | ||
Theorem | ntropn 21918 | 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 ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ∈ 𝐽) | ||
Theorem | clsval2 21919 | Express closure in terms of interior. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ 𝑆)))) | ||
Theorem | ntrval2 21920 | Interior expressed in terms of closure. (Contributed by NM, 1-Oct-2007.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑆)))) | ||
Theorem | ntrdif 21921 | 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 ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘𝐴))) | ||
Theorem | clsdif 21922 | A closure of a complement is the complement of the interior. (Contributed by Jeff Hankins, 31-Aug-2009.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((int‘𝐽)‘𝐴))) | ||
Theorem | clsss 21923 | Subset relationship for closure. (Contributed by NM, 10-Feb-2007.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((cls‘𝐽)‘𝑇) ⊆ ((cls‘𝐽)‘𝑆)) | ||
Theorem | ntrss 21924 | Subset relationship for interior. (Contributed by NM, 3-Oct-2007.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((int‘𝐽)‘𝑇) ⊆ ((int‘𝐽)‘𝑆)) | ||
Theorem | sscls 21925 | A subset of a topology's underlying set is included in its closure. (Contributed by NM, 22-Feb-2007.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆)) | ||
Theorem | ntrss2 21926 | A subset includes its interior. (Contributed by NM, 3-Oct-2007.) (Revised by Mario Carneiro, 11-Nov-2013.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ⊆ 𝑆) | ||
Theorem | ssntr 21927 | 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 ∧ 𝑆 ⊆ 𝑋) ∧ (𝑂 ∈ 𝐽 ∧ 𝑂 ⊆ 𝑆)) → 𝑂 ⊆ ((int‘𝐽)‘𝑆)) | ||
Theorem | clsss3 21928 | The closure of a subset of a topological space is included in the space. (Contributed by NM, 26-Feb-2007.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋) | ||
Theorem | ntrss3 21929 | The interior of a subset of a topological space is included in the space. (Contributed by NM, 1-Oct-2007.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ⊆ 𝑋) | ||
Theorem | ntrin 21930 | 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 ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝐴 ∩ 𝐵)) = (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵))) | ||
Theorem | cmclsopn 21931 | The complement of a closure is open. (Contributed by NM, 11-Sep-2006.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽) | ||
Theorem | cmntrcld 21932 | The complement of an interior is closed. (Contributed by NM, 1-Oct-2007.) (Proof shortened by OpenAI, 3-Jul-2020.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ((int‘𝐽)‘𝑆)) ∈ (Clsd‘𝐽)) | ||
Theorem | iscld3 21933 | A subset is closed iff it equals its own closure. (Contributed by NM, 2-Oct-2006.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝑆) = 𝑆)) | ||
Theorem | iscld4 21934 | A subset is closed iff it contains its own closure. (Contributed by NM, 31-Jan-2008.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝑆) ⊆ 𝑆)) | ||
Theorem | isopn3 21935 | 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 ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆)) | ||
Theorem | clsidm 21936 | The closure operation is idempotent. (Contributed by NM, 2-Oct-2007.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘((cls‘𝐽)‘𝑆)) = ((cls‘𝐽)‘𝑆)) | ||
Theorem | ntridm 21937 | The interior operation is idempotent. (Contributed by NM, 2-Oct-2007.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘((int‘𝐽)‘𝑆)) = ((int‘𝐽)‘𝑆)) | ||
Theorem | clstop 21938 | 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) | ||
Theorem | ntrtop 21939 | 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) | ||
Theorem | 0ntr 21940 | A subset with an empty interior cannot cover a whole (nonempty) topology. (Contributed by NM, 12-Sep-2006.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (((𝐽 ∈ Top ∧ 𝑋 ≠ ∅) ∧ (𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘𝑆) = ∅)) → (𝑋 ∖ 𝑆) ≠ ∅) | ||
Theorem | clsss2 21941 | 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) ∧ 𝑆 ⊆ 𝐶) → ((cls‘𝐽)‘𝑆) ⊆ 𝐶) | ||
Theorem | elcls 21942* | Membership in a closure. Theorem 6.5(a) of [Munkres] p. 95. (Contributed by NM, 22-Feb-2007.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → (𝑥 ∩ 𝑆) ≠ ∅))) | ||
Theorem | elcls2 21943* | Membership in a closure. (Contributed by NM, 5-Mar-2007.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ (𝑃 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 → (𝑥 ∩ 𝑆) ≠ ∅)))) | ||
Theorem | clsndisj 21944 | 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 ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((cls‘𝐽)‘𝑆)) ∧ (𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈)) → (𝑈 ∩ 𝑆) ≠ ∅) | ||
Theorem | ntrcls0 21945 | A subset whose closure has an empty interior also has an empty interior. (Contributed by NM, 4-Oct-2007.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) = ∅) → ((int‘𝐽)‘𝑆) = ∅) | ||
Theorem | ntreq0 21946* | 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 ∧ 𝑆 ⊆ 𝑋) → (((int‘𝐽)‘𝑆) = ∅ ↔ ∀𝑥 ∈ 𝐽 (𝑥 ⊆ 𝑆 → 𝑥 = ∅))) | ||
Theorem | cldmre 21947 | 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)) | ||
Theorem | mrccls 21948 | Moore closure generalizes closure in a topology. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ 𝐹 = (mrCls‘(Clsd‘𝐽)) ⇒ ⊢ (𝐽 ∈ Top → (cls‘𝐽) = 𝐹) | ||
Theorem | cls0 21949 | The closure of the empty set. (Contributed by NM, 2-Oct-2007.) (Proof shortened by Jim Kingdon, 12-Mar-2023.) |
⊢ (𝐽 ∈ Top → ((cls‘𝐽)‘∅) = ∅) | ||
Theorem | ntr0 21950 | The interior of the empty set. (Contributed by NM, 2-Oct-2007.) |
⊢ (𝐽 ∈ Top → ((int‘𝐽)‘∅) = ∅) | ||
Theorem | isopn3i 21951 | An open subset equals its own interior. (Contributed by Mario Carneiro, 30-Dec-2016.) |
⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → ((int‘𝐽)‘𝑆) = 𝑆) | ||
Theorem | elcls3 21952* | 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.) |
⊢ (𝜑 → 𝐽 = (topGen‘𝐵)) & ⊢ (𝜑 → 𝑋 = ∪ 𝐽) & ⊢ (𝜑 → 𝐵 ∈ TopBases) & ⊢ (𝜑 → 𝑆 ⊆ 𝑋) & ⊢ (𝜑 → 𝑃 ∈ 𝑋) ⇒ ⊢ (𝜑 → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑥 ∈ 𝐵 (𝑃 ∈ 𝑥 → (𝑥 ∩ 𝑆) ≠ ∅))) | ||
Theorem | opncldf1 21953* | 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 = (𝑢 ∈ 𝐽 ↦ (𝑋 ∖ 𝑢)) ⇒ ⊢ (𝐽 ∈ Top → (𝐹:𝐽–1-1-onto→(Clsd‘𝐽) ∧ ◡𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥)))) | ||
Theorem | opncldf2 21954* | 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 = (𝑢 ∈ 𝐽 ↦ (𝑋 ∖ 𝑢)) ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (𝐹‘𝐴) = (𝑋 ∖ 𝐴)) | ||
Theorem | opncldf3 21955* | 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 = (𝑢 ∈ 𝐽 ↦ (𝑋 ∖ 𝑢)) ⇒ ⊢ (𝐵 ∈ (Clsd‘𝐽) → (◡𝐹‘𝐵) = (𝑋 ∖ 𝐵)) | ||
Theorem | isclo 21956* | 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 ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))) | ||
Theorem | isclo2 21957* | 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 ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴)))) | ||
Theorem | discld 21958 | 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.) |
⊢ (𝐴 ∈ 𝑉 → (Clsd‘𝒫 𝐴) = 𝒫 𝐴) | ||
Theorem | sn0cld 21959 | The closed sets of the topology {∅}. (Contributed by FL, 5-Jan-2009.) |
⊢ (Clsd‘{∅}) = {∅} | ||
Theorem | indiscld 21960 | The closed sets of an indiscrete topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ (Clsd‘{∅, 𝐴}) = {∅, 𝐴} | ||
Theorem | mretopd 21961* | 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.) |
⊢ (𝜑 → 𝑀 ∈ (Moore‘𝐵)) & ⊢ (𝜑 → ∅ ∈ 𝑀) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀) → (𝑥 ∪ 𝑦) ∈ 𝑀) & ⊢ 𝐽 = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ 𝑀} ⇒ ⊢ (𝜑 → (𝐽 ∈ (TopOn‘𝐵) ∧ 𝑀 = (Clsd‘𝐽))) | ||
Theorem | toponmre 21962 | 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 21864. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
⊢ (𝐵 ∈ 𝑉 → (TopOn‘𝐵) ∈ (Moore‘𝒫 𝐵)) | ||
Theorem | cldmreon 21963 | 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)) | ||
Theorem | iscldtop 21964* | 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 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾)) | ||
Theorem | mreclatdemoBAD 21965 | The closed subspaces of a topology-bearing module form a complete lattice. Demonstration for mreclatBAD 18041. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 7159 update): This proof uses the old df-clat 17977 and references the required instance of mreclatBAD 18041 as a hypothesis. When mreclatBAD 18041 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) | ||
Syntax | cnei 21966 | Extend class notation with neighborhood relation for topologies. |
class nei | ||
Definition | df-nei 21967* | 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 ∣ ∃𝑔 ∈ 𝑗 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦)})) | ||
Theorem | neifval 21968* | Value of the neighborhood function on the subsets of the base set of a topology. (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 ∣ ∃𝑔 ∈ 𝐽 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣)})) | ||
Theorem | neif 21969 | The neighborhood function is a function from the set of the subsets of the base set of a topology. (Contributed by NM, 12-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ Top → (nei‘𝐽) Fn 𝒫 𝑋) | ||
Theorem | neiss2 21970 | A set with a neighborhood is a subset of the base set of a topology. (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)) → 𝑆 ⊆ 𝑋) | ||
Theorem | neival 21971* | Value of 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 ∧ 𝑆 ⊆ 𝑋) → ((nei‘𝐽)‘𝑆) = {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣)}) | ||
Theorem | isnei 21972* | The predicate "the class 𝑁 is a neighborhood of 𝑆". (Contributed by FL, 25-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)))) | ||
Theorem | neiint 21973 | 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 ∧ 𝑆 ⊆ 𝑋 ∧ 𝑁 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑁))) | ||
Theorem | isneip 21974* | The predicate "the class 𝑁 is a neighborhood of point 𝑃". (Contributed by NM, 26-Feb-2007.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘{𝑃}) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑃 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)))) | ||
Theorem | neii1 21975 | A neighborhood is included in the topology's base set. (Contributed by NM, 12-Feb-2007.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁 ⊆ 𝑋) | ||
Theorem | neisspw 21976 | 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) | ||
Theorem | neii2 21977* | Property of a neighborhood. (Contributed by NM, 12-Feb-2007.) |
⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)) | ||
Theorem | neiss 21978 | Any neighborhood of a set 𝑆 is also a neighborhood of any subset 𝑅 ⊆ 𝑆. Similar to Proposition 1 of [BourbakiTop1] p. I.2. (Contributed by FL, 25-Sep-2006.) |
⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → 𝑁 ∈ ((nei‘𝐽)‘𝑅)) | ||
Theorem | ssnei 21979 | A set is included in any of its neighborhoods. Generalization to subsets of elnei 21980. (Contributed by FL, 16-Nov-2006.) |
⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑁) | ||
Theorem | elnei 21980 | A point belongs to any of its neighborhoods. Property Viii of [BourbakiTop1] p. I.3. (Contributed by FL, 28-Sep-2006.) |
⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝐴 ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → 𝑃 ∈ 𝑁) | ||
Theorem | 0nnei 21981 | The empty set is not a neighborhood of a nonempty set. (Contributed by FL, 18-Sep-2007.) |
⊢ ((𝐽 ∈ Top ∧ 𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆)) | ||
Theorem | neips 21982* | A neighborhood of a set is a neighborhood of every point in the set. Proposition 1 of [BourbakiTop1] p. I.2. (Contributed by FL, 16-Nov-2006.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ ∀𝑝 ∈ 𝑆 𝑁 ∈ ((nei‘𝐽)‘{𝑝}))) | ||
Theorem | opnneissb 21983 | An open set is a neighborhood of any of its subsets. (Contributed by FL, 2-Oct-2006.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑋) → (𝑆 ⊆ 𝑁 ↔ 𝑁 ∈ ((nei‘𝐽)‘𝑆))) | ||
Theorem | opnssneib 21984 | Any superset of an open set is a neighborhood of it. (Contributed by NM, 14-Feb-2007.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) → (𝑆 ⊆ 𝑁 ↔ 𝑁 ∈ ((nei‘𝐽)‘𝑆))) | ||
Theorem | ssnei2 21985 | Any subset 𝑀 of 𝑋 containing a neighborhood 𝑁 of a set 𝑆 is a neighborhood of this set. Generalization to subsets of Property Vi of [BourbakiTop1] p. I.3. (Contributed by FL, 2-Oct-2006.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑁 ⊆ 𝑀 ∧ 𝑀 ⊆ 𝑋)) → 𝑀 ∈ ((nei‘𝐽)‘𝑆)) | ||
Theorem | neindisj 21986 | 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 ∧ 𝑆 ⊆ 𝑋) ∧ (𝑃 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃}))) → (𝑁 ∩ 𝑆) ≠ ∅) | ||
Theorem | opnneiss 21987 | An open set is a neighborhood of any of its subsets. (Contributed by NM, 13-Feb-2007.) |
⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑁) → 𝑁 ∈ ((nei‘𝐽)‘𝑆)) | ||
Theorem | opnneip 21988 | An open set is a neighborhood of any of its members. (Contributed by NM, 8-Mar-2007.) |
⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ 𝐽 ∧ 𝑃 ∈ 𝑁) → 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) | ||
Theorem | opnnei 21989* | A set is open iff it is a neighborhood of all of its points. (Contributed by Jeff Hankins, 15-Sep-2009.) |
⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))) | ||
Theorem | tpnei 21990 | The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 21987. (Contributed by FL, 2-Oct-2006.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ Top → (𝑆 ⊆ 𝑋 ↔ 𝑋 ∈ ((nei‘𝐽)‘𝑆))) | ||
Theorem | neiuni 21991 | 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 ∧ 𝑆 ⊆ 𝑋) → 𝑋 = ∪ ((nei‘𝐽)‘𝑆)) | ||
Theorem | neindisj2 21992* | 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 ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑃})(𝑛 ∩ 𝑆) ≠ ∅)) | ||
Theorem | topssnei 21993 | A finer topology has more neighborhoods. (Contributed by Mario Carneiro, 9-Apr-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ 𝐽 ⊆ 𝐾) → ((nei‘𝐽)‘𝑆) ⊆ ((nei‘𝐾)‘𝑆)) | ||
Theorem | innei 21994 | The intersection of two neighborhoods of a set is also a neighborhood of the set. Generalization to subsets of Property Vii of [BourbakiTop1] p. I.3 for binary intersections. (Contributed by FL, 28-Sep-2006.) |
⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁 ∩ 𝑀) ∈ ((nei‘𝐽)‘𝑆)) | ||
Theorem | opnneiid 21995 | Only an open set is a neighborhood of itself. (Contributed by FL, 2-Oct-2006.) |
⊢ (𝐽 ∈ Top → (𝑁 ∈ ((nei‘𝐽)‘𝑁) ↔ 𝑁 ∈ 𝐽)) | ||
Theorem | neissex 21996* | For any neighborhood 𝑁 of 𝑆, there is a neighborhood 𝑥 of 𝑆 such that 𝑁 is a neighborhood of all subsets of 𝑥. Generalization to subsets of Property Viv of [BourbakiTop1] p. I.3. (Contributed by FL, 2-Oct-2006.) |
⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑥 ∈ ((nei‘𝐽)‘𝑆)∀𝑦(𝑦 ⊆ 𝑥 → 𝑁 ∈ ((nei‘𝐽)‘𝑦))) | ||
Theorem | 0nei 21997 | The empty set is a neighborhood of itself. (Contributed by FL, 10-Dec-2006.) |
⊢ (𝐽 ∈ Top → ∅ ∈ ((nei‘𝐽)‘∅)) | ||
Theorem | neipeltop 21998* | Lemma for neiptopreu 22002. (Contributed by Thierry Arnoux, 6-Jan-2018.) |
⊢ 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)} ⇒ ⊢ (𝐶 ∈ 𝐽 ↔ (𝐶 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝐶 𝐶 ∈ (𝑁‘𝑝))) | ||
Theorem | neiptopuni 21999* | Lemma for neiptopreu 22002. (Contributed by Thierry Arnoux, 6-Jan-2018.) |
⊢ 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)} & ⊢ (𝜑 → 𝑁:𝑋⟶𝒫 𝒫 𝑋) & ⊢ ((((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑏 ∈ (𝑁‘𝑝)) & ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑋) → (fi‘(𝑁‘𝑝)) ⊆ (𝑁‘𝑝)) & ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑝 ∈ 𝑎) & ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑏 ∈ (𝑁‘𝑝)∀𝑞 ∈ 𝑏 𝑎 ∈ (𝑁‘𝑞)) & ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝)) ⇒ ⊢ (𝜑 → 𝑋 = ∪ 𝐽) | ||
Theorem | neiptoptop 22000* | Lemma for neiptopreu 22002. (Contributed by Thierry Arnoux, 7-Jan-2018.) |
⊢ 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)} & ⊢ (𝜑 → 𝑁:𝑋⟶𝒫 𝒫 𝑋) & ⊢ ((((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑏 ∈ (𝑁‘𝑝)) & ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑋) → (fi‘(𝑁‘𝑝)) ⊆ (𝑁‘𝑝)) & ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑝 ∈ 𝑎) & ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑏 ∈ (𝑁‘𝑝)∀𝑞 ∈ 𝑏 𝑎 ∈ (𝑁‘𝑞)) & ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝)) ⇒ ⊢ (𝜑 → 𝐽 ∈ Top) |
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