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Theorem List for Metamath Proof Explorer - 21501-21600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtg2 21501* Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
((𝐴 ∈ (topGen‘𝐵) ∧ 𝐶𝐴) → ∃𝑥𝐵 (𝐶𝑥𝑥𝐴))
 
Theorembastg 21502 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 21503 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 21504 Subset relation for generated topologies. (Contributed by NM, 7-May-2007.)
((𝐶𝑉𝐵𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶))
 
Theoremtgcl 21505 Show that a basis generates a topology. Remark in [Munkres] p. 79. (Contributed by NM, 17-Jul-2006.)
(𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top)
 
Theoremtgclb 21506 The property tgcl 21505 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 21507 A basis generates a topology on 𝐵. (Contributed by Mario Carneiro, 14-Aug-2015.)
(𝐵 ∈ TopBases → (topGen‘𝐵) ∈ (TopOn‘ 𝐵))
 
Theoremtopbas 21508 A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
(𝐽 ∈ Top → 𝐽 ∈ TopBases)
 
Theoremtgtop 21509 A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
(𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
 
Theoremeltop 21510 Membership in a topology, expressed without quantifiers. (Contributed by NM, 19-Jul-2006.)
(𝐽 ∈ Top → (𝐴𝐽𝐴 (𝐽 ∩ 𝒫 𝐴)))
 
Theoremeltop2 21511* Membership in a topology. (Contributed by NM, 19-Jul-2006.)
(𝐽 ∈ Top → (𝐴𝐽 ↔ ∀𝑥𝐴𝑦𝐽 (𝑥𝑦𝑦𝐴)))
 
Theoremeltop3 21512* Membership in a topology. (Contributed by NM, 19-Jul-2006.)
(𝐽 ∈ Top → (𝐴𝐽 ↔ ∃𝑥(𝑥𝐽𝐴 = 𝑥)))
 
Theoremfibas 21513 A collection of finite intersections is a basis. The initial set is a subbasis for the topology. (Contributed by Jeff Hankins, 25-Aug-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
(fi‘𝐴) ∈ TopBases
 
Theoremtgdom 21514 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 21515* 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 21516 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 21517 Two ways to express that a basis is a topology. (Contributed by NM, 18-Jul-2006.)
(𝐵 ∈ TopBases → (𝐵 ∈ Top ↔ (topGen‘𝐵) = 𝐵))
 
Theoremtgtop11 21518 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)
 
Theorem0top 21519 The singleton of the empty set is the only topology possible for an empty underlying set. (Contributed by NM, 9-Sep-2006.)
(𝐽 ∈ Top → ( 𝐽 = ∅ ↔ 𝐽 = {∅}))
 
Theoremen1top 21520 {∅} is the only topology with one element. (Contributed by FL, 18-Aug-2008.)
(𝐽 ∈ Top → (𝐽 ≈ 1o𝐽 = {∅}))
 
Theoremen2top 21521 If a topology has two elements, it is the indiscrete topology. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
(𝐽 ∈ (TopOn‘𝑋) → (𝐽 ≈ 2o ↔ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)))
 
Theoremtgss3 21522 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 21523* 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 21524 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 21525* 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)
 
Theorem2basgen 21526 Conditions that determine the equality of two generated topologies. (Contributed by NM, 8-May-2007.) (Revised by Mario Carneiro, 2-Sep-2015.)
((𝐵𝐶𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = (topGen‘𝐶))
 
Theoremtgfiss 21527 If a subbase is included into a topology, so is the generated topology. (Contributed by FL, 20-Apr-2012.) (Proof shortened by Mario Carneiro, 10-Jan-2015.)
((𝐽 ∈ Top ∧ 𝐴𝐽) → (topGen‘(fi‘𝐴)) ⊆ 𝐽)
 
Theoremtgdif0 21528 A generated topology is not affected by the addition or removal of the empty set from the base. (Contributed by Mario Carneiro, 28-Aug-2015.)
(topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵)
 
Theorembastop1 21529* 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 21530* A version of bastop1 21529 that doesn't have 𝐵𝐽 in the antecedent. (Contributed by NM, 3-Feb-2008.)
(𝐽 ∈ Top → ((topGen‘𝐵) = 𝐽 ↔ (𝐵𝐽 ∧ ∀𝑥𝐽𝑦(𝑦𝐵𝑥 = 𝑦))))
 
12.1.3  Examples of topologies
 
Theoremdistop 21531 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 21532 The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 7470; an alternate proof uses indiscrete topologies (see indistop 21538) and the analogue of pwnex 7470 with pairs {∅, 𝑥} instead of power sets 𝒫 𝑥 (that analogue is also a consequence of abnex 7468). (Contributed by BJ, 2-May-2021.)
Top ∉ V
 
Theoremdistopon 21533 The discrete topology on a set 𝐴, with base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐴𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
 
Theoremsn0topon 21534 The singleton of the empty set is a topology on the empty set. (Contributed by Mario Carneiro, 13-Aug-2015.)
{∅} ∈ (TopOn‘∅)
 
Theoremsn0top 21535 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
 
Theoremindislem 21536 A lemma to eliminate some sethood hypotheses when dealing with the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)
{∅, ( I ‘𝐴)} = {∅, 𝐴}
 
Theoremindistopon 21537 The indiscrete topology on a set 𝐴. Part of Example 2 in [Munkres] p. 77. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐴𝑉 → {∅, 𝐴} ∈ (TopOn‘𝐴))
 
Theoremindistop 21538 The indiscrete topology on a set 𝐴. Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 16-Jul-2006.) (Revised by Stefan Allan, 6-Nov-2008.) (Revised by Mario Carneiro, 13-Aug-2015.)
{∅, 𝐴} ∈ Top
 
Theoremindisuni 21539 The base set of the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)
( I ‘𝐴) = {∅, 𝐴}
 
Theoremfctop 21540* The finite complement topology on a set 𝐴. Example 3 in [Munkres] p. 77. (Contributed by FL, 15-Aug-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)
(𝐴𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ∈ Fin ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))
 
Theoremfctop2 21541* The finite complement topology on a set 𝐴. Example 3 in [Munkres] p. 77. (This version of fctop 21540 requires the Axiom of Infinity.) (Contributed by FL, 20-Aug-2006.)
(𝐴𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≺ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))
 
Theoremcctop 21542* The countable complement topology on a set 𝐴. Example 4 in [Munkres] p. 77. (Contributed by FL, 23-Aug-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)
(𝐴𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))
 
Theoremppttop 21543* The particular point topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ∈ (TopOn‘𝐴))
 
Theorempptbas 21544* The particular point topology is generated by a basis consisting of pairs {𝑥, 𝑃} for each 𝑥𝐴. (Contributed by Mario Carneiro, 3-Sep-2015.)
((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} = (topGen‘ran (𝑥𝐴 ↦ {𝑥, 𝑃})))
 
Theoremepttop 21545* The excluded point topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ (TopOn‘𝐴))
 
Theoremindistpsx 21546 The indiscrete topology on a set 𝐴 expressed as a topological space, using explicit structure component references. Compare with indistps 21547 and indistps2 21548. The advantage of this version is that the actual function for the structure is evident, and df-ndx 16474 is not needed, nor any other special definition outside of basic set theory. The disadvantage is that if the indices of the component definitions df-base 16477 and df-tset 16572 are changed in the future, this theorem will also have to be changed. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use; use indistps 21547 instead. (New usage is discouraged.) (Contributed by FL, 19-Jul-2006.)
𝐴 ∈ V    &   𝐾 = {⟨1, 𝐴⟩, ⟨9, {∅, 𝐴}⟩}       𝐾 ∈ TopSp
 
Theoremindistps 21547 The indiscrete topology on a set 𝐴 expressed as a topological space, using implicit structure indices. The advantage of this version over indistpsx 21546 is that it is independent of the indices of the component definitions df-base 16477 and df-tset 16572, and if they are changed in the future, this theorem will not be affected. The advantage over indistps2 21548 is that it is easy to eliminate the hypotheses with eqid 2818 and vtoclg 3565 to result in a closed theorem. Theorems indistpsALT 21549 and indistps2ALT 21550 show that the two forms can be derived from each other. (Contributed by FL, 19-Jul-2006.)
𝐴 ∈ V    &   𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), {∅, 𝐴}⟩}       𝐾 ∈ TopSp
 
Theoremindistps2 21548 The indiscrete topology on a set 𝐴 expressed as a topological space, using direct component assignments. Compare with indistps 21547. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 21549 and indistps2ALT 21550 show that the two forms can be derived from each other. (Contributed by NM, 24-Oct-2012.)
(Base‘𝐾) = 𝐴    &   (TopOpen‘𝐾) = {∅, 𝐴}       𝐾 ∈ TopSp
 
TheoremindistpsALT 21549 The indiscrete topology on a set 𝐴 expressed as a topological space. Here we show how to derive the structural version indistps 21547 from the direct component assignment version indistps2 21548. (Contributed by NM, 24-Oct-2012.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ V    &   𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), {∅, 𝐴}⟩}       𝐾 ∈ TopSp
 
Theoremindistps2ALT 21550 The indiscrete topology on a set 𝐴 expressed as a topological space, using direct component assignments. Here we show how to derive the direct component assignment version indistps2 21548 from the structural version indistps 21547. (Contributed by NM, 24-Oct-2012.) (New usage is discouraged.) (Proof modification is discouraged.)
(Base‘𝐾) = 𝐴    &   (TopOpen‘𝐾) = {∅, 𝐴}       𝐾 ∈ TopSp
 
Theoremdistps 21551 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
 
12.1.4  Closure and interior
 
Syntaxccld 21552 Extend class notation with the set of closed sets of a topology.
class Clsd
 
Syntaxcnt 21553 Extend class notation with interior of a subset of a topology base set.
class int
 
Syntaxccl 21554 Extend class notation with closure of a subset of a topology base set.
class cls
 
Definitiondf-cld 21555* 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 21556* Define a function on topologies whose value is the interior function on the subsets of the base set. See ntrval 21572. (Contributed by NM, 10-Sep-2006.)
int = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 (𝑗 ∩ 𝒫 𝑥)))
 
Definitiondf-cls 21557* Define a function on topologies whose value is the closure function on the subsets of the base set. See clsval 21573. (Contributed by NM, 3-Oct-2006.)
cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦}))
 
Theoremfncld 21558 The closed-set generator is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Clsd Fn Top
 
Theoremcldval 21559* 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 21560* 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 21561* 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 21562 Reverse closure of the closed set operation. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
 
Theoremiscld 21563 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 21564 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 21565 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 21566 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 21567 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 21568 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 21569 The complement of an open set is closed. (Contributed by NM, 6-Oct-2006.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝐽) → (𝑋𝑆) ∈ (Clsd‘𝐽))
 
Theoremdifopn 21570 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 21571 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 21572 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 21573* 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 21574 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 21575* 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 21576 The intersection of a set of closed sets is closed. (Contributed by NM, 5-Oct-2006.)
((𝐴 ≠ ∅ ∧ 𝐴 ⊆ (Clsd‘𝐽)) → 𝐴 ∈ (Clsd‘𝐽))
 
Theoremuncld 21577 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 21578 A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.)
(𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆)
 
Theoremincld 21579 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 21580* 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 21581* 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 21582 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 21583 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 21584 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 21585 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 21586 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 21587 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 21588 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 21589 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 21590 Subset relationship for closure. (Contributed by NM, 10-Feb-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((cls‘𝐽)‘𝑇) ⊆ ((cls‘𝐽)‘𝑆))
 
Theoremntrss 21591 Subset relationship for interior. (Contributed by NM, 3-Oct-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((int‘𝐽)‘𝑇) ⊆ ((int‘𝐽)‘𝑆))
 
Theoremsscls 21592 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 21593 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 21594 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 21595 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 21596 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 21597 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 21598 The complement of a closure is open. (Contributed by NM, 11-Sep-2006.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
 
Theoremcmntrcld 21599 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 21600 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))
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