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Theorem List for Metamath Proof Explorer - 20501-20600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem0top 20501 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 20502 {∅} is the only topology with one element. (Contributed by FL, 18-Aug-2008.)
(𝐽 ∈ Top → (𝐽 ≈ 1𝑜𝐽 = {∅}))
 
Theoremen2top 20503 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 20504 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 20505* 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 20506 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 20507* 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 20508 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 20509 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 20510 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 20511* 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 20512* A version of bastop1 20511 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 20513 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)
 
Theoremdistopon 20514 The discrete topology on a set 𝐴, with base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐴𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
 
Theoremsn0topon 20515 The singleton of the empty set is a topology on the empty set. (Contributed by Mario Carneiro, 13-Aug-2015.)
{∅} ∈ (TopOn‘∅)
 
Theoremsn0top 20516 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 20517 A lemma to eliminate some sethood hypotheses when dealing with the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)
{∅, ( I ‘𝐴)} = {∅, 𝐴}
 
Theoremindistopon 20518 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 20519 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 20520 The base set of the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)
( I ‘𝐴) = {∅, 𝐴}
 
Theoremfctop 20521* 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 20522* The finite complement topology on a set 𝐴. Example 3 in [Munkres] p. 77. (This version of fctop 20521 requires the Axiom of Infinity.) (Contributed by FL, 20-Aug-2006.)
(𝐴𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≺ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))
 
Theoremcctop 20523* 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 20524* The particular point topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ∈ (TopOn‘𝐴))
 
Theorempptbas 20525* 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 20526* The excluded point topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ (TopOn‘𝐴))
 
Theoremindistpsx 20527 The indiscrete topology on a set 𝐴 expressed as a topological space, using explicit structure component references. Compare with indistps 20528 and indistps2 20529. The advantage of this version is that the actual function for the structure is evident, and df-ndx 15582 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 15584 and df-tset 15671 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 20528 instead. (New usage is discouraged.) (Contributed by FL, 19-Jul-2006.)
𝐴 ∈ V    &   𝐾 = {⟨1, 𝐴⟩, ⟨9, {∅, 𝐴}⟩}       𝐾 ∈ TopSp
 
Theoremindistps 20528 The indiscrete topology on a set 𝐴 expressed as a topological space, using implicit structure indices. The advantage of this version over indistpsx 20527 is that it is independent of the indices of the component definitions df-base 15584 and df-tset 15671, and if they are changed in the future, this theorem will not be affected. The advantage over indistps2 20529 is that it is easy to eliminate the hypotheses with eqid 2514 and vtoclg 3143 to result in a closed theorem. Theorems indistpsALT 20530 and indistps2ALT 20531 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 20529 The indiscrete topology on a set 𝐴 expressed as a topological space, using direct component assignments. Compare with indistps 20528. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 20530 and indistps2ALT 20531 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 20530 The indiscrete topology on a set 𝐴 expressed as a topological space. Here we show how to derive the structural version indistps 20528 from the direct component assignment version indistps2 20529. (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 20531 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 20529 from the structural version indistps 20528. (Contributed by NM, 24-Oct-2012.) (New usage is discouraged.) (Proof modification is discouraged.)
(Base‘𝐾) = 𝐴    &   (TopOpen‘𝐾) = {∅, 𝐴}       𝐾 ∈ TopSp
 
Theoremdistps 20532 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 20533 Extend class notation with the set of closed sets of a topology.
class Clsd
 
Syntaxcnt 20534 Extend class notation with interior of a subset of a topology base set.
class int
 
Syntaxccl 20535 Extend class notation with closure of a subset of a topology base set.
class cls
 
Definitiondf-cld 20536* 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 20537* Define a function on topologies whose value is the interior function on the subsets of the base set. See ntrval 20553. (Contributed by NM, 10-Sep-2006.)
int = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 (𝑗 ∩ 𝒫 𝑥)))
 
Definitiondf-cls 20538* Define a function on topologies whose value is the closure function on the subsets of the base set. See clsval 20554. (Contributed by NM, 3-Oct-2006.)
cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦}))
 
Theoremfncld 20539 The closed-set generator is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Clsd Fn Top
 
Theoremcldval 20540* 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 20541* 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 20542* 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 20543 Reverse closure of the closed set operation. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
 
Theoremiscld 20544 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 20545 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 20546 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 20547 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 20548 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 20549 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 20550 The complement of an open set is closed. (Contributed by NM, 6-Oct-2006.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝐽) → (𝑋𝑆) ∈ (Clsd‘𝐽))
 
Theoremdifopn 20551 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 20552 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 20553 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 20554* 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 20555 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 20556* 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 20557 The intersection of a set of closed sets is closed. (Contributed by NM, 5-Oct-2006.)
((𝐴 ≠ ∅ ∧ 𝐴 ⊆ (Clsd‘𝐽)) → 𝐴 ∈ (Clsd‘𝐽))
 
Theoremuncld 20558 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 20559 A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.)
(𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆)
 
Theoremincld 20560 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 20561* 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 20562* 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 20563 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 20564 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 20565 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 20566 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 20567 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 20568 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 20569 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 20570 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 20571 Subset relationship for closure. (Contributed by NM, 10-Feb-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((cls‘𝐽)‘𝑇) ⊆ ((cls‘𝐽)‘𝑆))
 
Theoremntrss 20572 Subset relationship for interior. (Contributed by NM, 3-Oct-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((int‘𝐽)‘𝑇) ⊆ ((int‘𝐽)‘𝑆))
 
Theoremsscls 20573 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 20574 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 20575 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 20576 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 20577 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 20578 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 20579 The complement of a closure is open. (Contributed by NM, 11-Sep-2006.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
 
Theoremcmntrcld 20580 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 20581 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 20582 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 20583 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 20584 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 20585 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 20586 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 20587 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 20588 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 20589 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 20590* 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 20591* Membership in a closure. (Contributed by NM, 5-Mar-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ (𝑃𝑋 ∧ ∀𝑥𝐽 (𝑃𝑥 → (𝑥𝑆) ≠ ∅))))
 
Theoremclsndisj 20592 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 20593 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 20594* 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 20595 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 20596 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 20597 The closure of the empty set. (Contributed by NM, 2-Oct-2007.)
(𝐽 ∈ Top → ((cls‘𝐽)‘∅) = ∅)
 
Theoremntr0 20598 The interior of the empty set. (Contributed by NM, 2-Oct-2007.)
(𝐽 ∈ Top → ((int‘𝐽)‘∅) = ∅)
 
Theoremisopn3i 20599 An open subset equals its own interior. (Contributed by Mario Carneiro, 30-Dec-2016.)
((𝐽 ∈ Top ∧ 𝑆𝐽) → ((int‘𝐽)‘𝑆) = 𝑆)
 
Theoremelcls3 20600* 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) ≠ ∅)))
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