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Type | Label | Description |
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Statement | ||
Theorem | topnex 21601 | The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 7461; an alternate proof uses indiscrete topologies (see indistop 21607) and the analogue of pwnex 7461 with pairs {∅, 𝑥} instead of power sets 𝒫 𝑥 (that analogue is also a consequence of abnex 7459). (Contributed by BJ, 2-May-2021.) |
⊢ Top ∉ V | ||
Theorem | distopon 21602 | The discrete topology on a set 𝐴, with base set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴)) | ||
Theorem | sn0topon 21603 | The singleton of the empty set is a topology on the empty set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ {∅} ∈ (TopOn‘∅) | ||
Theorem | sn0top 21604 | 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 | ||
Theorem | indislem 21605 | A lemma to eliminate some sethood hypotheses when dealing with the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴} | ||
Theorem | indistopon 21606 | The indiscrete topology on a set 𝐴. Part of Example 2 in [Munkres] p. 77. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ (𝐴 ∈ 𝑉 → {∅, 𝐴} ∈ (TopOn‘𝐴)) | ||
Theorem | indistop 21607 | 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 | ||
Theorem | indisuni 21608 | The base set of the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ ( I ‘𝐴) = ∪ {∅, 𝐴} | ||
Theorem | fctop 21609* | 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.) |
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ∈ Fin ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴)) | ||
Theorem | fctop2 21610* | The finite complement topology on a set 𝐴. Example 3 in [Munkres] p. 77. (This version of fctop 21609 requires the Axiom of Infinity.) (Contributed by FL, 20-Aug-2006.) |
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≺ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴)) | ||
Theorem | cctop 21611* | 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.) |
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴)) | ||
Theorem | ppttop 21612* | The particular point topology. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴)) | ||
Theorem | pptbas 21613* | The particular point topology is generated by a basis consisting of pairs {𝑥, 𝑃} for each 𝑥 ∈ 𝐴. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} = (topGen‘ran (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃}))) | ||
Theorem | epttop 21614* | The excluded point topology. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ∈ (TopOn‘𝐴)) | ||
Theorem | indistpsx 21615 | The indiscrete topology on a set 𝐴 expressed as a topological space, using explicit structure component references. Compare with indistps 21616 and indistps2 21617. The advantage of this version is that the actual function for the structure is evident, and df-ndx 16478 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 16481 and df-tset 16576 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 21616 instead. (New usage is discouraged.) (Contributed by FL, 19-Jul-2006.) |
⊢ 𝐴 ∈ V & ⊢ 𝐾 = {〈1, 𝐴〉, 〈9, {∅, 𝐴}〉} ⇒ ⊢ 𝐾 ∈ TopSp | ||
Theorem | indistps 21616 | The indiscrete topology on a set 𝐴 expressed as a topological space, using implicit structure indices. The advantage of this version over indistpsx 21615 is that it is independent of the indices of the component definitions df-base 16481 and df-tset 16576, and if they are changed in the future, this theorem will not be affected. The advantage over indistps2 21617 is that it is easy to eliminate the hypotheses with eqid 2798 and vtoclg 3515 to result in a closed theorem. Theorems indistpsALT 21618 and indistps2ALT 21619 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 | ||
Theorem | indistps2 21617 | The indiscrete topology on a set 𝐴 expressed as a topological space, using direct component assignments. Compare with indistps 21616. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 21618 and indistps2ALT 21619 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 | ||
Theorem | indistpsALT 21618 | The indiscrete topology on a set 𝐴 expressed as a topological space. Here we show how to derive the structural version indistps 21616 from the direct component assignment version indistps2 21617. (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 | ||
Theorem | indistps2ALT 21619 | 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 21617 from the structural version indistps 21616. (Contributed by NM, 24-Oct-2012.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (Base‘𝐾) = 𝐴 & ⊢ (TopOpen‘𝐾) = {∅, 𝐴} ⇒ ⊢ 𝐾 ∈ TopSp | ||
Theorem | distps 21620 | 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 | ||
Syntax | ccld 21621 | Extend class notation with the set of closed sets of a topology. |
class Clsd | ||
Syntax | cnt 21622 | Extend class notation with interior of a subset of a topology base set. |
class int | ||
Syntax | ccl 21623 | Extend class notation with closure of a subset of a topology base set. |
class cls | ||
Definition | df-cld 21624* | 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 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗}) | ||
Definition | df-ntr 21625* | Define a function on topologies whose value is the interior function on the subsets of the base set. See ntrval 21641. (Contributed by NM, 10-Sep-2006.) |
⊢ int = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∪ (𝑗 ∩ 𝒫 𝑥))) | ||
Definition | df-cls 21626* | Define a function on topologies whose value is the closure function on the subsets of the base set. See clsval 21642. (Contributed by NM, 3-Oct-2006.) |
⊢ cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∩ {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥 ⊆ 𝑦})) | ||
Theorem | fncld 21627 | The closed-set generator is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ Clsd Fn Top | ||
Theorem | cldval 21628* | 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 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽}) | ||
Theorem | ntrfval 21629* | 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))) | ||
Theorem | clsfval 21630* | 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) ∣ 𝑥 ⊆ 𝑦})) | ||
Theorem | cldrcl 21631 | Reverse closure of the closed set operation. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | ||
Theorem | iscld 21632 | 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) ↔ (𝑆 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽))) | ||
Theorem | iscld2 21633 | 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 ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ 𝑆) ∈ 𝐽)) | ||
Theorem | cldss 21634 | 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) → 𝑆 ⊆ 𝑋) | ||
Theorem | cldss2 21635 | 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 | ||
Theorem | cldopn 21636 | 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) → (𝑋 ∖ 𝑆) ∈ 𝐽) | ||
Theorem | isopn2 21637 | 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 21638 | 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 21639 | 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 21640 | 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 21641 | 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 21642* | 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 21643 | 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 21644* | 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 21645 | 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 21646 | 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 21647 | A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.) |
⊢ (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆) | ||
Theorem | incld 21648 | 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 21649* | 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 21650* | 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 21651 | 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 21652 | 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 21653 | 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 21654 | 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 21655 | 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 21656 | 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 21657 | 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 21658 | 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 21659 | 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 21660 | 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 21661 | 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 21662 | 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 21663 | 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 21664 | 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 21665 | 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 21666 | 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 21667 | 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 21668 | 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 21669 | 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 21670 | 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 21671 | 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 21672 | 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 21673 | 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 21674 | 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 21675 | 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 21676 | 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 21677 | 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 21678* | 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 21679* | 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 21680 | 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 21681 | 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 21682* | 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 21683 | 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 21684 | 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 21685 | 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 21686 | The interior of the empty set. (Contributed by NM, 2-Oct-2007.) |
⊢ (𝐽 ∈ Top → ((int‘𝐽)‘∅) = ∅) | ||
Theorem | isopn3i 21687 | An open subset equals its own interior. (Contributed by Mario Carneiro, 30-Dec-2016.) |
⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → ((int‘𝐽)‘𝑆) = 𝑆) | ||
Theorem | elcls3 21688* | 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 21689* | 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 21690* | 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 21691* | 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 21692* | 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 21693* | 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 21694 | 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 21695 | The closed sets of the topology {∅}. (Contributed by FL, 5-Jan-2009.) |
⊢ (Clsd‘{∅}) = {∅} | ||
Theorem | indiscld 21696 | 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 21697* | 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 21698 | 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 21600. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
⊢ (𝐵 ∈ 𝑉 → (TopOn‘𝐵) ∈ (Moore‘𝒫 𝐵)) | ||
Theorem | cldmreon 21699 | 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 21700* | 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)) |
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