![]() |
Metamath
Proof Explorer Theorem List (p. 230 of 483) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-30741) |
![]() (30742-32264) |
![]() (32265-48238) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | en2top 22901 | 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 ≠ ∅))) | ||
Theorem | tgss3 22902 | 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.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶))) | ||
Theorem | tgss2 22903* | 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.) |
⊢ ((𝐵 ∈ 𝑉 ∧ ∪ 𝐵 = ∪ 𝐶) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ ∀𝑥 ∈ ∪ 𝐵∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐶 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)))) | ||
Theorem | basgen 22904 | 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 ∧ 𝐵 ⊆ 𝐽 ∧ 𝐽 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = 𝐽) | ||
Theorem | basgen2 22905* | 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 ∧ 𝐵 ⊆ 𝐽 ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)) → (topGen‘𝐵) = 𝐽) | ||
Theorem | 2basgen 22906 | Conditions that determine the equality of two generated topologies. (Contributed by NM, 8-May-2007.) (Revised by Mario Carneiro, 2-Sep-2015.) |
⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = (topGen‘𝐶)) | ||
Theorem | tgfiss 22907 | 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 ∧ 𝐴 ⊆ 𝐽) → (topGen‘(fi‘𝐴)) ⊆ 𝐽) | ||
Theorem | tgdif0 22908 | 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) | ||
Theorem | bastop1 22909* | 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 ∧ 𝐵 ⊆ 𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥 ∈ 𝐽 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦))) | ||
Theorem | bastop2 22910* | A version of bastop1 22909 that doesn't have 𝐵 ⊆ 𝐽 in the antecedent. (Contributed by NM, 3-Feb-2008.) |
⊢ (𝐽 ∈ Top → ((topGen‘𝐵) = 𝐽 ↔ (𝐵 ⊆ 𝐽 ∧ ∀𝑥 ∈ 𝐽 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))) | ||
Theorem | distop 22911 | 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.) |
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top) | ||
Theorem | topnex 22912 | The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 7761; an alternate proof uses indiscrete topologies (see indistop 22918) and the analogue of pwnex 7761 with pairs {∅, 𝑥} instead of power sets 𝒫 𝑥 (that analogue is also a consequence of abnex 7759). (Contributed by BJ, 2-May-2021.) |
⊢ Top ∉ V | ||
Theorem | distopon 22913 | 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 22914 | The singleton of the empty set is a topology on the empty set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ {∅} ∈ (TopOn‘∅) | ||
Theorem | sn0top 22915 | 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 22916 | 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 22917 | 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 22918 | 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 22919 | The base set of the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ ( I ‘𝐴) = ∪ {∅, 𝐴} | ||
Theorem | fctop 22920* | 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 22921* | The finite complement topology on a set 𝐴. Example 3 in [Munkres] p. 77. (This version of fctop 22920 requires the Axiom of Infinity.) (Contributed by FL, 20-Aug-2006.) |
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≺ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴)) | ||
Theorem | cctop 22922* | 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 22923* | 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 22924* | 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 22925* | 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 22926 | The indiscrete topology on a set 𝐴 expressed as a topological space, using explicit structure component references. Compare with indistps 22927 and indistps2 22928. The advantage of this version is that the actual function for the structure is evident, and df-ndx 17163 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 17181 and df-tset 17252 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 22927 instead. (New usage is discouraged.) (Contributed by FL, 19-Jul-2006.) |
⊢ 𝐴 ∈ V & ⊢ 𝐾 = {〈1, 𝐴〉, 〈9, {∅, 𝐴}〉} ⇒ ⊢ 𝐾 ∈ TopSp | ||
Theorem | indistps 22927 | The indiscrete topology on a set 𝐴 expressed as a topological space, using implicit structure indices. The advantage of this version over indistpsx 22926 is that it is independent of the indices of the component definitions df-base 17181 and df-tset 17252, and if they are changed in the future, this theorem will not be affected. The advantage over indistps2 22928 is that it is easy to eliminate the hypotheses with eqid 2728 and vtoclg 3540 to result in a closed theorem. Theorems indistpsALT 22929 and indistps2ALT 22931 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 22928 | The indiscrete topology on a set 𝐴 expressed as a topological space, using direct component assignments. Compare with indistps 22927. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 22929 and indistps2ALT 22931 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 22929 | The indiscrete topology on a set 𝐴 expressed as a topological space. Here we show how to derive the structural version indistps 22927 from the direct component assignment version indistps2 22928. (Contributed by NM, 24-Oct-2012.) (Revised by AV, 31-Oct-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ V & ⊢ 𝐾 = {〈(Base‘ndx), 𝐴〉, 〈(TopSet‘ndx), {∅, 𝐴}〉} ⇒ ⊢ 𝐾 ∈ TopSp | ||
Theorem | indistpsALTOLD 22930 | Obsolete version of indistpsALT 22929 as of 31-Oct-2024. The indiscrete topology on a set 𝐴 expressed as a topological space. Here we show how to derive the structural version indistps 22927 from the direct component assignment version indistps2 22928. (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 22931 | 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 22928 from the structural version indistps 22927. (Contributed by NM, 24-Oct-2012.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (Base‘𝐾) = 𝐴 & ⊢ (TopOpen‘𝐾) = {∅, 𝐴} ⇒ ⊢ 𝐾 ∈ TopSp | ||
Theorem | distps 22932 | 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 22933 | Extend class notation with the set of closed sets of a topology. |
class Clsd | ||
Syntax | cnt 22934 | Extend class notation with interior of a subset of a topology base set. |
class int | ||
Syntax | ccl 22935 | Extend class notation with closure of a subset of a topology base set. |
class cls | ||
Definition | df-cld 22936* | 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 22937* | Define a function on topologies whose value is the interior function on the subsets of the base set. See ntrval 22953. (Contributed by NM, 10-Sep-2006.) |
⊢ int = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∪ (𝑗 ∩ 𝒫 𝑥))) | ||
Definition | df-cls 22938* | Define a function on topologies whose value is the closure function on the subsets of the base set. See clsval 22954. (Contributed by NM, 3-Oct-2006.) |
⊢ cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∩ {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥 ⊆ 𝑦})) | ||
Theorem | fncld 22939 | The closed-set generator is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ Clsd Fn Top | ||
Theorem | cldval 22940* | 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 22941* | 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 22942* | 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 22943 | Reverse closure of the closed set operation. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | ||
Theorem | iscld 22944 | 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 22945 | 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 22946 | 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 22947 | 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 22948 | 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 22949 | 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 22950 | 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 22951 | 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 22952 | 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 22953 | 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 22954* | 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 22955 | 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 22956* | 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 22957 | 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 22958 | 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 22959 | A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.) |
⊢ (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆) | ||
Theorem | incld 22960 | 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 22961* | 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 22962* | 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 22963 | 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 22964 | 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 22965 | 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 22966 | 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 22967 | 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 22968 | 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 22969 | 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 22970 | 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 22971 | 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 22972 | 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 22973 | 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 22974 | 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 22975 | 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 22976 | 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 22977 | 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 22978 | 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 22979 | 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 22980 | 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 22981 | 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 22982 | 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 22983 | 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 22984 | 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 22985 | 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 22986 | 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 22987 | 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 22988 | 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 22989 | 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 22990* | 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 22991* | 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 22992 | 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 22993 | 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 22994* | 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 22995 | 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 22996 | 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 22997 | 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 22998 | The interior of the empty set. (Contributed by NM, 2-Oct-2007.) |
⊢ (𝐽 ∈ Top → ((int‘𝐽)‘∅) = ∅) | ||
Theorem | isopn3i 22999 | An open subset equals its own interior. (Contributed by Mario Carneiro, 30-Dec-2016.) |
⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → ((int‘𝐽)‘𝑆) = 𝑆) | ||
Theorem | elcls3 23000* | 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) ≠ ∅))) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |