Home | Metamath
Proof Explorer Theorem List (p. 216 of 450) | < 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: | Metamath Proof Explorer
(1-28695) |
Hilbert Space Explorer
(28696-30218) |
Users' Mathboxes
(30219-44926) |
Type | Label | Description |
---|---|---|
Statement | ||
A topology on a set is a set of subsets of that set, called open sets, which satisfy certain conditions. One condition is that the whole set be an open set. Therefore, a set is recoverable from a topology on it (as its union, see toponuni 21522), and it may sometimes be more convenient to consider topologies without reference to the underlying set. This is why we define successively the class of topologies (df-top 21502), then the function which associates with a set the set of topologies on it (df-topon 21519), and finally the class of topological spaces, as extensible structures having an underlying set and a topology on it (df-topsp 21541). Of course, a topology is the same thing as a topology on a set (see toprntopon 21533). | ||
Syntax | ctop 21501 | Syntax for the class of topologies. |
class Top | ||
Definition | df-top 21502* |
Define the class of topologies. It is a proper class (see topnex 21604).
See istopg 21503 and istop2g 21504 for the corresponding characterizations,
using respectively binary intersections like in this definition and
nonempty finite intersections.
The final form of the definition is due to Bourbaki (Def. 1 of [BourbakiTop1] p. I.1), while the idea of defining a topology in terms of its open sets is due to Aleksandrov. For the convoluted history of the definitions of these notions, see Gregory H. Moore, The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica 35 (2008) 220--241. (Contributed by NM, 3-Mar-2006.) (Revised by BJ, 20-Oct-2018.) |
⊢ Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)} | ||
Theorem | istopg 21503* |
Express the predicate "𝐽 is a topology". See istop2g 21504 for another
characterization using nonempty finite intersections instead of binary
intersections.
Note: In the literature, a topology is often represented by a calligraphic letter T, which resembles the letter J. This confusion may have led to J being used by some authors (e.g., K. D. Joshi, Introduction to General Topology (1983), p. 114) and it is convenient for us since we later use 𝑇 to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
⊢ (𝐽 ∈ 𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽))) | ||
Theorem | istop2g 21504* | Express the predicate "𝐽 is a topology" using nonempty finite intersections instead of binary intersections as in istopg 21503. (Contributed by NM, 19-Jul-2006.) |
⊢ (𝐽 ∈ 𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥((𝑥 ⊆ 𝐽 ∧ 𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → ∩ 𝑥 ∈ 𝐽)))) | ||
Theorem | uniopn 21505 | The union of a subset of a topology (that is, the union of any family of open sets of a topology) is an open set. (Contributed by Stefan Allan, 27-Feb-2006.) |
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → ∪ 𝐴 ∈ 𝐽) | ||
Theorem | iunopn 21506* | The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.) |
⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) | ||
Theorem | inopn 21507 | The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.) |
⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∩ 𝐵) ∈ 𝐽) | ||
Theorem | fitop 21508 | A topology is closed under finite intersections. (Contributed by Jeff Hankins, 7-Oct-2009.) |
⊢ (𝐽 ∈ Top → (fi‘𝐽) = 𝐽) | ||
Theorem | fiinopn 21509 | The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.) |
⊢ (𝐽 ∈ Top → ((𝐴 ⊆ 𝐽 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → ∩ 𝐴 ∈ 𝐽)) | ||
Theorem | iinopn 21510* | The intersection of a nonempty finite family of open sets is open. (Contributed by Mario Carneiro, 14-Sep-2014.) |
⊢ ((𝐽 ∈ Top ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) | ||
Theorem | unopn 21511 | The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∪ 𝐵) ∈ 𝐽) | ||
Theorem | 0opn 21512 | The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.) |
⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | ||
Theorem | 0ntop 21513 | The empty set is not a topology. (Contributed by FL, 1-Jun-2008.) |
⊢ ¬ ∅ ∈ Top | ||
Theorem | topopn 21514 | The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) | ||
Theorem | eltopss 21515 | A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋) | ||
Theorem | riinopn 21516* | A finite indexed relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → (𝑋 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) ∈ 𝐽) | ||
Theorem | rintopn 21517 | A finite relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽 ∧ 𝐴 ∈ Fin) → (𝑋 ∩ ∩ 𝐴) ∈ 𝐽) | ||
Syntax | ctopon 21518 | Syntax for the function of topologies on sets. |
class TopOn | ||
Definition | df-topon 21519* | Define the function that associates with a set the set of topologies on it. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗}) | ||
Theorem | istopon 21520 | Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.) |
⊢ (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽)) | ||
Theorem | topontop 21521 | A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top) | ||
Theorem | toponuni 21522 | The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽) | ||
Theorem | topontopi 21523 | A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ 𝐽 ∈ (TopOn‘𝐵) ⇒ ⊢ 𝐽 ∈ Top | ||
Theorem | toponunii 21524 | The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ 𝐽 ∈ (TopOn‘𝐵) ⇒ ⊢ 𝐵 = ∪ 𝐽 | ||
Theorem | toptopon 21525 | Alternative definition of Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) | ||
Theorem | toptopon2 21526 | A topology is the same thing as a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.) |
⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | ||
Theorem | topontopon 21527 | A topology on a set is a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.) |
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ (TopOn‘∪ 𝐽)) | ||
Theorem | funtopon 21528 | The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.) |
⊢ Fun TopOn | ||
Theorem | toponrestid 21529 | Given a topology on a set, restricting it to that same set has no effect. (Contributed by Jim Kingdon, 6-Jul-2022.) |
⊢ 𝐴 ∈ (TopOn‘𝐵) ⇒ ⊢ 𝐴 = (𝐴 ↾t 𝐵) | ||
Theorem | toponsspwpw 21530 | The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.) |
⊢ (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴 | ||
Theorem | dmtopon 21531 | The domain of TopOn is the universal class V. (Contributed by BJ, 29-Apr-2021.) |
⊢ dom TopOn = V | ||
Theorem | fntopon 21532 | The class TopOn is a function with domain the universal class V. Analogue for topologies of fnmre 16862 for Moore collections. (Contributed by BJ, 29-Apr-2021.) |
⊢ TopOn Fn V | ||
Theorem | toprntopon 21533 | A topology is the same thing as a topology on a set (variable-free version). (Contributed by BJ, 27-Apr-2021.) |
⊢ Top = ∪ ran TopOn | ||
Theorem | toponmax 21534 | The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ 𝐽) | ||
Theorem | toponss 21535 | A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋) | ||
Theorem | toponcom 21536 | If 𝐾 is a topology on the base set of topology 𝐽, then 𝐽 is a topology on the base of 𝐾. (Contributed by Mario Carneiro, 22-Aug-2015.) |
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘∪ 𝐽)) → 𝐽 ∈ (TopOn‘∪ 𝐾)) | ||
Theorem | toponcomb 21537 | Biconditional form of toponcom 21536. (Contributed by BJ, 5-Dec-2021.) |
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘∪ 𝐾) ↔ 𝐾 ∈ (TopOn‘∪ 𝐽))) | ||
Theorem | topgele 21538 | The topologies over the same set have the greatest element (the discrete topology) and the least element (the indiscrete topology). (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 16-Sep-2015.) |
⊢ (𝐽 ∈ (TopOn‘𝑋) → ({∅, 𝑋} ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 𝑋)) | ||
Theorem | topsn 21539 | The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 4830). (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.) |
⊢ (𝐽 ∈ (TopOn‘{𝐴}) → 𝐽 = 𝒫 {𝐴}) | ||
Syntax | ctps 21540 | Syntax for the class of topological spaces. |
class TopSp | ||
Definition | df-topsp 21541 | Define the class of topological spaces (as extensible structures). (Contributed by Stefan O'Rear, 13-Aug-2015.) |
⊢ TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))} | ||
Theorem | istps 21542 | Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ 𝐴 = (Base‘𝐾) & ⊢ 𝐽 = (TopOpen‘𝐾) ⇒ ⊢ (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐴)) | ||
Theorem | istps2 21543 | Express the predicate "is a topological space." (Contributed by NM, 20-Oct-2012.) |
⊢ 𝐴 = (Base‘𝐾) & ⊢ 𝐽 = (TopOpen‘𝐾) ⇒ ⊢ (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = ∪ 𝐽)) | ||
Theorem | tpsuni 21544 | The base set of a topological space. (Contributed by FL, 27-Jun-2014.) |
⊢ 𝐴 = (Base‘𝐾) & ⊢ 𝐽 = (TopOpen‘𝐾) ⇒ ⊢ (𝐾 ∈ TopSp → 𝐴 = ∪ 𝐽) | ||
Theorem | tpstop 21545 | The topology extractor on a topological space is a topology. (Contributed by FL, 27-Jun-2014.) |
⊢ 𝐽 = (TopOpen‘𝐾) ⇒ ⊢ (𝐾 ∈ TopSp → 𝐽 ∈ Top) | ||
Theorem | tpspropd 21546 | A topological space depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 13-Aug-2015.) |
⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) & ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿)) ⇒ ⊢ (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp)) | ||
Theorem | tpsprop2d 21547 | A topological space depends only on the base and topology components. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) & ⊢ (𝜑 → (TopSet‘𝐾) = (TopSet‘𝐿)) ⇒ ⊢ (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp)) | ||
Theorem | topontopn 21548 | Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ 𝐴 = (Base‘𝐾) & ⊢ 𝐽 = (TopSet‘𝐾) ⇒ ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopOpen‘𝐾)) | ||
Theorem | tsettps 21549 | If the topology component is already correctly truncated, then it forms a topological space (with the topology extractor function coming out the same as the component). (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ 𝐴 = (Base‘𝐾) & ⊢ 𝐽 = (TopSet‘𝐾) ⇒ ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp) | ||
Theorem | istpsi 21550 | Properties that determine a topological space. (Contributed by NM, 20-Oct-2012.) |
⊢ (Base‘𝐾) = 𝐴 & ⊢ (TopOpen‘𝐾) = 𝐽 & ⊢ 𝐴 = ∪ 𝐽 & ⊢ 𝐽 ∈ Top ⇒ ⊢ 𝐾 ∈ TopSp | ||
Theorem | eltpsg 21551 | Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ 𝐾 = {〈(Base‘ndx), 𝐴〉, 〈(TopSet‘ndx), 𝐽〉} ⇒ ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp) | ||
Theorem | eltpsi 21552 | Properties that determine a topological space from a construction (using no explicit indices). (Contributed by NM, 20-Oct-2012.) (Revised by Mario Carneiro, 13-Aug-2015.) |
⊢ 𝐾 = {〈(Base‘ndx), 𝐴〉, 〈(TopSet‘ndx), 𝐽〉} & ⊢ 𝐴 = ∪ 𝐽 & ⊢ 𝐽 ∈ Top ⇒ ⊢ 𝐾 ∈ TopSp | ||
Syntax | ctb 21553 | Syntax for the class of topological bases. |
class TopBases | ||
Definition | df-bases 21554* | Define the class of topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 21556). Note that "bases" is the plural of "basis". (Contributed by NM, 17-Jul-2006.) |
⊢ TopBases = {𝑥 ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ⊆ ∪ (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧))} | ||
Theorem | isbasisg 21555* | Express the predicate "the set 𝐵 is a basis for a topology". (Contributed by NM, 17-Jul-2006.) |
⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))) | ||
Theorem | isbasis2g 21556* | Express the predicate "the set 𝐵 is a basis for a topology". (Contributed by NM, 17-Jul-2006.) |
⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))) | ||
Theorem | isbasis3g 21557* | Express the predicate "the set 𝐵 is a basis for a topology". Definition of basis in [Munkres] p. 78. (Contributed by NM, 17-Jul-2006.) |
⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ (∀𝑥 ∈ 𝐵 𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ ∪ 𝐵∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))) | ||
Theorem | basis1 21558 | Property of a basis. (Contributed by NM, 16-Jul-2006.) |
⊢ ((𝐵 ∈ TopBases ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐶 ∩ 𝐷) ⊆ ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝐷))) | ||
Theorem | basis2 21559* | Property of a basis. (Contributed by NM, 17-Jul-2006.) |
⊢ (((𝐵 ∈ TopBases ∧ 𝐶 ∈ 𝐵) ∧ (𝐷 ∈ 𝐵 ∧ 𝐴 ∈ (𝐶 ∩ 𝐷))) → ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))) | ||
Theorem | fiinbas 21560* | If a set is closed under finite intersection, then it is a basis for a topology. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝐵 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases) | ||
Theorem | basdif0 21561 | A basis is not affected by the addition or removal of the empty set. (Contributed by Mario Carneiro, 28-Aug-2015.) |
⊢ ((𝐵 ∖ {∅}) ∈ TopBases ↔ 𝐵 ∈ TopBases) | ||
Theorem | baspartn 21562* | A disjoint system of sets is a basis for a topology. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ ((𝑃 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) → 𝑃 ∈ TopBases) | ||
Theorem | tgval 21563* | The topology generated by a basis. See also tgval2 21564 and tgval3 21571. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.) |
⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)}) | ||
Theorem | tgval2 21564* | Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 21577) that (topGen‘𝐵) is indeed a topology (on ∪ 𝐵, see unitg 21575). See also tgval 21563 and tgval3 21571. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.) |
⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))}) | ||
Theorem | eltg 21565 | Membership in a topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴))) | ||
Theorem | eltg2 21566* | Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ (𝐴 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))) | ||
Theorem | eltg2b 21567* | Membership in a topology generated by a basis. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 10-Jan-2015.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))) | ||
Theorem | eltg4i 21568 | An open set in a topology generated by a basis is the union of all basic open sets contained in it. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐴 = ∪ (𝐵 ∩ 𝒫 𝐴)) | ||
Theorem | eltg3i 21569 | The union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 30-Aug-2015.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → ∪ 𝐴 ∈ (topGen‘𝐵)) | ||
Theorem | eltg3 21570* | Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Mario Carneiro, 30-Aug-2015.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 = ∪ 𝑥))) | ||
Theorem | tgval3 21571* | Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80. See also tgval 21563 and tgval2 21564. (Contributed by NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.) |
⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)}) | ||
Theorem | tg1 21572 | Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.) |
⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐴 ⊆ ∪ 𝐵) | ||
Theorem | tg2 21573* | Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.) |
⊢ ((𝐴 ∈ (topGen‘𝐵) ∧ 𝐶 ∈ 𝐴) → ∃𝑥 ∈ 𝐵 (𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) | ||
Theorem | bastg 21574 | 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.) |
⊢ (𝐵 ∈ 𝑉 → 𝐵 ⊆ (topGen‘𝐵)) | ||
Theorem | unitg 21575 | 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.) |
⊢ (𝐵 ∈ 𝑉 → ∪ (topGen‘𝐵) = ∪ 𝐵) | ||
Theorem | tgss 21576 | Subset relation for generated topologies. (Contributed by NM, 7-May-2007.) |
⊢ ((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶)) | ||
Theorem | tgcl 21577 | Show that a basis generates a topology. Remark in [Munkres] p. 79. (Contributed by NM, 17-Jul-2006.) |
⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top) | ||
Theorem | tgclb 21578 | The property tgcl 21577 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) | ||
Theorem | tgtopon 21579 | A basis generates a topology on ∪ 𝐵. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ (TopOn‘∪ 𝐵)) | ||
Theorem | topbas 21580 | A topology is its own basis. (Contributed by NM, 17-Jul-2006.) |
⊢ (𝐽 ∈ Top → 𝐽 ∈ TopBases) | ||
Theorem | tgtop 21581 | A topology is its own basis. (Contributed by NM, 18-Jul-2006.) |
⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) | ||
Theorem | eltop 21582 | Membership in a topology, expressed without quantifiers. (Contributed by NM, 19-Jul-2006.) |
⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ 𝐴 ⊆ ∪ (𝐽 ∩ 𝒫 𝐴))) | ||
Theorem | eltop2 21583* | Membership in a topology. (Contributed by NM, 19-Jul-2006.) |
⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))) | ||
Theorem | eltop3 21584* | Membership in a topology. (Contributed by NM, 19-Jul-2006.) |
⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∃𝑥(𝑥 ⊆ 𝐽 ∧ 𝐴 = ∪ 𝑥))) | ||
Theorem | fibas 21585 | 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 | ||
Theorem | tgdom 21586 | 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.) |
⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) ≼ 𝒫 𝐵) | ||
Theorem | tgiun 21587* | The indexed union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 2-Sep-2015.) |
⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐶 ∈ (topGen‘𝐵)) | ||
Theorem | tgidm 21588 | The topology generator function is idempotent. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.) |
⊢ (𝐵 ∈ 𝑉 → (topGen‘(topGen‘𝐵)) = (topGen‘𝐵)) | ||
Theorem | bastop 21589 | Two ways to express that a basis is a topology. (Contributed by NM, 18-Jul-2006.) |
⊢ (𝐵 ∈ TopBases → (𝐵 ∈ Top ↔ (topGen‘𝐵) = 𝐵)) | ||
Theorem | tgtop11 21590 | 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) | ||
Theorem | 0top 21591 | 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 = {∅})) | ||
Theorem | en1top 21592 | {∅} is the only topology with one element. (Contributed by FL, 18-Aug-2008.) |
⊢ (𝐽 ∈ Top → (𝐽 ≈ 1o ↔ 𝐽 = {∅})) | ||
Theorem | en2top 21593 | 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 21594 | 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 21595* | 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 21596 | 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 21597* | 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 21598 | 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 21599 | 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 21600 | 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) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |