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Theorem List for Intuitionistic Logic Explorer - 12201-12300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-pt 12201* Define the product topology on a collection of topologies. For convenience, it is defined on arbitrary collections of sets, expressed as a function from some index set to the subbases of each factor space. (Contributed by Mario Carneiro, 3-Feb-2015.)
t = (𝑓 ∈ V ↦ (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn dom 𝑓 ∧ ∀𝑦 ∈ dom 𝑓(𝑔𝑦) ∈ (𝑓𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓𝑧)(𝑔𝑦) = (𝑓𝑦)) ∧ 𝑥 = X𝑦 ∈ dom 𝑓(𝑔𝑦))}))

Syntaxcprds 12202 The function constructing structure products.
class Xs

Syntaxcpws 12203 The function constructing structure powers.
class s

Definitiondf-prds 12204* Define a structure product. This can be a product of groups, rings, modules, or ordered topological fields; any unused components will have garbage in them but this is usually not relevant for the purpose of inheriting the structures present in the factors. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.)
Xs = (𝑠 ∈ V, 𝑟 ∈ V ↦ X𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) / 𝑣(𝑓𝑣, 𝑔𝑣X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥))) / (({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(·𝑖‘(𝑟𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓𝑥)(le‘(𝑟𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐𝑣 ↦ (𝑑 ∈ (𝑐(2nd𝑎)), 𝑒 ∈ (𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑟𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩})))

Theoremreldmprds 12205 The structure product is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.)
Rel dom Xs

Definitiondf-pws 12206* Define a structure power, which is just a structure product where all the factors are the same. (Contributed by Mario Carneiro, 11-Jan-2015.)
s = (𝑟 ∈ V, 𝑖 ∈ V ↦ ((Scalar‘𝑟)Xs(𝑖 × {𝑟})))

6.2  The complex numbers as an algebraic extensible structure

6.2.1  Definition and basic properties

Syntaxcpsmet 12207 Extend class notation with the class of all pseudometric spaces.
class PsMet

Syntaxcxmet 12208 Extend class notation with the class of all extended metric spaces.
class ∞Met

Syntaxcmet 12209 Extend class notation with the class of all metrics.
class Met

Syntaxcbl 12210 Extend class notation with the metric space ball function.
class ball

Syntaxcfbas 12211 Extend class definition to include the class of filter bases.
class fBas

Syntaxcfg 12212 Extend class definition to include the filter generating function.
class filGen

Syntaxcmopn 12213 Extend class notation with a function mapping each metric space to the family of its open sets.
class MetOpen

Syntaxcmetu 12214 Extend class notation with the function mapping metrics to the uniform structure generated by that metric.
class metUnif

Definitiondf-psmet 12215* Define the set of all pseudometrics on a given base set. In a pseudo metric, two distinct points may have a distance zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧𝑥𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})

Definitiondf-xmet 12216* Define the set of all extended metrics on a given base set. The definition is similar to df-met 12217, but we also allow the metric to take on the value +∞. (Contributed by Mario Carneiro, 20-Aug-2015.)
∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})

Definitiondf-met 12217* Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric. However, we will often also call the metric itself a "metric space".) Equivalent to Definition 14-1.1 of [Gleason] p. 223. (Contributed by NM, 25-Aug-2006.)
Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))})

Definitiondf-bl 12218* Define the metric space ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Thierry Arnoux, 11-Feb-2018.)
ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑧 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑧}))

Definitiondf-mopn 12219 Define a function whose value is the family of open sets of a metric space. (Contributed by NM, 1-Sep-2006.)
MetOpen = (𝑑 ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))

Definitiondf-fbas 12220* Define the class of all filter bases. Note that a filter base on one set is also a filter base for any superset, so there is not a unique base set that can be recovered. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
fBas = (𝑤 ∈ V ↦ {𝑥 ∈ 𝒫 𝒫 𝑤 ∣ (𝑥 ≠ ∅ ∧ ∅ ∉ 𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑥 ∩ 𝒫 (𝑦𝑧)) ≠ ∅)})

Definitiondf-fg 12221* Define the filter generating function. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
filGen = (𝑤 ∈ V, 𝑥 ∈ (fBas‘𝑤) ↦ {𝑦 ∈ 𝒫 𝑤 ∣ (𝑥 ∩ 𝒫 𝑦) ≠ ∅})

Definitiondf-metu 12222* Define the function mapping metrics to the uniform structure generated by that metric. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
metUnif = (𝑑 ran PsMet ↦ ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (𝑑 “ (0[,)𝑎)))))

PART 7  BASIC TOPOLOGY

7.1  Topology

7.1.1  Topological spaces

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), and it may sometimes be more convenient to consider topologies without reference to the underlying set.

7.1.1.1  Topologies

Syntaxctop 12223 Syntax for the class of topologies.
class Top

Definitiondf-top 12224* Define the class of topologies. It is a proper class. See istopg 12225 and istopfin 12226 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 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}

Theoremistopg 12225* Express the predicate "𝐽 is a topology". See istopfin 12226 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.)

(𝐽𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))

Theoremistopfin 12226* Express the predicate "𝐽 is a topology" using nonempty finite intersections instead of binary intersections as in istopg 12225. It is not clear we can prove the converse without adding additional conditions. (Contributed by NM, 19-Jul-2006.) (Revised by Jim Kingdon, 14-Jan-2023.)
(𝐽 ∈ Top → (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥((𝑥𝐽𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → 𝑥𝐽)))

Theoremuniopn 12227 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 ∧ 𝐴𝐽) → 𝐴𝐽)

Theoremiunopn 12228* The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)
((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵𝐽) → 𝑥𝐴 𝐵𝐽)

Theoreminopn 12229 The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)

Theoremfiinopn 12230 The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)
(𝐽 ∈ Top → ((𝐴𝐽𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → 𝐴𝐽))

Theoremunopn 12231 The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)

Theorem0opn 12232 The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.)
(𝐽 ∈ Top → ∅ ∈ 𝐽)

Theorem0ntop 12233 The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)
¬ ∅ ∈ Top

Theoremtopopn 12234 The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
𝑋 = 𝐽       (𝐽 ∈ Top → 𝑋𝐽)

Theoremeltopss 12235 A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴𝑋)

7.1.1.2  Topologies on sets

Syntaxctopon 12236 Syntax for the function of topologies on sets.
class TopOn

Definitiondf-topon 12237* 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})

Theoremfuntopon 12238 The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.)
Fun TopOn

Theoremistopon 12239 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))

Theoremtopontop 12240 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)

Theoremtoponuni 12241 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOn‘𝐵) → 𝐵 = 𝐽)

Theoremtopontopi 12242 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐽 ∈ (TopOn‘𝐵)       𝐽 ∈ Top

Theoremtoponunii 12243 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐽 ∈ (TopOn‘𝐵)       𝐵 = 𝐽

Theoremtoptopon 12244 Alternative definition of Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝑋 = 𝐽       (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))

Theoremtoptopon2 12245 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))

Theoremtopontopon 12246 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))

Theoremtoponrestid 12247 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 = (𝐴t 𝐵)

Theoremtoponsspwpwg 12248 The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.) (Revised by Jim Kingdon, 16-Jan-2023.)
(𝐴𝑉 → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)

Theoremdmtopon 12249 The domain of TopOn is V. (Contributed by BJ, 29-Apr-2021.)
dom TopOn = V

Theoremfntopon 12250 The class TopOn is a function with domain V. (Contributed by BJ, 29-Apr-2021.)
TopOn Fn V

Theoremtoponmax 12251 The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOn‘𝐵) → 𝐵𝐽)

Theoremtoponss 12252 A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝐴𝑋)

Theoremtoponcom 12253 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))

Theoremtoponcomb 12254 Biconditional form of toponcom 12253. (Contributed by BJ, 5-Dec-2021.)
((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘ 𝐾) ↔ 𝐾 ∈ (TopOn‘ 𝐽)))

Theoremtopgele 12255 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} ⊆ 𝐽𝐽 ⊆ 𝒫 𝑋))

7.1.1.3  Topological spaces

Syntaxctps 12256 Syntax for the class of topological spaces.
class TopSp

Definitiondf-topsp 12257 Define the class of topological spaces (as extensible structures). (Contributed by Stefan O'Rear, 13-Aug-2015.)
TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}

Theoremistps 12258 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))

Theoremistps2 12259 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))

Theoremtpsuni 12260 The base set of a topological space. (Contributed by FL, 27-Jun-2014.)
𝐴 = (Base‘𝐾)    &   𝐽 = (TopOpen‘𝐾)       (𝐾 ∈ TopSp → 𝐴 = 𝐽)

Theoremtpstop 12261 The topology extractor on a topological space is a topology. (Contributed by FL, 27-Jun-2014.)
𝐽 = (TopOpen‘𝐾)       (𝐾 ∈ TopSp → 𝐽 ∈ Top)

Theoremtpspropd 12262 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))

Theoremtopontopn 12263 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))

Theoremtsettps 12264 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)

Theoremistpsi 12265 Properties that determine a topological space. (Contributed by NM, 20-Oct-2012.)
(Base‘𝐾) = 𝐴    &   (TopOpen‘𝐾) = 𝐽    &   𝐴 = 𝐽    &   𝐽 ∈ Top       𝐾 ∈ TopSp

Theoremeltpsg 12266 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)

Theoremeltpsi 12267 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

7.1.2  Topological bases

Syntaxctb 12268 Syntax for the class of topological bases.
class TopBases

Definitiondf-bases 12269* Define the class of topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 12271). Note that "bases" is the plural of "basis". (Contributed by NM, 17-Jul-2006.)
TopBases = {𝑥 ∣ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ⊆ (𝑥 ∩ 𝒫 (𝑦𝑧))}

Theoremisbasisg 12270* Express the predicate "the set 𝐵 is a basis for a topology". (Contributed by NM, 17-Jul-2006.)
(𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))

Theoremisbasis2g 12271* Express the predicate "the set 𝐵 is a basis for a topology". (Contributed by NM, 17-Jul-2006.)
(𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))

Theoremisbasis3g 12272* 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.)
(𝐵𝐶 → (𝐵 ∈ TopBases ↔ (∀𝑥𝐵 𝑥 𝐵 ∧ ∀𝑥 𝐵𝑦𝐵 𝑥𝑦 ∧ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))))

Theorembasis1 12273 Property of a basis. (Contributed by NM, 16-Jul-2006.)
((𝐵 ∈ TopBases ∧ 𝐶𝐵𝐷𝐵) → (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷)))

Theorembasis2 12274* Property of a basis. (Contributed by NM, 17-Jul-2006.)
(((𝐵 ∈ TopBases ∧ 𝐶𝐵) ∧ (𝐷𝐵𝐴 ∈ (𝐶𝐷))) → ∃𝑥𝐵 (𝐴𝑥𝑥 ⊆ (𝐶𝐷)))

Theoremfiinbas 12275* If a set is closed under finite intersection, then it is a basis for a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases)

Theorembaspartn 12276* A disjoint system of sets is a basis for a topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝑃𝑉 ∧ ∀𝑥𝑃𝑦𝑃 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)) → 𝑃 ∈ TopBases)

Theoremtgval 12277* The topology generated by a basis. See also tgval2 12279 and tgval3 12286. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})

Theoremtgvalex 12278 The topology generated by a basis is a set. (Contributed by Jim Kingdon, 4-Mar-2023.)
(𝐵𝑉 → (topGen‘𝐵) ∈ V)

Theoremtgval2 12279* Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 12292) that (topGen‘𝐵) is indeed a topology (on 𝐵, see unitg 12290). See also tgval 12277 and tgval3 12286. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (topGen‘𝐵) = {𝑥 ∣ (𝑥 𝐵 ∧ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))})

Theoremeltg 12280 Membership in a topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 (𝐵 ∩ 𝒫 𝐴)))

Theoremeltg2 12281* Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ (𝐴 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝑦𝐴))))

Theoremeltg2b 12282* Membership in a topology generated by a basis. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝑦𝐴)))

Theoremeltg4i 12283 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))

Theoremeltg3i 12284 The union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 30-Aug-2015.)
((𝐵𝑉𝐴𝐵) → 𝐴 ∈ (topGen‘𝐵))

Theoremeltg3 12285* Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Jim Kingdon, 4-Mar-2023.)
(𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∃𝑥(𝑥𝐵𝐴 = 𝑥)))

Theoremtgval3 12286* Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80. See also tgval 12277 and tgval2 12279. (Contributed by NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.)
(𝐵𝑉 → (topGen‘𝐵) = {𝑥 ∣ ∃𝑦(𝑦𝐵𝑥 = 𝑦)})

Theoremtg1 12287 Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
(𝐴 ∈ (topGen‘𝐵) → 𝐴 𝐵)

Theoremtg2 12288* Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
((𝐴 ∈ (topGen‘𝐵) ∧ 𝐶𝐴) → ∃𝑥𝐵 (𝐶𝑥𝑥𝐴))

Theorembastg 12289 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 12290 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 12291 Subset relation for generated topologies. (Contributed by NM, 7-May-2007.)
((𝐶𝑉𝐵𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶))

Theoremtgcl 12292 Show that a basis generates a topology. Remark in [Munkres] p. 79. (Contributed by NM, 17-Jul-2006.)
(𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top)

Theoremtgclb 12293 The property tgcl 12292 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 12294 A basis generates a topology on 𝐵. (Contributed by Mario Carneiro, 14-Aug-2015.)
(𝐵 ∈ TopBases → (topGen‘𝐵) ∈ (TopOn‘ 𝐵))

Theoremtopbas 12295 A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
(𝐽 ∈ Top → 𝐽 ∈ TopBases)

Theoremtgtop 12296 A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
(𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)

Theoremeltop 12297 Membership in a topology, expressed without quantifiers. (Contributed by NM, 19-Jul-2006.)
(𝐽 ∈ Top → (𝐴𝐽𝐴 (𝐽 ∩ 𝒫 𝐴)))

Theoremeltop2 12298* Membership in a topology. (Contributed by NM, 19-Jul-2006.)
(𝐽 ∈ Top → (𝐴𝐽 ↔ ∀𝑥𝐴𝑦𝐽 (𝑥𝑦𝑦𝐴)))

Theoremeltop3 12299* Membership in a topology. (Contributed by NM, 19-Jul-2006.)
(𝐽 ∈ Top → (𝐴𝐽 ↔ ∃𝑥(𝑥𝐽𝐴 = 𝑥)))

Theoremtgdom 12300 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)

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