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Theorem List for Metamath Proof Explorer - 22401-22500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremordthmeolem 22401 Lemma for ordthmeo 22402. (Contributed by Mario Carneiro, 9-Sep-2015.)
𝑋 = dom 𝑅    &   𝑌 = dom 𝑆       ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝐹 ∈ ((ordTop‘𝑅) Cn (ordTop‘𝑆)))

Theoremordthmeo 22402 An order isomorphism is a homeomorphism on the respective order topologies. (Contributed by Mario Carneiro, 9-Sep-2015.)
𝑋 = dom 𝑅    &   𝑌 = dom 𝑆       ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝐹 ∈ ((ordTop‘𝑅)Homeo(ordTop‘𝑆)))

Theoremtxhmeo 22403* Lift a pair of homeomorphisms on the factors to a homeomorphism of product topologies. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝑋 = 𝐽    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐽Homeo𝐿))    &   (𝜑𝐺 ∈ (𝐾Homeo𝑀))       (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩) ∈ ((𝐽 ×t 𝐾)Homeo(𝐿 ×t 𝑀)))

Theoremtxswaphmeolem 22404* Show inverse for the "swap components" operation on a Cartesian product. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)) = ( I ↾ (𝑋 × 𝑌))

Theoremtxswaphmeo 22405* There is a homeomorphism from 𝑋 × 𝑌 to 𝑌 × 𝑋. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ ((𝐽 ×t 𝐾)Homeo(𝐾 ×t 𝐽)))

Theorempt1hmeo 22406* The canonical homeomorphism from a topological product on a singleton to the topology of the factor. (Contributed by Mario Carneiro, 3-Feb-2015.) (Proof shortened by AV, 18-Apr-2021.)
𝐾 = (∏t‘{⟨𝐴, 𝐽⟩})    &   (𝜑𝐴𝑉)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))       (𝜑 → (𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽Homeo𝐾))

Theoremptuncnv 22407* Exhibit the converse function of the map 𝐺 which joins two product topologies on disjoint index sets. (Contributed by Mario Carneiro, 8-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
𝑋 = 𝐾    &   𝑌 = 𝐿    &   𝐽 = (∏t𝐹)    &   𝐾 = (∏t‘(𝐹𝐴))    &   𝐿 = (∏t‘(𝐹𝐵))    &   𝐺 = (𝑥𝑋, 𝑦𝑌 ↦ (𝑥𝑦))    &   (𝜑𝐶𝑉)    &   (𝜑𝐹:𝐶⟶Top)    &   (𝜑𝐶 = (𝐴𝐵))    &   (𝜑 → (𝐴𝐵) = ∅)       (𝜑𝐺 = (𝑧 𝐽 ↦ ⟨(𝑧𝐴), (𝑧𝐵)⟩))

Theoremptunhmeo 22408* Define a homeomorphism from a binary product of indexed product topologies to an indexed product topology on the union of the index sets. This is the topological analogue of (𝐴𝐵) · (𝐴𝐶) = 𝐴↑(𝐵 + 𝐶). (Contributed by Mario Carneiro, 8-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
𝑋 = 𝐾    &   𝑌 = 𝐿    &   𝐽 = (∏t𝐹)    &   𝐾 = (∏t‘(𝐹𝐴))    &   𝐿 = (∏t‘(𝐹𝐵))    &   𝐺 = (𝑥𝑋, 𝑦𝑌 ↦ (𝑥𝑦))    &   (𝜑𝐶𝑉)    &   (𝜑𝐹:𝐶⟶Top)    &   (𝜑𝐶 = (𝐴𝐵))    &   (𝜑 → (𝐴𝐵) = ∅)       (𝜑𝐺 ∈ ((𝐾 ×t 𝐿)Homeo𝐽))

Theoremxpstopnlem1 22409* The function 𝐹 used in xpsval 16835 is a homeomorphism from the binary product topology to the indexed product topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐹 = (𝑥𝑋, 𝑦𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))       (𝜑𝐹 ∈ ((𝐽 ×t 𝐾)Homeo(∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩})))

Theoremxpstps 22410 A binary product of topologies is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)       ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑇 ∈ TopSp)

Theoremxpstopnlem2 22411* Lemma for xpstopn 22412. (Contributed by Mario Carneiro, 27-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)    &   𝐽 = (TopOpen‘𝑅)    &   𝐾 = (TopOpen‘𝑆)    &   𝑂 = (TopOpen‘𝑇)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   𝐹 = (𝑥𝑋, 𝑦𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})       ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑂 = (𝐽 ×t 𝐾))

Theoremxpstopn 22412 The topology on a binary product of topological spaces, as we have defined it (transferring the indexed product topology on functions on {∅, 1o} to (𝑋 × 𝑌) by the canonical bijection), coincides with the usual topological product (generated by a base of rectangles). (Contributed by Mario Carneiro, 27-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)    &   𝐽 = (TopOpen‘𝑅)    &   𝐾 = (TopOpen‘𝑆)    &   𝑂 = (TopOpen‘𝑇)       ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑂 = (𝐽 ×t 𝐾))

Theoremptcmpfi 22413 A topological product of finitely many compact spaces is compact. This weak version of Tychonoff's theorem does not require the axiom of choice. (Contributed by Mario Carneiro, 8-Feb-2015.)
((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t𝐹) ∈ Comp)

Theoremxkocnv 22414* The inverse of the "currying" function 𝐹 is the uncurrying function. (Contributed by Mario Carneiro, 13-Apr-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   𝐹 = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↦ (𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑥𝑓𝑦))))    &   (𝜑𝐽 ∈ 𝑛-Locally Comp)    &   (𝜑𝐾 ∈ 𝑛-Locally Comp)    &   (𝜑𝐿 ∈ Top)       (𝜑𝐹 = (𝑔 ∈ (𝐽 Cn (𝐿ko 𝐾)) ↦ (𝑥𝑋, 𝑦𝑌 ↦ ((𝑔𝑥)‘𝑦))))

Theoremxkohmeo 22415* The Exponential Law for topological spaces. The "currying" function 𝐹 is a homeomorphism on function spaces when 𝐽 and 𝐾 are exponentiable spaces (by xkococn 22260, it is sufficient to assume that 𝐽, 𝐾 are locally compact to ensure exponentiability). (Contributed by Mario Carneiro, 13-Apr-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   𝐹 = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↦ (𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑥𝑓𝑦))))    &   (𝜑𝐽 ∈ 𝑛-Locally Comp)    &   (𝜑𝐾 ∈ 𝑛-Locally Comp)    &   (𝜑𝐿 ∈ Top)       (𝜑𝐹 ∈ ((𝐿ko (𝐽 ×t 𝐾))Homeo((𝐿ko 𝐾) ↑ko 𝐽)))

Theoremqtopf1 22416 If a quotient map is injective, then it is a homeomorphism. (Contributed by Mario Carneiro, 25-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹:𝑋1-1𝑌)       (𝜑𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)))

Theoremqtophmeo 22417* If two functions on a base topology 𝐽 make the same identifications in order to create quotient spaces 𝐽 qTop 𝐹 and 𝐽 qTop 𝐺, then not only are 𝐽 qTop 𝐹 and 𝐽 qTop 𝐺 homeomorphic, but there is a unique homeomorphism that makes the diagram commute. (Contributed by Mario Carneiro, 24-Mar-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹:𝑋onto𝑌)    &   (𝜑𝐺:𝑋onto𝑌)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐺𝑥) = (𝐺𝑦)))       (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓𝐹))

Theoremt0kq 22418* A topological space is T0 iff the quotient map is a homeomorphism onto the space's Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹 ∈ (𝐽Homeo(KQ‘𝐽))))

Theoremkqhmph 22419 A topological space is T0 iff it is homeomorphic to its Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
(𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽))

Theoremist1-5lem 22420 Lemma for ist1-5 22422 and similar theorems. If 𝐴 is a topological property which implies T0, such as T1 or T2, the property can be "decomposed" into T0 and a non-T0 version of property 𝐴 (which is defined as stating that the Kolmogorov quotient of the space has property 𝐴). For example, if 𝐴 is T1, then the theorem states that a space is T1 iff it is T0 and its Kolmogorov quotient is T1 (we call this property R0). (Contributed by Mario Carneiro, 25-Aug-2015.)
(𝐽𝐴𝐽 ∈ Kol2)    &   (𝐽 ≃ (KQ‘𝐽) → (𝐽𝐴 → (KQ‘𝐽) ∈ 𝐴))    &   ((KQ‘𝐽) ≃ 𝐽 → ((KQ‘𝐽) ∈ 𝐴𝐽𝐴))       (𝐽𝐴 ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴))

Theoremt1r0 22421 A T1 space is R0. That is, the Kolmogorov quotient of a T1 space is also T1 (because they are homeomorphic). (Contributed by Mario Carneiro, 25-Aug-2015.)
(𝐽 ∈ Fre → (KQ‘𝐽) ∈ Fre)

Theoremist1-5 22422 A topological space is T1 iff it is both T0 and R0. (Contributed by Mario Carneiro, 25-Aug-2015.)
(𝐽 ∈ Fre ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ Fre))

Theoremishaus3 22423 A topological space is Hausdorff iff it is both T0 and R1 (where R1 means that any two topologically distinct points are separated by neighborhoods). (Contributed by Mario Carneiro, 25-Aug-2015.)
(𝐽 ∈ Haus ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ Haus))

Theoremnrmreg 22424 A normal T1 space is regular Hausdorff. In other words, a T4 space is T3 . One can get away with slightly weaker assumptions; see nrmr0reg 22349. (Contributed by Mario Carneiro, 25-Aug-2015.)
((𝐽 ∈ Nrm ∧ 𝐽 ∈ Fre) → 𝐽 ∈ Reg)

Theoremreghaus 22425 A regular T0 space is Hausdorff. In other words, a T3 space is T2 . A regular Hausdorff or T0 space is also known as a T3 space. (Contributed by Mario Carneiro, 24-Aug-2015.)
(𝐽 ∈ Reg → (𝐽 ∈ Haus ↔ 𝐽 ∈ Kol2))

Theoremnrmhaus 22426 A T1 normal space is Hausdorff. A Hausdorff or T1 normal space is also known as a T4 space. (Contributed by Mario Carneiro, 24-Aug-2015.)
(𝐽 ∈ Nrm → (𝐽 ∈ Haus ↔ 𝐽 ∈ Fre))

12.2  Filters and filter bases

12.2.1  Filter bases

Theoremelmptrab 22427* Membership in a one-parameter class of sets. (Contributed by Stefan O'Rear, 28-Jul-2015.)
𝐹 = (𝑥𝐷 ↦ {𝑦𝐵𝜑})    &   ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑𝜓))    &   (𝑥 = 𝑋𝐵 = 𝐶)    &   (𝑥𝐷𝐵𝑉)       (𝑌 ∈ (𝐹𝑋) ↔ (𝑋𝐷𝑌𝐶𝜓))

Theoremelmptrab2 22428* Membership in a one-parameter class of sets, indexed by arbitrary base sets. (Contributed by Stefan O'Rear, 28-Jul-2015.) (Revised by AV, 26-Mar-2021.)
𝐹 = (𝑥 ∈ V ↦ {𝑦𝐵𝜑})    &   ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑𝜓))    &   (𝑥 = 𝑋𝐵 = 𝐶)    &   𝐵 ∈ V    &   (𝑌𝐶𝑋𝑊)       (𝑌 ∈ (𝐹𝑋) ↔ (𝑌𝐶𝜓))

Theoremisfbas 22429* The predicate "𝐹 is a filter base." Note that some authors require filter bases to be closed under pairwise intersections, but that is not necessary under our definition. One advantage of this definition is that tails in a directed set form a filter base under our meaning. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
(𝐵𝐴 → (𝐹 ∈ (fBas‘𝐵) ↔ (𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥𝐹𝑦𝐹 (𝐹 ∩ 𝒫 (𝑥𝑦)) ≠ ∅))))

Theoremfbasne0 22430 There are no empty filter bases. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
(𝐹 ∈ (fBas‘𝐵) → 𝐹 ≠ ∅)

Theorem0nelfb 22431 No filter base contains the empty set. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
(𝐹 ∈ (fBas‘𝐵) → ¬ ∅ ∈ 𝐹)

Theoremfbsspw 22432 A filter base on a set is a subset of the power set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
(𝐹 ∈ (fBas‘𝐵) → 𝐹 ⊆ 𝒫 𝐵)

Theoremfbelss 22433 An element of the filter base is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
((𝐹 ∈ (fBas‘𝐵) ∧ 𝑋𝐹) → 𝑋𝐵)

Theoremfbdmn0 22434 The domain of a filter base is nonempty. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
(𝐹 ∈ (fBas‘𝐵) → 𝐵 ≠ ∅)

Theoremisfbas2 22435* The predicate "𝐹 is a filter base." (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)
(𝐵𝐴 → (𝐹 ∈ (fBas‘𝐵) ↔ (𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥𝐹𝑦𝐹𝑧𝐹 𝑧 ⊆ (𝑥𝑦)))))

Theoremfbasssin 22436* A filter base contains subsets of its pairwise intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Jeff Hankins, 1-Dec-2010.)
((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵))

Theoremfbssfi 22437* A filter base contains subsets of its finite intersections. (Contributed by Mario Carneiro, 26-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ (fi‘𝐹)) → ∃𝑥𝐹 𝑥𝐴)

Theoremfbssint 22438* A filter base contains subsets of its finite intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)
((𝐹 ∈ (fBas‘𝐵) ∧ 𝐴𝐹𝐴 ∈ Fin) → ∃𝑥𝐹 𝑥 𝐴)

Theoremfbncp 22439 A filter base does not contain complements of its elements. (Contributed by Mario Carneiro, 26-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹) → ¬ (𝐵𝐴) ∈ 𝐹)

Theoremfbun 22440* A necessary and sufficient condition for the union of two filter bases to also be a filter base. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝐹𝐺) ∈ (fBas‘𝑋) ↔ ∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))

Theoremfbfinnfr 22441 No filter base containing a finite element is free. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)
((𝐹 ∈ (fBas‘𝐵) ∧ 𝑆𝐹𝑆 ∈ Fin) → 𝐹 ≠ ∅)

Theoremopnfbas 22442* The collection of open supersets of a nonempty set in a topology is a neighborhoods of the set, one of the motivations for the filter concept. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Aug-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋𝑆 ≠ ∅) → {𝑥𝐽𝑆𝑥} ∈ (fBas‘𝑋))

Theoremtrfbas2 22443 Conditions for the trace of a filter base 𝐹 to be a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)
((𝐹 ∈ (fBas‘𝑌) ∧ 𝐴𝑌) → ((𝐹t 𝐴) ∈ (fBas‘𝐴) ↔ ¬ ∅ ∈ (𝐹t 𝐴)))

Theoremtrfbas 22444* Conditions for the trace of a filter base 𝐹 to be a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)
((𝐹 ∈ (fBas‘𝑌) ∧ 𝐴𝑌) → ((𝐹t 𝐴) ∈ (fBas‘𝐴) ↔ ∀𝑣𝐹 (𝑣𝐴) ≠ ∅))

12.2.2  Filters

Syntaxcfil 22445 Extend class notation with the set of filters on a set.
class Fil

Definitiondf-fil 22446* The set of filters on a set. Definition 1 (axioms FI, FIIa, FIIb, FIII) of [BourbakiTop1] p. I.36. Filters are used to define the concept of limit in the general case. They are a generalization of the idea of neighborhoods. Suppose you are in . With neighborhoods you can express the idea of a variable that tends to a specific number but you can't express the idea of a variable that tends to infinity. Filters relax the "axioms" of neighborhoods and then succeed in expressing the idea of something that tends to infinity. Filters were invented by Cartan in 1937 and made famous by Bourbaki in his treatise. A notion similar to the notion of filter is the concept of net invented by Moore and Smith in 1922. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
Fil = (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)})

Theoremisfil 22447* The predicate "is a filter." (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.)
(𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹)))

Theoremfilfbas 22448 A filter is a filter base. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
(𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))

Theorem0nelfil 22449 The empty set doesn't belong to a filter. (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.)
(𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)

Theoremfileln0 22450 An element of a filter is nonempty. (Contributed by FL, 24-May-2011.) (Revised by Mario Carneiro, 28-Jul-2015.)
((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝐴 ≠ ∅)

Theoremfilsspw 22451 A filter is a subset of the power set of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
(𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)

Theoremfilelss 22452 An element of a filter is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝐴𝑋)

Theoremfilss 22453 A filter is closed under taking supersets. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐹𝐵𝑋𝐴𝐵)) → 𝐵𝐹)

Theoremfilin 22454 A filter is closed under taking intersections. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → (𝐴𝐵) ∈ 𝐹)

Theoremfiltop 22455 The underlying set belongs to the filter. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
(𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)

Theoremisfil2 22456* Derive the standard axioms of a filter. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
(𝐹 ∈ (Fil‘𝑋) ↔ ((𝐹 ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ 𝐹𝑋𝐹) ∧ ∀𝑥 ∈ 𝒫 𝑋(∃𝑦𝐹 𝑦𝑥𝑥𝐹) ∧ ∀𝑥𝐹𝑦𝐹 (𝑥𝑦) ∈ 𝐹))

Theoremisfildlem 22457* Lemma for isfild 22458. (Contributed by Mario Carneiro, 1-Dec-2013.)
(𝜑 → (𝑥𝐹 ↔ (𝑥𝐴𝜓)))    &   (𝜑𝐴 ∈ V)       (𝜑 → (𝐵𝐹 ↔ (𝐵𝐴[𝐵 / 𝑥]𝜓)))

Theoremisfild 22458* Sufficient condition for a set of the form {𝑥 ∈ 𝒫 𝐴𝜑} to be a filter. (Contributed by Mario Carneiro, 1-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
(𝜑 → (𝑥𝐹 ↔ (𝑥𝐴𝜓)))    &   (𝜑𝐴 ∈ V)    &   (𝜑[𝐴 / 𝑥]𝜓)    &   (𝜑 → ¬ [∅ / 𝑥]𝜓)    &   ((𝜑𝑦𝐴𝑧𝑦) → ([𝑧 / 𝑥]𝜓[𝑦 / 𝑥]𝜓))    &   ((𝜑𝑦𝐴𝑧𝐴) → (([𝑦 / 𝑥]𝜓[𝑧 / 𝑥]𝜓) → [(𝑦𝑧) / 𝑥]𝜓))       (𝜑𝐹 ∈ (Fil‘𝐴))

Theoremfilfi 22459 A filter is closed under taking intersections. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
(𝐹 ∈ (Fil‘𝑋) → (fi‘𝐹) = 𝐹)

Theoremfilinn0 22460 The intersection of two elements of a filter can't be empty. (Contributed by FL, 16-Sep-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → (𝐴𝐵) ≠ ∅)

Theoremfilintn0 22461 A filter has the finite intersection property. Remark below Definition 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 20-Sep-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐹𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐴 ≠ ∅)

Theoremfiln0 22462 The empty set is not a filter. Remark below Definition 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 30-Oct-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
(𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)

Theoreminfil 22463 The intersection of two filters is a filter. Use fiint 8787 to extend this property to the intersection of a finite set of filters. Paragraph 3 of [BourbakiTop1] p. I.36. (Contributed by FL, 17-Sep-2007.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝐹𝐺) ∈ (Fil‘𝑋))

Theoremsnfil 22464 A singleton is a filter. Example 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 16-Sep-2007.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝐴𝐵𝐴 ≠ ∅) → {𝐴} ∈ (Fil‘𝐴))

Theoremfbasweak 22465 A filter base on any set is also a filter base on any larger set. (Contributed by Stefan O'Rear, 2-Aug-2015.)
((𝐹 ∈ (fBas‘𝑋) ∧ 𝐹 ⊆ 𝒫 𝑌𝑌𝑉) → 𝐹 ∈ (fBas‘𝑌))

Theoremsnfbas 22466 Condition for a singleton to be a filter base. (Contributed by Stefan O'Rear, 2-Aug-2015.)
((𝐴𝐵𝐴 ≠ ∅ ∧ 𝐵𝑉) → {𝐴} ∈ (fBas‘𝐵))

Theoremfsubbas 22467 A condition for a set to generate a filter base. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
(𝑋𝑉 → ((fi‘𝐴) ∈ (fBas‘𝑋) ↔ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))))

Theoremfbasfip 22468 A filter base has the finite intersection property. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
(𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ (fi‘𝐹))

Theoremfbunfip 22469* A helpful lemma for showing that certain sets generate filters. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (¬ ∅ ∈ (fi‘(𝐹𝐺)) ↔ ∀𝑥𝐹𝑦𝐺 (𝑥𝑦) ≠ ∅))

Theoremfgval 22470* The filter generating class gives a filter for every filter base. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
(𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅})

Theoremelfg 22471* A condition for elements of a generated filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
(𝐹 ∈ (fBas‘𝑋) → (𝐴 ∈ (𝑋filGen𝐹) ↔ (𝐴𝑋 ∧ ∃𝑥𝐹 𝑥𝐴)))

Theoremssfg 22472 A filter base is a subset of its generated filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
(𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))

Theoremfgss 22473 A bigger base generates a bigger filter. (Contributed by NM, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋) ∧ 𝐹𝐺) → (𝑋filGen𝐹) ⊆ (𝑋filGen𝐺))

Theoremfgss2 22474* A condition for a filter to be finer than another involving their filter bases. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) ↔ ∀𝑥𝐹𝑦𝐺 𝑦𝑥))

Theoremfgfil 22475 A filter generates itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
(𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)

Theoremelfilss 22476* An element belongs to a filter iff any element below it does. (Contributed by Stefan O'Rear, 2-Aug-2015.)
((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝑋) → (𝐴𝐹 ↔ ∃𝑡𝐹 𝑡𝐴))

Theoremfilfinnfr 22477 No filter containing a finite element is free. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝐹 ∈ (Fil‘𝑋) ∧ 𝑆𝐹𝑆 ∈ Fin) → 𝐹 ≠ ∅)

Theoremfgcl 22478 A generated filter is a filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
(𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))

Theoremfgabs 22479 Absorption law for filter generation. (Contributed by Mario Carneiro, 15-Oct-2015.)
((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌𝑋) → (𝑋filGen(𝑌filGen𝐹)) = (𝑋filGen𝐹))

Theoremneifil 22480 The neighborhoods of a nonempty set is a filter. Example 2 of [BourbakiTop1] p. I.36. (Contributed by FL, 18-Sep-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ((nei‘𝐽)‘𝑆) ∈ (Fil‘𝑋))

Theoremfilunibas 22481 Recover the base set from a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.)
(𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)

Theoremfilunirn 22482 Two ways to express a filter on an unspecified base. (Contributed by Stefan O'Rear, 2-Aug-2015.)
(𝐹 ran Fil ↔ 𝐹 ∈ (Fil‘ 𝐹))

Theoremfilconn 22483 A filter gives rise to a connected topology. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
(𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {∅}) ∈ Conn)

Theoremfbasrn 22484* Given a filter on a domain, produce a filter on the range. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
𝐶 = ran (𝑥𝐵 ↦ (𝐹𝑥))       ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → 𝐶 ∈ (fBas‘𝑌))

Theoremfiluni 22485* The union of a nonempty set of filters with a common base and closed under pairwise union is a filter. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅ ∧ ∀𝑓𝐹𝑔𝐹 (𝑓𝑔) ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))

Theoremtrfil1 22486 Conditions for the trace of a filter 𝐿 to be a filter. (Contributed by FL, 2-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → 𝐴 = (𝐿t 𝐴))

Theoremtrfil2 22487* Conditions for the trace of a filter 𝐿 to be a filter. (Contributed by FL, 2-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → ((𝐿t 𝐴) ∈ (Fil‘𝐴) ↔ ∀𝑣𝐿 (𝑣𝐴) ≠ ∅))

Theoremtrfil3 22488 Conditions for the trace of a filter 𝐿 to be a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.)
((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → ((𝐿t 𝐴) ∈ (Fil‘𝐴) ↔ ¬ (𝑌𝐴) ∈ 𝐿))

Theoremtrfilss 22489 If 𝐴 is a member of the filter, then the filter truncated to 𝐴 is a subset of the original filter. (Contributed by Mario Carneiro, 15-Oct-2015.)
((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) ⊆ 𝐹)

Theoremfgtr 22490 If 𝐴 is a member of the filter, then truncating 𝐹 to 𝐴 and regenerating the behavior outside 𝐴 using filGen recovers the original filter. (Contributed by Mario Carneiro, 15-Oct-2015.)
((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑋filGen(𝐹t 𝐴)) = 𝐹)

Theoremtrfg 22491 The trace operation and the filGen operation are inverses to one another in some sense, with filGen growing the base set and t shrinking it. See fgtr 22490 for the converse cancellation law. (Contributed by Mario Carneiro, 15-Oct-2015.)
((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴𝑋𝑋𝑉) → ((𝑋filGen𝐹) ↾t 𝐴) = 𝐹)

Theoremtrnei 22492 The trace, over a set 𝐴, of the filter of the neighborhoods of a point 𝑃 is a filter iff 𝑃 belongs to the closure of 𝐴. (This is trfil2 22487 applied to a filter of neighborhoods.) (Contributed by FL, 15-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑃}) ↾t 𝐴) ∈ (Fil‘𝐴)))

Theoremcfinfil 22493* Relative complements of the finite parts of an infinite set is a filter. When 𝐴 = ℕ the set of the relative complements is called Frechet's filter and is used to define the concept of limit of a sequence. (Contributed by FL, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ∈ (Fil‘𝑋))

Theoremcsdfil 22494* The set of all elements whose complement is dominated by the base set is a filter. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ≺ 𝑋} ∈ (Fil‘𝑋))

Theoremsupfil 22495* The supersets of a nonempty set which are also subsets of a given base set form a filter. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
((𝐴𝑉𝐵𝐴𝐵 ≠ ∅) → {𝑥 ∈ 𝒫 𝐴𝐵𝑥} ∈ (Fil‘𝐴))

Theoremzfbas 22496 The set of upper sets of integers is a filter base on , which corresponds to convergence of sequences on . (Contributed by Mario Carneiro, 13-Oct-2015.)
ran ℤ ∈ (fBas‘ℤ)

Theoremuzrest 22497 The restriction of the set of upper sets of integers to an upper set of integers is the set of upper sets of integers based at a point above the cutoff. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑍 = (ℤ𝑀)       (𝑀 ∈ ℤ → (ran ℤt 𝑍) = (ℤ𝑍))

Theoremuzfbas 22498 The set of upper sets of integers based at a point in a fixed upper integer set like is a filter base on , which corresponds to convergence of sequences on . (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑍 = (ℤ𝑀)       (𝑀 ∈ ℤ → (ℤ𝑍) ∈ (fBas‘𝑍))

12.2.3  Ultrafilters

Syntaxcufil 22499 Extend class notation with the ultrafilters-on-a-set function.
class UFil

Syntaxcufl 22500 Extend class notation with the ultrafilter lemma.
class UFL

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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