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Theorem List for Metamath Proof Explorer - 21301-21400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxclocfin 21301 Extend class definition to include the class of locally finite covers.
class LocFin

Definitiondf-ref 21302* Define the refinement relation. (Contributed by Jeff Hankins, 18-Jan-2010.)
Ref = {⟨𝑥, 𝑦⟩ ∣ ( 𝑦 = 𝑥 ∧ ∀𝑧𝑥𝑤𝑦 𝑧𝑤)}

Definitiondf-ptfin 21303* Define "point-finite." (Contributed by Jeff Hankins, 21-Jan-2010.)
PtFin = {𝑥 ∣ ∀𝑦 𝑥{𝑧𝑥𝑦𝑧} ∈ Fin}

Definitiondf-locfin 21304* Define "locally finite." (Contributed by Jeff Hankins, 21-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
LocFin = (𝑥 ∈ Top ↦ {𝑦 ∣ ( 𝑥 = 𝑦 ∧ ∀𝑝 𝑥𝑛𝑥 (𝑝𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))})

Theoremrefrel 21305 Refinement is a relation. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
Rel Ref

Theoremisref 21306* The property of being a refinement of a cover. Dr. Nyikos once commented in class that the term "refinement" is actually misleading and that people are inclined to confuse it with the notion defined in isfne 32318. On the other hand, the two concepts do seem to have a dual relationship. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
𝑋 = 𝐴    &   𝑌 = 𝐵       (𝐴𝐶 → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)))

Theoremrefbas 21307 A refinement covers the same set. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
𝑋 = 𝐴    &   𝑌 = 𝐵       (𝐴Ref𝐵𝑌 = 𝑋)

Theoremrefssex 21308* Every set in a refinement has a superset in the original cover. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
((𝐴Ref𝐵𝑆𝐴) → ∃𝑥𝐵 𝑆𝑥)

Theoremssref 21309 A subcover is a refinement of the original cover. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
𝑋 = 𝐴    &   𝑌 = 𝐵       ((𝐴𝐶𝐴𝐵𝑋 = 𝑌) → 𝐴Ref𝐵)

Theoremrefref 21310 Reflexivity of refinement. (Contributed by Jeff Hankins, 18-Jan-2010.)
(𝐴𝑉𝐴Ref𝐴)

Theoremreftr 21311 Refinement is transitive. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
((𝐴Ref𝐵𝐵Ref𝐶) → 𝐴Ref𝐶)

Theoremrefun0 21312 Adding the empty set preserves refinements. (Contributed by Thierry Arnoux, 31-Jan-2020.)
((𝐴Ref𝐵𝐵 ≠ ∅) → (𝐴 ∪ {∅})Ref𝐵)

Theoremisptfin 21313* The statement "is a point-finite cover." (Contributed by Jeff Hankins, 21-Jan-2010.)
𝑋 = 𝐴       (𝐴𝐵 → (𝐴 ∈ PtFin ↔ ∀𝑥𝑋 {𝑦𝐴𝑥𝑦} ∈ Fin))

Theoremislocfin 21314* The statement "is a locally finite cover." (Contributed by Jeff Hankins, 21-Jan-2010.)
𝑋 = 𝐽    &   𝑌 = 𝐴       (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))

Theoremfinptfin 21315 A finite cover is a point-finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.)
(𝐴 ∈ Fin → 𝐴 ∈ PtFin)

Theoremptfinfin 21316* A point covered by a point-finite cover is only covered by finitely many elements. (Contributed by Jeff Hankins, 21-Jan-2010.)
𝑋 = 𝐴       ((𝐴 ∈ PtFin ∧ 𝑃𝑋) → {𝑥𝐴𝑃𝑥} ∈ Fin)

Theoremfinlocfin 21317 A finite cover of a topological space is a locally finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.)
𝑋 = 𝐽    &   𝑌 = 𝐴       ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐴 ∈ (LocFin‘𝐽))

Theoremlocfintop 21318 A locally finite cover covers a topological space. (Contributed by Jeff Hankins, 21-Jan-2010.)
(𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)

Theoremlocfinbas 21319 A locally finite cover must cover the base set of its corresponding topological space. (Contributed by Jeff Hankins, 21-Jan-2010.)
𝑋 = 𝐽    &   𝑌 = 𝐴       (𝐴 ∈ (LocFin‘𝐽) → 𝑋 = 𝑌)

Theoremlocfinnei 21320* A point covered by a locally finite cover has a neighborhood which intersects only finitely many elements of the cover. (Contributed by Jeff Hankins, 21-Jan-2010.)
𝑋 = 𝐽       ((𝐴 ∈ (LocFin‘𝐽) ∧ 𝑃𝑋) → ∃𝑛𝐽 (𝑃𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))

Theoremlfinpfin 21321 A locally finite cover is point-finite. (Contributed by Jeff Hankins, 21-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
(𝐴 ∈ (LocFin‘𝐽) → 𝐴 ∈ PtFin)

Theoremlfinun 21322 Adding a finite set preserves locally finite covers. (Contributed by Thierry Arnoux, 31-Jan-2020.)
((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin ∧ 𝐵 𝐽) → (𝐴𝐵) ∈ (LocFin‘𝐽))

Theoremlocfincmp 21323 For a compact space, the locally finite covers are precisely the finite covers. Sadly, this property does not properly characterize all compact spaces. (Contributed by Jeff Hankins, 22-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
𝑋 = 𝐽    &   𝑌 = 𝐶       (𝐽 ∈ Comp → (𝐶 ∈ (LocFin‘𝐽) ↔ (𝐶 ∈ Fin ∧ 𝑋 = 𝑌)))

Theoremunisngl 21324* Taking the union of the set of singletons recovers the initial set. (Contributed by Thierry Arnoux, 9-Jan-2020.)
𝐶 = {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}       𝑋 = 𝐶

Theoremdissnref 21325* The set of singletons is a refinement of any open covering of the discrete topology. (Contributed by Thierry Arnoux, 9-Jan-2020.)
𝐶 = {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}       ((𝑋𝑉 𝑌 = 𝑋) → 𝐶Ref𝑌)

Theoremdissnlocfin 21326* The set of singletons is locally finite in the discrete topology. (Contributed by Thierry Arnoux, 9-Jan-2020.)
𝐶 = {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}       (𝑋𝑉𝐶 ∈ (LocFin‘𝒫 𝑋))

Theoremlocfindis 21327 The locally finite covers of a discrete space are precisely the point-finite covers. (Contributed by Jeff Hankins, 22-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
𝑌 = 𝐶       (𝐶 ∈ (LocFin‘𝒫 𝑋) ↔ (𝐶 ∈ PtFin ∧ 𝑋 = 𝑌))

Theoremlocfincf 21328 A locally finite cover in a coarser topology is locally finite in a finer topology. (Contributed by Jeff Hankins, 22-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
𝑋 = 𝐽       ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (LocFin‘𝐽) ⊆ (LocFin‘𝐾))

Theoremcomppfsc 21329* A space where every open cover has a point-finite subcover is compact. This is significant in part because it shows half of the proposition that if only half the generalization in the definition of metacompactness (and consequently paracompactness) is performed, one does not obtain any more spaces. (Contributed by Jeff Hankins, 21-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
𝑋 = 𝐽       (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑))))

12.1.17  Compactly generated spaces

Syntaxckgen 21330 Extend class notation with the compact generator operation.
class 𝑘Gen

Definitiondf-kgen 21331* Define the "compact generator", the functor from topological spaces to compactly generated spaces. Also known as the k-ification operation. A set is k-open, i.e. 𝑥 ∈ (𝑘Gen‘𝑗), iff the preimage of 𝑥 is open in all compact Hausdorff spaces. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 𝑗 ∣ ∀𝑘 ∈ 𝒫 𝑗((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘))})

Theoremkgenval 21332* Value of the compact generator. (The "k" in 𝑘Gen comes from the name "k-space" for these spaces, after the German word kompakt.) (Contributed by Mario Carneiro, 20-Mar-2015.)
(𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))})

Theoremelkgen 21333* Value of the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.)
(𝐽 ∈ (TopOn‘𝑋) → (𝐴 ∈ (𝑘Gen‘𝐽) ↔ (𝐴𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐴𝑘) ∈ (𝐽t 𝑘)))))

Theoremkgeni 21334 Property of the open sets in the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.)
((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝐴𝐾) ∈ (𝐽t 𝐾))

Theoremkgentopon 21335 The compact generator generates a topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
(𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ (TopOn‘𝑋))

Theoremkgenuni 21336 The base set of the compact generator is the same as the original topology. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑋 = 𝐽       (𝐽 ∈ Top → 𝑋 = (𝑘Gen‘𝐽))

Theoremkgenftop 21337 The compact generator generates a topology. (Contributed by Mario Carneiro, 20-Mar-2015.)
(𝐽 ∈ Top → (𝑘Gen‘𝐽) ∈ Top)

Theoremkgenf 21338 The compact generator is a function on topologies. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen:Top⟶Top

Theoremkgentop 21339 A compactly generated space is a topology. (Note: henceforth we will use the idiom "𝐽 ∈ ran 𝑘Gen " to denote "𝐽 is compactly generated", since as we will show a space is compactly generated iff it is in the range of the compact generator.) (Contributed by Mario Carneiro, 20-Mar-2015.)
(𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)

Theoremkgenss 21340 The compact generator generates a finer topology than the original. (Contributed by Mario Carneiro, 20-Mar-2015.)
(𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽))

Theoremkgenhaus 21341 The compact generator generates another Hausdorff topology given a Hausdorff topology to start from. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝐽 ∈ Haus → (𝑘Gen‘𝐽) ∈ Haus)

Theoremkgencmp 21342 The compact generator topology is the same as the original topology on compact subspaces. (Contributed by Mario Carneiro, 20-Mar-2015.)
((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t 𝐾) = ((𝑘Gen‘𝐽) ↾t 𝐾))

Theoremkgencmp2 21343 The compact generator topology has the same compact sets as the original topology. (Contributed by Mario Carneiro, 20-Mar-2015.)
(𝐽 ∈ Top → ((𝐽t 𝐾) ∈ Comp ↔ ((𝑘Gen‘𝐽) ↾t 𝐾) ∈ Comp))

Theoremkgenidm 21344 The compact generator is idempotent on compactly generated spaces. (Contributed by Mario Carneiro, 20-Mar-2015.)
(𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽)

Theoremiskgen2 21345 A space is compactly generated iff it contains its image under the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.)
(𝐽 ∈ ran 𝑘Gen ↔ (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽))

Theoremiskgen3 21346* Derive the usual definition of "compactly generated". A topology is compactly generated if every subset of 𝑋 that is open in every compact subset is open. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑋 = 𝐽       (𝐽 ∈ ran 𝑘Gen ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝑋(∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘)) → 𝑥𝐽)))

Theoremllycmpkgen2 21347* A locally compact space is compactly generated. (This variant of llycmpkgen 21349 uses the weaker definition of locally compact, "every point has a compact neighborhood", instead of "every point has a local base of compact neighborhoods".) (Contributed by Mario Carneiro, 21-Mar-2015.)
𝑋 = 𝐽    &   (𝜑𝐽 ∈ Top)    &   ((𝜑𝑥𝑋) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽t 𝑘) ∈ Comp)       (𝜑𝐽 ∈ ran 𝑘Gen)

Theoremcmpkgen 21348 A compact space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝐽 ∈ Comp → 𝐽 ∈ ran 𝑘Gen)

Theoremllycmpkgen 21349 A locally compact space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ ran 𝑘Gen)

Theorem1stckgenlem 21350 The one-point compactification of is compact. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹:ℕ⟶𝑋)    &   (𝜑𝐹(⇝𝑡𝐽)𝐴)       (𝜑 → (𝐽t (ran 𝐹 ∪ {𝐴})) ∈ Comp)

Theorem1stckgen 21351 A first-countable space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝐽 ∈ 1st𝜔 → 𝐽 ∈ ran 𝑘Gen)

Theoremkgen2ss 21352 The compact generator preserves the subset (fineness) relationship on topologies. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑘Gen‘𝐽) ⊆ (𝑘Gen‘𝐾))

Theoremkgencn 21353* A function from a compactly generated space is continuous iff it is continuous "on compacta". (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝐹𝑘) ∈ ((𝐽t 𝑘) Cn 𝐾)))))

Theoremkgencn2 21354* A function 𝐹:𝐽𝐾 from a compactly generated space is continuous iff for all compact spaces 𝑧 and continuous 𝑔:𝑧𝐽, the composite 𝐹𝑔:𝑧𝐾 is continuous. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑧 ∈ Comp ∀𝑔 ∈ (𝑧 Cn 𝐽)(𝐹𝑔) ∈ (𝑧 Cn 𝐾))))

Theoremkgencn3 21355 The set of continuous functions from 𝐽 to 𝐾 is unaffected by k-ification of 𝐾, if 𝐽 is already compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝐽 ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) = (𝐽 Cn (𝑘Gen‘𝐾)))

Theoremkgen2cn 21356 A continuous function is also continuous with the domain and codomain replaced by their compact generator topologies. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹 ∈ ((𝑘Gen‘𝐽) Cn (𝑘Gen‘𝐾)))

12.1.18  Product topologies

Syntaxctx 21357 Extend class notation with the binary topological product operation.
class ×t

Syntaxcxko 21358 Extend class notation with a function whose value is the compact-open topology.
class ^ko

Definitiondf-tx 21359* Define the binary topological product, which is homeomorphic to the general topological product over a two element set, but is more convenient to use. (Contributed by Jeff Madsen, 2-Sep-2009.)
×t = (𝑟 ∈ V, 𝑠 ∈ V ↦ (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))))

Definitiondf-xko 21360* Define the compact-open topology, which is the natural topology on the set of continuous functions between two topological spaces. (Contributed by Mario Carneiro, 19-Mar-2015.)
^ko = (𝑠 ∈ Top, 𝑟 ∈ Top ↦ (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp}, 𝑣𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣}))))

Theoremtxval 21361* Value of the binary topological product operation. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 30-Aug-2015.)
𝐵 = ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))       ((𝑅𝑉𝑆𝑊) → (𝑅 ×t 𝑆) = (topGen‘𝐵))

Theoremtxuni2 21362* The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 31-Aug-2015.)
𝐵 = ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))    &   𝑋 = 𝑅    &   𝑌 = 𝑆       (𝑋 × 𝑌) = 𝐵

Theoremtxbasex 21363* The basis for the product topology is a set. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐵 = ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))       ((𝑅𝑉𝑆𝑊) → 𝐵 ∈ V)

Theoremtxbas 21364* The set of Cartesian products of elements from two topological bases is a basis. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐵 = ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))       ((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) → 𝐵 ∈ TopBases)

Theoremeltx 21365* A set in a product is open iff each point is surrounded by an open rectangle. (Contributed by Stefan O'Rear, 25-Jan-2015.)
((𝐽𝑉𝐾𝑊) → (𝑆 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑝𝑆𝑥𝐽𝑦𝐾 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆)))

Theoremtxtop 21366 The product of two topologies is a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)

Theoremptval 21367* The value of the product topology function. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}       ((𝐴𝑉𝐹 Fn 𝐴) → (∏t𝐹) = (topGen‘𝐵))

Theoremptpjpre1 21368* The preimage of a projection function can be expressed as an indexed cartesian product. (Contributed by Mario Carneiro, 6-Feb-2015.)
𝑋 = X𝑘𝐴 (𝐹𝑘)       (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝐼𝐴𝑈 ∈ (𝐹𝐼))) → ((𝑤𝑋 ↦ (𝑤𝐼)) “ 𝑈) = X𝑘𝐴 if(𝑘 = 𝐼, 𝑈, (𝐹𝑘)))

Theoremelpt 21369* Elementhood in the bases of a product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}       (𝑆𝐵 ↔ ∃(( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦)) ∧ 𝑆 = X𝑦𝐴 (𝑦)))

Theoremelptr 21370* A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}       ((𝐴𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) ∈ (𝐹𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑊)(𝐺𝑦) = (𝐹𝑦))) → X𝑦𝐴 (𝐺𝑦) ∈ 𝐵)

Theoremelptr2 21371* A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}    &   (𝜑𝐴𝑉)    &   (𝜑𝑊 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝑆 ∈ (𝐹𝑘))    &   ((𝜑𝑘 ∈ (𝐴𝑊)) → 𝑆 = (𝐹𝑘))       (𝜑X𝑘𝐴 𝑆𝐵)

Theoremptbasid 21372* The base set of the product topology is a basic open set. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}       ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) ∈ 𝐵)

Theoremptuni2 21373* The base set for the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}       ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) = 𝐵)

Theoremptbasin 21374* The basis for a product topology is closed under intersections. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}       (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝑌) ∈ 𝐵)

Theoremptbasin2 21375* The basis for a product topology is closed under intersections. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}       ((𝐴𝑉𝐹:𝐴⟶Top) → (fi‘𝐵) = 𝐵)

Theoremptbas 21376* The basis for a product topology is a basis. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}       ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵 ∈ TopBases)

Theoremptpjpre2 21377* The basis for a product topology is a basis. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}    &   𝑋 = X𝑛𝐴 (𝐹𝑛)       (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝐼𝐴𝑈 ∈ (𝐹𝐼))) → ((𝑤𝑋 ↦ (𝑤𝐼)) “ 𝑈) ∈ 𝐵)

Theoremptbasfi 21378* The basis for the product topology can also be written as the set of finite intersections of "cylinder sets", the preimages of projections into one factor from open sets in the factor. (We have to add 𝑋 itself to the list because if 𝐴 is empty we get (fi‘∅) = ∅ while 𝐵 = {∅}.) (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}    &   𝑋 = X𝑛𝐴 (𝐹𝑛)       ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵 = (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))

Theorempttop 21379 The product topology is a topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
((𝐴𝑉𝐹:𝐴⟶Top) → (∏t𝐹) ∈ Top)

Theoremptopn 21380* A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶Top)    &   (𝜑𝑊 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝑆 ∈ (𝐹𝑘))    &   ((𝜑𝑘 ∈ (𝐴𝑊)) → 𝑆 = (𝐹𝑘))       (𝜑X𝑘𝐴 𝑆 ∈ (∏t𝐹))

Theoremptopn2 21381* A sub-basic open set in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶Top)    &   (𝜑𝑂 ∈ (𝐹𝑌))       (𝜑X𝑘𝐴 if(𝑘 = 𝑌, 𝑂, (𝐹𝑘)) ∈ (∏t𝐹))

Theoremxkotf 21382* Functionality of function 𝑇. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑋 = 𝑅    &   𝐾 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}    &   𝑇 = (𝑘𝐾, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})       𝑇:(𝐾 × 𝑆)⟶𝒫 (𝑅 Cn 𝑆)

Theoremxkobval 21383* Alternative expression for the subbase of the compact-open topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
𝑋 = 𝑅    &   𝐾 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}    &   𝑇 = (𝑘𝐾, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})       ran 𝑇 = {𝑠 ∣ ∃𝑘 ∈ 𝒫 𝑋𝑣𝑆 ((𝑅t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})}

Theoremxkoval 21384* Value of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑋 = 𝑅    &   𝐾 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}    &   𝑇 = (𝑘𝐾, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})       ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) = (topGen‘(fi‘ran 𝑇)))

Theoremxkotop 21385 The compact-open topology is a topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ Top)

Theoremxkoopn 21386* A basic open set of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑋 = 𝑅    &   (𝜑𝑅 ∈ Top)    &   (𝜑𝑆 ∈ Top)    &   (𝜑𝐴𝑋)    &   (𝜑 → (𝑅t 𝐴) ∈ Comp)    &   (𝜑𝑈𝑆)       (𝜑 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} ∈ (𝑆 ^ko 𝑅))

Theoremtxtopi 21387 The product of two topologies is a topology. (Contributed by Jeff Madsen, 15-Jun-2010.)
𝑅 ∈ Top    &   𝑆 ∈ Top       (𝑅 ×t 𝑆) ∈ Top

Theoremtxtopon 21388 The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 22-Aug-2015.) (Revised by Mario Carneiro, 2-Sep-2015.)
((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))

Theoremtxuni 21389 The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝑋 = 𝑅    &   𝑌 = 𝑆       ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = (𝑅 ×t 𝑆))

Theoremtxunii 21390 The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 15-Jun-2010.)
𝑅 ∈ Top    &   𝑆 ∈ Top    &   𝑋 = 𝑅    &   𝑌 = 𝑆       (𝑋 × 𝑌) = (𝑅 ×t 𝑆)

Theoremptuni 21391* The base set for the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐽 = (∏t𝐹)       ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑥𝐴 (𝐹𝑥) = 𝐽)

Theoremptunimpt 21392* Base set of a product topology given by substitution. (Contributed by Stefan O'Rear, 22-Feb-2015.)
𝐽 = (∏t‘(𝑥𝐴𝐾))       ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐾 ∈ Top) → X𝑥𝐴 𝐾 = 𝐽)

Theorempttopon 21393* The base set for the product topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
𝐽 = (∏t‘(𝑥𝐴𝐾))       ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐾 ∈ (TopOn‘𝐵)) → 𝐽 ∈ (TopOn‘X𝑥𝐴 𝐵))

Theorempttoponconst 21394 The base set for a product topology when all factors are the same. (Contributed by Mario Carneiro, 22-Aug-2015.)
𝐽 = (∏t‘(𝐴 × {𝑅}))       ((𝐴𝑉𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘(𝑋𝑚 𝐴)))

Theoremptuniconst 21395 The base set for a product topology when all factors are the same. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐽 = (∏t‘(𝐴 × {𝑅}))    &   𝑋 = 𝑅       ((𝐴𝑉𝑅 ∈ Top) → (𝑋𝑚 𝐴) = 𝐽)

Theoremxkouni 21396 The base set of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝐽 = (𝑆 ^ko 𝑅)       ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = 𝐽)

Theoremxkotopon 21397 The base set of the compact-open topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
𝐽 = (𝑆 ^ko 𝑅)       ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝐽 ∈ (TopOn‘(𝑅 Cn 𝑆)))

Theoremptval2 21398* The value of the product topology function. (Contributed by Mario Carneiro, 7-Feb-2015.)
𝐽 = (∏t𝐹)    &   𝑋 = 𝐽    &   𝐺 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))       ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐽 = (topGen‘(fi‘({𝑋} ∪ ran 𝐺))))

Theoremtxopn 21399 The product of two open sets is open in the product topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
(((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → (𝐴 × 𝐵) ∈ (𝑅 ×t 𝑆))

Theoremtxcld 21400 The product of two closed sets is closed in the product topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ∈ (Clsd‘(𝑅 ×t 𝑆)))

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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42322
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