HomeHome Metamath Proof Explorer
Theorem List (p. 211 of 425)
< 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  Metamath Proof Explorer
(1-26947)
  Hilbert Space Explorer  Hilbert Space Explorer
(26948-28472)
  Users' Mathboxes  Users' Mathboxes
(28473-42426)
 

Theorem List for Metamath Proof Explorer - 21001-21100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremllyrest 21001 An open subspace of a locally 𝐴 space is also locally 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)
((𝐽 ∈ Locally 𝐴𝐵𝐽) → (𝐽t 𝐵) ∈ Locally 𝐴)
 
Theoremnllyrest 21002 An open subspace of an n-locally 𝐴 space is also n-locally 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)
((𝐽 ∈ 𝑛-Locally 𝐴𝐵𝐽) → (𝐽t 𝐵) ∈ 𝑛-Locally 𝐴)
 
Theoremloclly 21003 If 𝐴 is a local property, then both Locally 𝐴 and 𝑛-Locally 𝐴 simplify to 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)
(Locally 𝐴 = 𝐴 ↔ 𝑛-Locally 𝐴 = 𝐴)
 
Theoremllyidm 21004 Idempotence of the "locally" predicate, i.e. being "locally 𝐴 " is a local property. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally Locally 𝐴 = Locally 𝐴
 
Theoremnllyidm 21005 Idempotence of the "n-locally" predicate, i.e. being "n-locally 𝐴 " is a local property. (Use loclly 21003 to show 𝑛-Locally 𝑛-Locally 𝐴 = 𝑛-Locally 𝐴.) (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally 𝑛-Locally 𝐴 = 𝑛-Locally 𝐴
 
Theoremtoplly 21006 A topology is locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally Top = Top
 
Theoremtopnlly 21007 A topology is n-locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛-Locally Top = Top
 
Theoremhauslly 21008 A Hausdorff space is locally Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐽 ∈ Haus → 𝐽 ∈ Locally Haus)
 
Theoremhausnlly 21009 A Hausdorff space is n-locally Hausdorff iff it is locally Hausdorff (both notions are thus referred to as "locally Hausdorff"). (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐽 ∈ 𝑛-Locally Haus ↔ 𝐽 ∈ Locally Haus)
 
Theoremhausllycmp 21010 A compact Hausdorff space is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → 𝐽 ∈ 𝑛-Locally Comp)
 
Theoremcldllycmp 21011 A closed subspace of a locally compact space is also locally compact. (The analogous result for open subspaces follows from the more general nllyrest 21002.) (Contributed by Mario Carneiro, 2-Mar-2015.)
((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽t 𝐴) ∈ 𝑛-Locally Comp)
 
Theoremlly1stc 21012 First-countability is a local property (unlike second-countability). (Contributed by Mario Carneiro, 21-Mar-2015.)
Locally 1st𝜔 = 1st𝜔
 
Theoremdislly 21013* The discrete space 𝒫 𝑋 is locally 𝐴 if and only if every singleton space has property 𝐴. (Contributed by Mario Carneiro, 20-Mar-2015.)
(𝑋𝑉 → (𝒫 𝑋 ∈ Locally 𝐴 ↔ ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴))
 
Theoremdisllycmp 21014 A discrete space is locally compact. (Contributed by Mario Carneiro, 20-Mar-2015.)
(𝑋𝑉 → 𝒫 𝑋 ∈ Locally Comp)
 
Theoremdis1stc 21015 A discrete space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝑋𝑉 → 𝒫 𝑋 ∈ 1st𝜔)
 
Theoremhausmapdom 21016 If 𝑋 is a first-countable Hausdorff space, then the cardinality of the closure of a set 𝐴 is bounded by to the power 𝐴. In particular, a first-countable Hausdorff space with a dense subset 𝐴 has cardinality at most 𝐴↑ℕ, and a separable first-countable Hausdorff space has cardinality at most 𝒫 ℕ. (Compare hauspwpwdom 21505 to see a weaker result if the assumption of first-countability is omitted.) (Contributed by Mario Carneiro, 9-Apr-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ≼ (𝐴𝑚 ℕ))
 
Theoremhauspwdom 21017 Simplify the cardinal 𝐴↑ℕ of hausmapdom 21016 to 𝒫 𝐵 = 2↑𝐵 when 𝐵 is an infinite cardinal greater than 𝐴. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝑋 = 𝐽       (((𝐽 ∈ Haus ∧ 𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) ∧ (𝐴 ≼ 𝒫 𝐵 ∧ ℕ ≼ 𝐵)) → ((cls‘𝐽)‘𝐴) ≼ 𝒫 𝐵)
 
12.1.16  Refinements
 
Syntaxcref 21018 Extend class definition to include the refinement relation.
class Ref
 
Syntaxcptfin 21019 Extend class definition to include the class of point-finite covers.
class PtFin
 
Syntaxclocfin 21020 Extend class definition to include the class of locally finite covers.
class LocFin
 
Definitiondf-ref 21021* Define the refinement relation. (Contributed by Jeff Hankins, 18-Jan-2010.)
Ref = {⟨𝑥, 𝑦⟩ ∣ ( 𝑦 = 𝑥 ∧ ∀𝑧𝑥𝑤𝑦 𝑧𝑤)}
 
Definitiondf-ptfin 21022* Define "point-finite." (Contributed by Jeff Hankins, 21-Jan-2010.)
PtFin = {𝑥 ∣ ∀𝑦 𝑥{𝑧𝑥𝑦𝑧} ∈ Fin}
 
Definitiondf-locfin 21023* 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 21024 Refinement is a relation. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
Rel Ref
 
Theoremisref 21025* 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 31338. 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 21026 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 21027* 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 21028 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 21029 Reflexivity of refinement. (Contributed by Jeff Hankins, 18-Jan-2010.)
(𝐴𝑉𝐴Ref𝐴)
 
Theoremreftr 21030 Refinement is transitive. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
((𝐴Ref𝐵𝐵Ref𝐶) → 𝐴Ref𝐶)
 
Theoremrefun0 21031 Adding the empty set preserves refinements. (Contributed by Thierry Arnoux, 31-Jan-2020.)
((𝐴Ref𝐵𝐵 ≠ ∅) → (𝐴 ∪ {∅})Ref𝐵)
 
Theoremisptfin 21032* The statement "is a point-finite cover." (Contributed by Jeff Hankins, 21-Jan-2010.)
𝑋 = 𝐴       (𝐴𝐵 → (𝐴 ∈ PtFin ↔ ∀𝑥𝑋 {𝑦𝐴𝑥𝑦} ∈ Fin))
 
Theoremislocfin 21033* 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 21034 A finite cover is a point-finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.)
(𝐴 ∈ Fin → 𝐴 ∈ PtFin)
 
Theoremptfinfin 21035* 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 21036 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 21037 A locally finite cover covers a topological space. (Contributed by Jeff Hankins, 21-Jan-2010.)
(𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
 
Theoremlocfinbas 21038 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 21039* 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 21040 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 21041 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 21042 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 21043* 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 21044* 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 21045* 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 21046 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 21047 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 21048* 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 21049 Extend class notation with the compact generator operation.
class 𝑘Gen
 
Definitiondf-kgen 21050* 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 21051* 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 21052* 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 21053 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 21054 The compact generator generates a topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
(𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ (TopOn‘𝑋))
 
Theoremkgenuni 21055 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 21056 The compact generator generates a topology. (Contributed by Mario Carneiro, 20-Mar-2015.)
(𝐽 ∈ Top → (𝑘Gen‘𝐽) ∈ Top)
 
Theoremkgenf 21057 The compact generator is a function on topologies. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen:Top⟶Top
 
Theoremkgentop 21058 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 21059 The compact generator generates a finer topology than the original. (Contributed by Mario Carneiro, 20-Mar-2015.)
(𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽))
 
Theoremkgenhaus 21060 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 21061 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 21062 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 21063 The compact generator is idempotent on compactly generated spaces. (Contributed by Mario Carneiro, 20-Mar-2015.)
(𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽)
 
Theoremiskgen2 21064 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 21065* 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 21066* A locally compact space is compactly generated. (This variant of llycmpkgen 21068 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 21067 A compact space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝐽 ∈ Comp → 𝐽 ∈ ran 𝑘Gen)
 
Theoremllycmpkgen 21068 A locally compact space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ ran 𝑘Gen)
 
Theorem1stckgenlem 21069 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 21070 A first-countable space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝐽 ∈ 1st𝜔 → 𝐽 ∈ ran 𝑘Gen)
 
Theoremkgen2ss 21071 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 21072* 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 21073* 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 21074 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 21075 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 21076 Extend class notation with the binary topological product operation.
class ×t
 
Syntaxcxko 21077 Extend class notation with a function whose value is the compact-open topology.
class ^ko
 
Definitiondf-tx 21078* 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 21079* 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 21080* 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 21081* 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 21082* 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 21083* 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 21084* 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 21085 The product of two topologies is a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
 
Theoremptval 21086* 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 21087* 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 21088* 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 21089* 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 21090* 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 21091* 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 21092* 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 21093* 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 21094* 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 21095* 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 21096* 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 21097* 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 21098 The product topology is a topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
((𝐴𝑉𝐹:𝐴⟶Top) → (∏t𝐹) ∈ Top)
 
Theoremptopn 21099* 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 21100* 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))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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 42301-42400 425 42401-42426
  Copyright terms: Public domain < Previous  Next >