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Theorem List for Metamath Proof Explorer - 22001-22100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem1stccn 22001* A mapping 𝑋𝑌, where 𝑋 is first-countable, is continuous iff it is sequentially continuous, meaning that for any sequence 𝑓(𝑛) converging to 𝑥, its image under 𝐹 converges to 𝐹(𝑥). (Contributed by Mario Carneiro, 7-Sep-2015.)
(𝜑𝐽 ∈ 1stω)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐹:𝑋𝑌)       (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑓(𝑓:ℕ⟶𝑋 → ∀𝑥(𝑓(⇝𝑡𝐽)𝑥 → (𝐹𝑓)(⇝𝑡𝐾)(𝐹𝑥)))))
 
12.1.15  Local topological properties
 
Syntaxclly 22002 Extend class notation with the "locally 𝐴 " predicate of a topological space.
class Locally 𝐴
 
Syntaxcnlly 22003 Extend class notation with the "N-locally 𝐴 " predicate of a topological space.
class 𝑛-Locally 𝐴
 
Definitiondf-lly 22004* Define a space that is locally 𝐴, where 𝐴 is a topological property like "compact", "connected", or "path-connected". A topological space is locally 𝐴 if every neighborhood of a point contains an open subneighborhood that is 𝐴 in the subspace topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴)}
 
Definitiondf-nlly 22005* Define a space that is n-locally 𝐴, where 𝐴 is a topological property like "compact", "connected", or "path-connected". A topological space is n-locally 𝐴 if every neighborhood of a point contains a subneighborhood that is 𝐴 in the subspace topology.

The terminology "n-locally", where 'n' stands for "neighborhood", is not standard, although this is sometimes called "weakly locally 𝐴". The reason for the distinction is that some notions only make sense for arbitrary neighborhoods (such as "locally compact", which is actually 𝑛-Locally Comp in our terminology - open compact sets are not very useful), while others such as "locally connected" are strictly weaker notions if the neighborhoods are not required to be open. (Contributed by Mario Carneiro, 2-Mar-2015.)

𝑛-Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴}
 
Theoremislly 22006* The property of being a locally 𝐴 topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
 
Theoremisnlly 22007* The property of being an n-locally 𝐴 topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴))
 
Theoremllyeq 22008 Equality theorem for the Locally 𝐴 predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐴 = 𝐵 → Locally 𝐴 = Locally 𝐵)
 
Theoremnllyeq 22009 Equality theorem for the Locally 𝐴 predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐴 = 𝐵 → 𝑛-Locally 𝐴 = 𝑛-Locally 𝐵)
 
Theoremllytop 22010 A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐽 ∈ Locally 𝐴𝐽 ∈ Top)
 
Theoremnllytop 22011 A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐽 ∈ 𝑛-Locally 𝐴𝐽 ∈ Top)
 
Theoremllyi 22012* The property of a locally 𝐴 topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
((𝐽 ∈ Locally 𝐴𝑈𝐽𝑃𝑈) → ∃𝑢𝐽 (𝑢𝑈𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
 
Theoremnllyi 22013* The property of an n-locally 𝐴 topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
((𝐽 ∈ 𝑛-Locally 𝐴𝑈𝐽𝑃𝑈) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑃})(𝑢𝑈 ∧ (𝐽t 𝑢) ∈ 𝐴))
 
Theoremnlly2i 22014* Eliminate the neighborhood symbol from nllyi 22013. (Contributed by Mario Carneiro, 2-Mar-2015.)
((𝐽 ∈ 𝑛-Locally 𝐴𝑈𝐽𝑃𝑈) → ∃𝑠 ∈ 𝒫 𝑈𝑢𝐽 (𝑃𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))
 
Theoremllynlly 22015 A locally 𝐴 space is n-locally 𝐴: the "n-locally" predicate is the weaker notion. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐽 ∈ Locally 𝐴𝐽 ∈ 𝑛-Locally 𝐴)
 
Theoremllyssnlly 22016 A locally 𝐴 space is n-locally 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally 𝐴 ⊆ 𝑛-Locally 𝐴
 
Theoremllyss 22017 The "locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐴𝐵 → Locally 𝐴 ⊆ Locally 𝐵)
 
Theoremnllyss 22018 The "n-locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐴𝐵 → 𝑛-Locally 𝐴 ⊆ 𝑛-Locally 𝐵)
 
Theoremsubislly 22019* The property of a subspace being locally 𝐴. (Contributed by Mario Carneiro, 10-Mar-2015.)
((𝐽 ∈ Top ∧ 𝐵𝑉) → ((𝐽t 𝐵) ∈ Locally 𝐴 ↔ ∀𝑥𝐽𝑦 ∈ (𝑥𝐵)∃𝑢𝐽 ((𝑢𝐵) ⊆ 𝑥𝑦𝑢 ∧ (𝐽t (𝑢𝐵)) ∈ 𝐴)))
 
Theoremrestnlly 22020* If the property 𝐴 passes to open subspaces, then a space is n-locally 𝐴 iff it is locally 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)
((𝜑 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝐴)       (𝜑 → 𝑛-Locally 𝐴 = Locally 𝐴)
 
Theoremrestlly 22021* If the property 𝐴 passes to open subspaces, then a space which is 𝐴 is also locally 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)
((𝜑 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝐴)    &   (𝜑𝐴 ⊆ Top)       (𝜑𝐴 ⊆ Locally 𝐴)
 
Theoremislly2 22022* An alternative expression for 𝐽 ∈ Locally 𝐴 when 𝐴 passes to open subspaces: A space is locally 𝐴 if every point is contained in an open neighborhood with property 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)
((𝜑 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝐴)    &   𝑋 = 𝐽       (𝜑 → (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦𝑋𝑢𝐽 (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))))
 
Theoremllyrest 22023 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 22024 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 22025 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 22026 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 22027 Idempotence of the "n-locally" predicate, i.e. being "n-locally 𝐴 " is a local property. (Use loclly 22025 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 22028 A topology is locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally Top = Top
 
Theoremtopnlly 22029 A topology is n-locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛-Locally Top = Top
 
Theoremhauslly 22030 A Hausdorff space is locally Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐽 ∈ Haus → 𝐽 ∈ Locally Haus)
 
Theoremhausnlly 22031 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 22032 A compact Hausdorff space is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → 𝐽 ∈ 𝑛-Locally Comp)
 
Theoremcldllycmp 22033 A closed subspace of a locally compact space is also locally compact. (The analogous result for open subspaces follows from the more general nllyrest 22024.) (Contributed by Mario Carneiro, 2-Mar-2015.)
((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽t 𝐴) ∈ 𝑛-Locally Comp)
 
Theoremlly1stc 22034 First-countability is a local property (unlike second-countability). (Contributed by Mario Carneiro, 21-Mar-2015.)
Locally 1stω = 1stω
 
Theoremdislly 22035* 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 22036 A discrete space is locally compact. (Contributed by Mario Carneiro, 20-Mar-2015.)
(𝑋𝑉 → 𝒫 𝑋 ∈ Locally Comp)
 
Theoremdis1stc 22037 A discrete space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝑋𝑉 → 𝒫 𝑋 ∈ 1stω)
 
Theoremhausmapdom 22038 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 22526 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 22039 Simplify the cardinal 𝐴↑ℕ of hausmapdom 22038 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 22040 Extend class definition to include the refinement relation.
class Ref
 
Syntaxcptfin 22041 Extend class definition to include the class of point-finite covers.
class PtFin
 
Syntaxclocfin 22042 Extend class definition to include the class of locally finite covers.
class LocFin
 
Definitiondf-ref 22043* Define the refinement relation. (Contributed by Jeff Hankins, 18-Jan-2010.)
Ref = {⟨𝑥, 𝑦⟩ ∣ ( 𝑦 = 𝑥 ∧ ∀𝑧𝑥𝑤𝑦 𝑧𝑤)}
 
Definitiondf-ptfin 22044* Define "point-finite." (Contributed by Jeff Hankins, 21-Jan-2010.)
PtFin = {𝑥 ∣ ∀𝑦 𝑥{𝑧𝑥𝑦𝑧} ∈ Fin}
 
Definitiondf-locfin 22045* 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 22046 Refinement is a relation. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
Rel Ref
 
Theoremisref 22047* 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 33585. 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 22048 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 22049* 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 22050 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 22051 Reflexivity of refinement. (Contributed by Jeff Hankins, 18-Jan-2010.)
(𝐴𝑉𝐴Ref𝐴)
 
Theoremreftr 22052 Refinement is transitive. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
((𝐴Ref𝐵𝐵Ref𝐶) → 𝐴Ref𝐶)
 
Theoremrefun0 22053 Adding the empty set preserves refinements. (Contributed by Thierry Arnoux, 31-Jan-2020.)
((𝐴Ref𝐵𝐵 ≠ ∅) → (𝐴 ∪ {∅})Ref𝐵)
 
Theoremisptfin 22054* The statement "is a point-finite cover." (Contributed by Jeff Hankins, 21-Jan-2010.)
𝑋 = 𝐴       (𝐴𝐵 → (𝐴 ∈ PtFin ↔ ∀𝑥𝑋 {𝑦𝐴𝑥𝑦} ∈ Fin))
 
Theoremislocfin 22055* 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 22056 A finite cover is a point-finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.)
(𝐴 ∈ Fin → 𝐴 ∈ PtFin)
 
Theoremptfinfin 22057* 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 22058 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 22059 A locally finite cover covers a topological space. (Contributed by Jeff Hankins, 21-Jan-2010.)
(𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
 
Theoremlocfinbas 22060 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 22061* 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 22062 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 22063 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 22064 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 22065* 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 22066* 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 22067* 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 22068 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 22069 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 22070* 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 22071 Extend class notation with the compact generator operation.
class 𝑘Gen
 
Definitiondf-kgen 22072* 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 22073* 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 22074* 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 22075 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 22076 The compact generator generates a topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
(𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ (TopOn‘𝑋))
 
Theoremkgenuni 22077 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 22078 The compact generator generates a topology. (Contributed by Mario Carneiro, 20-Mar-2015.)
(𝐽 ∈ Top → (𝑘Gen‘𝐽) ∈ Top)
 
Theoremkgenf 22079 The compact generator is a function on topologies. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen:Top⟶Top
 
Theoremkgentop 22080 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 22081 The compact generator generates a finer topology than the original. (Contributed by Mario Carneiro, 20-Mar-2015.)
(𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽))
 
Theoremkgenhaus 22082 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 22083 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 22084 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 22085 The compact generator is idempotent on compactly generated spaces. (Contributed by Mario Carneiro, 20-Mar-2015.)
(𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽)
 
Theoremiskgen2 22086 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 22087* 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 22088* A locally compact space is compactly generated. (This variant of llycmpkgen 22090 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 22089 A compact space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝐽 ∈ Comp → 𝐽 ∈ ran 𝑘Gen)
 
Theoremllycmpkgen 22090 A locally compact space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ ran 𝑘Gen)
 
Theorem1stckgenlem 22091 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 22092 A first-countable space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝐽 ∈ 1stω → 𝐽 ∈ ran 𝑘Gen)
 
Theoremkgen2ss 22093 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 22094* 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 22095* 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 22096 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 22097 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 22098 Extend class notation with the binary topological product operation.
class ×t
 
Syntaxcxko 22099 Extend class notation with a function whose value is the compact-open topology.
class ko
 
Definitiondf-tx 22100* 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))))
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