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

Theorem2ndcredom 21301 A second-countable space has at most the cardinality of the continuum. (Contributed by Mario Carneiro, 9-Apr-2015.)
(𝐽 ∈ 2nd𝜔 → 𝐽 ≼ ℝ)

Theorem2ndc1stc 21302 A second-countable space is first-countable. (Contributed by Jeff Hankins, 17-Jan-2010.)
(𝐽 ∈ 2nd𝜔 → 𝐽 ∈ 1st𝜔)

Theorem1stcrestlem 21303* Lemma for 1stcrest 21304. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)
(𝐵 ≼ ω → ran (𝑥𝐵𝐶) ≼ ω)

Theorem1stcrest 21304 A subspace of a first-countable space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ 1st𝜔)

Theorem2ndcrest 21305 A subspace of a second-countable space is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝐽 ∈ 2nd𝜔 ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ 2nd𝜔)

Theorem2ndcctbss 21306* If a topology is second-countable, every base has a countable subset which is a base. Exercise 16B2 in Willard. (Contributed by Jeff Hankins, 28-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
𝑋 = 𝐵    &   𝐽 = (topGen‘𝐵)    &   𝑆 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝑐𝑣𝑐 ∧ ∃𝑤𝐵 (𝑢𝑤𝑤𝑣))}       ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏𝐵𝐽 = (topGen‘𝑏)))

Theorem2ndcdisj 21307* Any disjoint family of open sets in a second-countable space is countable. (The sets are required to be nonempty because otherwise there could be many empty sets in the family.) (Contributed by Mario Carneiro, 21-Mar-2015.) (Proof shortened by Mario Carneiro, 9-Apr-2015.) (Revised by NM, 17-Jun-2017.)
((𝐽 ∈ 2nd𝜔 ∧ ∀𝑥𝐴 𝐵 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃*𝑥𝐴 𝑦𝐵) → 𝐴 ≼ ω)

Theorem2ndcdisj2 21308* Any disjoint collection of open sets in a second-countable space is countable. (Contributed by Mario Carneiro, 21-Mar-2015.) (Proof shortened by Mario Carneiro, 9-Apr-2015.) (Revised by NM, 17-Jun-2017.)
((𝐽 ∈ 2nd𝜔 ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥𝐴 𝑦𝑥) → 𝐴 ≼ ω)

Theorem2ndcomap 21309* A surjective continuous open map maps second-countable spaces to second-countable spaces. (Contributed by Mario Carneiro, 9-Apr-2015.)
𝑌 = 𝐾    &   (𝜑𝐽 ∈ 2nd𝜔)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑 → ran 𝐹 = 𝑌)    &   ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ 𝐾)       (𝜑𝐾 ∈ 2nd𝜔)

Theorem2ndcsep 21310* A second-countable topology is separable, which is to say it contains a countable dense subset. (Contributed by Mario Carneiro, 13-Apr-2015.)
𝑋 = 𝐽       (𝐽 ∈ 2nd𝜔 → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))

Theoremdis2ndc 21311 A discrete space is second-countable iff it is countable. (Contributed by Mario Carneiro, 13-Apr-2015.)
(𝑋 ≼ ω ↔ 𝒫 𝑋 ∈ 2nd𝜔)

Theorem1stcelcls 21312* A point belongs to the closure of a subset iff there is a sequence in the subset converging to it. Theorem 1.4-6(a) of [Kreyszig] p. 30. This proof uses countable choice ax-cc 9295. A space satisfying the conclusion of this theorem is called a sequential space, so the theorem can also be stated as "every first-countable space is a sequential space". (Contributed by Mario Carneiro, 21-Mar-2015.)
𝑋 = 𝐽       ((𝐽 ∈ 1st𝜔 ∧ 𝑆𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑓(𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)))

Theorem1stccnp 21313* A mapping is continuous at 𝑃 in a first-countable space 𝑋 iff it is sequentially continuous at 𝑃, meaning that the image under 𝐹 of every sequence converging at 𝑃 converges to 𝐹(𝑃). This proof uses ax-cc 9295, but only via 1stcelcls 21312, so it could be refactored into a proof that continuity and sequential continuity are the same in sequential spaces. (Contributed by Mario Carneiro, 7-Sep-2015.)
(𝜑𝐽 ∈ 1st𝜔)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝑃𝑋)       (𝜑 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑓((𝑓:ℕ⟶𝑋𝑓(⇝𝑡𝐽)𝑃) → (𝐹𝑓)(⇝𝑡𝐾)(𝐹𝑃)))))

Theorem1stccn 21314* 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 21315 Extend class notation with the "locally 𝐴 " predicate of a topological space.
class Locally 𝐴

Syntaxcnlly 21316 Extend class notation with the "N-locally 𝐴 " predicate of a topological space.
class 𝑛-Locally 𝐴

Definitiondf-lly 21317* 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 sub-neighborhood 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 21318* 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 sub-neighborhood 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 21319* 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 21320* 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 21321 Equality theorem for the Locally 𝐴 predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐴 = 𝐵 → Locally 𝐴 = Locally 𝐵)

Theoremnllyeq 21322 Equality theorem for the Locally 𝐴 predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐴 = 𝐵 → 𝑛-Locally 𝐴 = 𝑛-Locally 𝐵)

Theoremllytop 21323 A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐽 ∈ Locally 𝐴𝐽 ∈ Top)

Theoremnllytop 21324 A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐽 ∈ 𝑛-Locally 𝐴𝐽 ∈ Top)

Theoremllyi 21325* The property of a locally 𝐴 topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
((𝐽 ∈ Locally 𝐴𝑈𝐽𝑃𝑈) → ∃𝑢𝐽 (𝑢𝑈𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))

Theoremnllyi 21326* 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 21327* Eliminate the neighborhood symbol from nllyi 21326. (Contributed by Mario Carneiro, 2-Mar-2015.)
((𝐽 ∈ 𝑛-Locally 𝐴𝑈𝐽𝑃𝑈) → ∃𝑠 ∈ 𝒫 𝑈𝑢𝐽 (𝑃𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))

Theoremllynlly 21328 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 21329 A locally 𝐴 space is n-locally 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally 𝐴 ⊆ 𝑛-Locally 𝐴

Theoremllyss 21330 The "locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐴𝐵 → Locally 𝐴 ⊆ Locally 𝐵)

Theoremnllyss 21331 The "n-locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐴𝐵 → 𝑛-Locally 𝐴 ⊆ 𝑛-Locally 𝐵)

Theoremsubislly 21332* 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 21333* 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 21334* 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 21335* 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 21336 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 21337 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 21338 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 21339 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 21340 Idempotence of the "n-locally" predicate, i.e. being "n-locally 𝐴 " is a local property. (Use loclly 21338 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 21341 A topology is locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally Top = Top

Theoremtopnlly 21342 A topology is n-locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛-Locally Top = Top

Theoremhauslly 21343 A Hausdorff space is locally Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐽 ∈ Haus → 𝐽 ∈ Locally Haus)

Theoremhausnlly 21344 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 21345 A compact Hausdorff space is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → 𝐽 ∈ 𝑛-Locally Comp)

Theoremcldllycmp 21346 A closed subspace of a locally compact space is also locally compact. (The analogous result for open subspaces follows from the more general nllyrest 21337.) (Contributed by Mario Carneiro, 2-Mar-2015.)
((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽t 𝐴) ∈ 𝑛-Locally Comp)

Theoremlly1stc 21347 First-countability is a local property (unlike second-countability). (Contributed by Mario Carneiro, 21-Mar-2015.)
Locally 1st𝜔 = 1st𝜔

Theoremdislly 21348* 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 21349 A discrete space is locally compact. (Contributed by Mario Carneiro, 20-Mar-2015.)
(𝑋𝑉 → 𝒫 𝑋 ∈ Locally Comp)

Theoremdis1stc 21350 A discrete space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝑋𝑉 → 𝒫 𝑋 ∈ 1st𝜔)

Theoremhausmapdom 21351 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 21839 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 21352 Simplify the cardinal 𝐴↑ℕ of hausmapdom 21351 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 21353 Extend class definition to include the refinement relation.
class Ref

Syntaxcptfin 21354 Extend class definition to include the class of point-finite covers.
class PtFin

Syntaxclocfin 21355 Extend class definition to include the class of locally finite covers.
class LocFin

Definitiondf-ref 21356* Define the refinement relation. (Contributed by Jeff Hankins, 18-Jan-2010.)
Ref = {⟨𝑥, 𝑦⟩ ∣ ( 𝑦 = 𝑥 ∧ ∀𝑧𝑥𝑤𝑦 𝑧𝑤)}

Definitiondf-ptfin 21357* Define "point-finite." (Contributed by Jeff Hankins, 21-Jan-2010.)
PtFin = {𝑥 ∣ ∀𝑦 𝑥{𝑧𝑥𝑦𝑧} ∈ Fin}

Definitiondf-locfin 21358* 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 21359 Refinement is a relation. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
Rel Ref

Theoremisref 21360* 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 32459. 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 21361 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 21362* 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 21363 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 21364 Reflexivity of refinement. (Contributed by Jeff Hankins, 18-Jan-2010.)
(𝐴𝑉𝐴Ref𝐴)

Theoremreftr 21365 Refinement is transitive. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
((𝐴Ref𝐵𝐵Ref𝐶) → 𝐴Ref𝐶)

Theoremrefun0 21366 Adding the empty set preserves refinements. (Contributed by Thierry Arnoux, 31-Jan-2020.)
((𝐴Ref𝐵𝐵 ≠ ∅) → (𝐴 ∪ {∅})Ref𝐵)

Theoremisptfin 21367* The statement "is a point-finite cover." (Contributed by Jeff Hankins, 21-Jan-2010.)
𝑋 = 𝐴       (𝐴𝐵 → (𝐴 ∈ PtFin ↔ ∀𝑥𝑋 {𝑦𝐴𝑥𝑦} ∈ Fin))

Theoremislocfin 21368* 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 21369 A finite cover is a point-finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.)
(𝐴 ∈ Fin → 𝐴 ∈ PtFin)

Theoremptfinfin 21370* 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 21371 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 21372 A locally finite cover covers a topological space. (Contributed by Jeff Hankins, 21-Jan-2010.)
(𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)

Theoremlocfinbas 21373 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 21374* 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 21375 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 21376 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 21377 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 21378* 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 21379* 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 21380* 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 21381 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 21382 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 21383* 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 21384 Extend class notation with the compact generator operation.
class 𝑘Gen

Definitiondf-kgen 21385* 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 21386* 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 21387* 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 21388 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 21389 The compact generator generates a topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
(𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ (TopOn‘𝑋))

Theoremkgenuni 21390 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 21391 The compact generator generates a topology. (Contributed by Mario Carneiro, 20-Mar-2015.)
(𝐽 ∈ Top → (𝑘Gen‘𝐽) ∈ Top)

Theoremkgenf 21392 The compact generator is a function on topologies. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑘Gen:Top⟶Top

Theoremkgentop 21393 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 21394 The compact generator generates a finer topology than the original. (Contributed by Mario Carneiro, 20-Mar-2015.)
(𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽))

Theoremkgenhaus 21395 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 21396 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 21397 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 21398 The compact generator is idempotent on compactly generated spaces. (Contributed by Mario Carneiro, 20-Mar-2015.)
(𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽)

Theoremiskgen2 21399 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 21400* 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)))

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