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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 1stccn 22001* | A mapping 𝑋⟶𝑌, 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.) |
⊢ (𝜑 → 𝐽 ∈ 1stω) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑓(𝑓:ℕ⟶𝑋 → ∀𝑥(𝑓(⇝𝑡‘𝐽)𝑥 → (𝐹 ∘ 𝑓)(⇝𝑡‘𝐾)(𝐹‘𝑥))))) | ||
Syntax | clly 22002 | Extend class notation with the "locally 𝐴 " predicate of a topological space. |
class Locally 𝐴 | ||
Syntax | cnlly 22003 | Extend class notation with the "N-locally 𝐴 " predicate of a topological space. |
class 𝑛-Locally 𝐴 | ||
Definition | df-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 ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴)} | ||
Definition | df-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 ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴} | ||
Theorem | islly 22006* | The property of being a locally 𝐴 topological space. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) | ||
Theorem | isnlly 22007* | The property of being an n-locally 𝐴 topological space. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴)) | ||
Theorem | llyeq 22008 | Equality theorem for the Locally 𝐴 predicate. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ (𝐴 = 𝐵 → Locally 𝐴 = Locally 𝐵) | ||
Theorem | nllyeq 22009 | Equality theorem for the Locally 𝐴 predicate. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ (𝐴 = 𝐵 → 𝑛-Locally 𝐴 = 𝑛-Locally 𝐵) | ||
Theorem | llytop 22010 | A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ Top) | ||
Theorem | nllytop 22011 | A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ (𝐽 ∈ 𝑛-Locally 𝐴 → 𝐽 ∈ Top) | ||
Theorem | llyi 22012* | The property of a locally 𝐴 topological space. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → ∃𝑢 ∈ 𝐽 (𝑢 ⊆ 𝑈 ∧ 𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)) | ||
Theorem | nllyi 22013* | The property of an n-locally 𝐴 topological space. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑃})(𝑢 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)) | ||
Theorem | nlly2i 22014* | Eliminate the neighborhood symbol from nllyi 22013. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → ∃𝑠 ∈ 𝒫 𝑈∃𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝐽 ↾t 𝑠) ∈ 𝐴)) | ||
Theorem | llynlly 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 𝐴 → 𝐽 ∈ 𝑛-Locally 𝐴) | ||
Theorem | llyssnlly 22016 | A locally 𝐴 space is n-locally 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ Locally 𝐴 ⊆ 𝑛-Locally 𝐴 | ||
Theorem | llyss 22017 | The "locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ (𝐴 ⊆ 𝐵 → Locally 𝐴 ⊆ Locally 𝐵) | ||
Theorem | nllyss 22018 | The "n-locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ (𝐴 ⊆ 𝐵 → 𝑛-Locally 𝐴 ⊆ 𝑛-Locally 𝐵) | ||
Theorem | subislly 22019* | The property of a subspace being locally 𝐴. (Contributed by Mario Carneiro, 10-Mar-2015.) |
⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) → ((𝐽 ↾t 𝐵) ∈ Locally 𝐴 ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ (𝑥 ∩ 𝐵)∃𝑢 ∈ 𝐽 ((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t (𝑢 ∩ 𝐵)) ∈ 𝐴))) | ||
Theorem | restnlly 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 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝑛-Locally 𝐴 = Locally 𝐴) | ||
Theorem | restlly 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 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝐴) & ⊢ (𝜑 → 𝐴 ⊆ Top) ⇒ ⊢ (𝜑 → 𝐴 ⊆ Locally 𝐴) | ||
Theorem | islly2 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 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝐴) & ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝜑 → (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))) | ||
Theorem | llyrest 22023 | An open subspace of a locally 𝐴 space is also locally 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ Locally 𝐴) | ||
Theorem | nllyrest 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 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ 𝑛-Locally 𝐴) | ||
Theorem | loclly 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) | ||
Theorem | llyidm 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 | ||
Theorem | nllyidm 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 | ||
Theorem | toplly 22028 | A topology is locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ Locally Top = Top | ||
Theorem | topnlly 22029 | A topology is n-locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ 𝑛-Locally Top = Top | ||
Theorem | hauslly 22030 | A Hausdorff space is locally Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ (𝐽 ∈ Haus → 𝐽 ∈ Locally Haus) | ||
Theorem | hausnlly 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) | ||
Theorem | hausllycmp 22032 | A compact Hausdorff space is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → 𝐽 ∈ 𝑛-Locally Comp) | ||
Theorem | cldllycmp 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)) → (𝐽 ↾t 𝐴) ∈ 𝑛-Locally Comp) | ||
Theorem | lly1stc 22034 | First-countability is a local property (unlike second-countability). (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ Locally 1stω = 1stω | ||
Theorem | dislly 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.) |
⊢ (𝑋 ∈ 𝑉 → (𝒫 𝑋 ∈ Locally 𝐴 ↔ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 𝐴)) | ||
Theorem | disllycmp 22036 | A discrete space is locally compact. (Contributed by Mario Carneiro, 20-Mar-2015.) |
⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Locally Comp) | ||
Theorem | dis1stc 22037 | A discrete space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ 1stω) | ||
Theorem | hausmapdom 22038 | If 𝑋 is a first-countable Hausdorff space, then the cardinality of the closure of a set 𝐴 is bounded by ℕ 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 ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ≼ (𝐴 ↑m ℕ)) | ||
Theorem | hauspwdom 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 ∧ 𝐴 ⊆ 𝑋) ∧ (𝐴 ≼ 𝒫 𝐵 ∧ ℕ ≼ 𝐵)) → ((cls‘𝐽)‘𝐴) ≼ 𝒫 𝐵) | ||
Syntax | cref 22040 | Extend class definition to include the refinement relation. |
class Ref | ||
Syntax | cptfin 22041 | Extend class definition to include the class of point-finite covers. |
class PtFin | ||
Syntax | clocfin 22042 | Extend class definition to include the class of locally finite covers. |
class LocFin | ||
Definition | df-ref 22043* | Define the refinement relation. (Contributed by Jeff Hankins, 18-Jan-2010.) |
⊢ Ref = {〈𝑥, 𝑦〉 ∣ (∪ 𝑦 = ∪ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)} | ||
Definition | df-ptfin 22044* | Define "point-finite." (Contributed by Jeff Hankins, 21-Jan-2010.) |
⊢ PtFin = {𝑥 ∣ ∀𝑦 ∈ ∪ 𝑥{𝑧 ∈ 𝑥 ∣ 𝑦 ∈ 𝑧} ∈ Fin} | ||
Definition | df-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 ∈ ∪ 𝑥∃𝑛 ∈ 𝑥 (𝑝 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))}) | ||
Theorem | refrel 22046 | Refinement is a relation. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.) |
⊢ Rel Ref | ||
Theorem | isref 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 ⇒ ⊢ (𝐴 ∈ 𝐶 → (𝐴Ref𝐵 ↔ (𝑌 = 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))) | ||
Theorem | refbas 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 ⇒ ⊢ (𝐴Ref𝐵 → 𝑌 = 𝑋) | ||
Theorem | refssex 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.) |
⊢ ((𝐴Ref𝐵 ∧ 𝑆 ∈ 𝐴) → ∃𝑥 ∈ 𝐵 𝑆 ⊆ 𝑥) | ||
Theorem | ssref 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 ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → 𝐴Ref𝐵) | ||
Theorem | refref 22051 | Reflexivity of refinement. (Contributed by Jeff Hankins, 18-Jan-2010.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴Ref𝐴) | ||
Theorem | reftr 22052 | Refinement is transitive. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.) |
⊢ ((𝐴Ref𝐵 ∧ 𝐵Ref𝐶) → 𝐴Ref𝐶) | ||
Theorem | refun0 22053 | Adding the empty set preserves refinements. (Contributed by Thierry Arnoux, 31-Jan-2020.) |
⊢ ((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) → (𝐴 ∪ {∅})Ref𝐵) | ||
Theorem | isptfin 22054* | The statement "is a point-finite cover." (Contributed by Jeff Hankins, 21-Jan-2010.) |
⊢ 𝑋 = ∪ 𝐴 ⇒ ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ PtFin ↔ ∀𝑥 ∈ 𝑋 {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin)) | ||
Theorem | islocfin 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 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))) | ||
Theorem | finptfin 22056 | A finite cover is a point-finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.) |
⊢ (𝐴 ∈ Fin → 𝐴 ∈ PtFin) | ||
Theorem | ptfinfin 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 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥} ∈ Fin) | ||
Theorem | finlocfin 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)) | ||
Theorem | locfintop 22059 | A locally finite cover covers a topological space. (Contributed by Jeff Hankins, 21-Jan-2010.) |
⊢ (𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top) | ||
Theorem | locfinbas 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) | ||
Theorem | locfinnei 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) ∧ 𝑃 ∈ 𝑋) → ∃𝑛 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)) | ||
Theorem | lfinpfin 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) | ||
Theorem | lfinun 22063 | Adding a finite set preserves locally finite covers. (Contributed by Thierry Arnoux, 31-Jan-2020.) |
⊢ ((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → (𝐴 ∪ 𝐵) ∈ (LocFin‘𝐽)) | ||
Theorem | locfincmp 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))) | ||
Theorem | unisngl 22065* | Taking the union of the set of singletons recovers the initial set. (Contributed by Thierry Arnoux, 9-Jan-2020.) |
⊢ 𝐶 = {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ⇒ ⊢ 𝑋 = ∪ 𝐶 | ||
Theorem | dissnref 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 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) → 𝐶Ref𝑌) | ||
Theorem | dissnlocfin 22067* | The set of singletons is locally finite in the discrete topology. (Contributed by Thierry Arnoux, 9-Jan-2020.) |
⊢ 𝐶 = {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ⇒ ⊢ (𝑋 ∈ 𝑉 → 𝐶 ∈ (LocFin‘𝒫 𝑋)) | ||
Theorem | locfindis 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)) | ||
Theorem | locfincf 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) ∧ 𝐽 ⊆ 𝐾) → (LocFin‘𝐽) ⊆ (LocFin‘𝐾)) | ||
Theorem | comppfsc 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 (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)))) | ||
Syntax | ckgen 22071 | Extend class notation with the compact generator operation. |
class 𝑘Gen | ||
Definition | df-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((𝑗 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘))}) | ||
Theorem | kgenval 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((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))}) | ||
Theorem | elkgen 22074* | Value of the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.) |
⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐴 ∈ (𝑘Gen‘𝐽) ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐴 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))))) | ||
Theorem | kgeni 22075 | Property of the open sets in the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.) |
⊢ ((𝐴 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐴 ∩ 𝐾) ∈ (𝐽 ↾t 𝐾)) | ||
Theorem | kgentopon 22076 | The compact generator generates a topology. (Contributed by Mario Carneiro, 22-Aug-2015.) |
⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ (TopOn‘𝑋)) | ||
Theorem | kgenuni 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)) | ||
Theorem | kgenftop 22078 | The compact generator generates a topology. (Contributed by Mario Carneiro, 20-Mar-2015.) |
⊢ (𝐽 ∈ Top → (𝑘Gen‘𝐽) ∈ Top) | ||
Theorem | kgenf 22079 | The compact generator is a function on topologies. (Contributed by Mario Carneiro, 20-Mar-2015.) |
⊢ 𝑘Gen:Top⟶Top | ||
Theorem | kgentop 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) | ||
Theorem | kgenss 22081 | The compact generator generates a finer topology than the original. (Contributed by Mario Carneiro, 20-Mar-2015.) |
⊢ (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽)) | ||
Theorem | kgenhaus 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) | ||
Theorem | kgencmp 22083 | The compact generator topology is the same as the original topology on compact subspaces. (Contributed by Mario Carneiro, 20-Mar-2015.) |
⊢ ((𝐽 ∈ Top ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐽 ↾t 𝐾) = ((𝑘Gen‘𝐽) ↾t 𝐾)) | ||
Theorem | kgencmp2 22084 | The compact generator topology has the same compact sets as the original topology. (Contributed by Mario Carneiro, 20-Mar-2015.) |
⊢ (𝐽 ∈ Top → ((𝐽 ↾t 𝐾) ∈ Comp ↔ ((𝑘Gen‘𝐽) ↾t 𝐾) ∈ Comp)) | ||
Theorem | kgenidm 22085 | The compact generator is idempotent on compactly generated spaces. (Contributed by Mario Carneiro, 20-Mar-2015.) |
⊢ (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽) | ||
Theorem | iskgen2 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)) | ||
Theorem | iskgen3 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((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) → 𝑥 ∈ 𝐽))) | ||
Theorem | llycmpkgen2 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 & ⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽 ↾t 𝑘) ∈ Comp) ⇒ ⊢ (𝜑 → 𝐽 ∈ ran 𝑘Gen) | ||
Theorem | cmpkgen 22089 | A compact space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ (𝐽 ∈ Comp → 𝐽 ∈ ran 𝑘Gen) | ||
Theorem | llycmpkgen 22090 | A locally compact space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ ran 𝑘Gen) | ||
Theorem | 1stckgenlem 22091 | The one-point compactification of ℕ is compact. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) & ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝐴) ⇒ ⊢ (𝜑 → (𝐽 ↾t (ran 𝐹 ∪ {𝐴})) ∈ Comp) | ||
Theorem | 1stckgen 22092 | A first-countable space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ (𝐽 ∈ 1stω → 𝐽 ∈ ran 𝑘Gen) | ||
Theorem | kgen2ss 22093 | The compact generator preserves the subset (fineness) relationship on topologies. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑘Gen‘𝐽) ⊆ (𝑘Gen‘𝐾)) | ||
Theorem | kgencn 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:𝑋⟶𝑌 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾))))) | ||
Theorem | kgencn2 22095* | A function 𝐹:𝐽⟶𝐾 from a compactly generated space is continuous iff for all compact spaces 𝑧 and continuous 𝑔:𝑧⟶𝐽, the composite 𝐹 ∘ 𝑔:𝑧⟶𝐾 is continuous. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑧 ∈ Comp ∀𝑔 ∈ (𝑧 Cn 𝐽)(𝐹 ∘ 𝑔) ∈ (𝑧 Cn 𝐾)))) | ||
Theorem | kgencn3 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))) | ||
Theorem | kgen2cn 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))) | ||
Syntax | ctx 22098 | Extend class notation with the binary topological product operation. |
class ×t | ||
Syntax | cxko 22099 | Extend class notation with a function whose value is the compact-open topology. |
class ↑ko | ||
Definition | df-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 (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)))) |
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