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Theorem List for Metamath Proof Explorer - 21601-21700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremconnsuba 21601* Connectedness for a subspace. See connsub 21602. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → ((𝐽t 𝐴) ∈ Conn ↔ ∀𝑥𝐽𝑦𝐽 (((𝑥𝐴) ≠ ∅ ∧ (𝑦𝐴) ≠ ∅ ∧ ((𝑥𝑦) ∩ 𝐴) = ∅) → ((𝑥𝑦) ∩ 𝐴) ≠ 𝐴)))

Theoremconnsub 21602* Two equivalent ways of saying that a subspace topology is connected. (Contributed by Jeff Hankins, 9-Jul-2009.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → ((𝐽t 𝑆) ∈ Conn ↔ ∀𝑥𝐽𝑦𝐽 (((𝑥𝑆) ≠ ∅ ∧ (𝑦𝑆) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋𝑆)) → ¬ 𝑆 ⊆ (𝑥𝑦))))

Theoremcnconn 21603 Connectedness is respected by a continuous onto map. (Contributed by Jeff Hankins, 12-Jul-2009.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
𝑌 = 𝐾       ((𝐽 ∈ Conn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Conn)

Theoremnconnsubb 21604 Disconnectedness for a subspace. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐴𝑋)    &   (𝜑𝑈𝐽)    &   (𝜑𝑉𝐽)    &   (𝜑 → (𝑈𝐴) ≠ ∅)    &   (𝜑 → (𝑉𝐴) ≠ ∅)    &   (𝜑 → ((𝑈𝑉) ∩ 𝐴) = ∅)    &   (𝜑𝐴 ⊆ (𝑈𝑉))       (𝜑 → ¬ (𝐽t 𝐴) ∈ Conn)

Theoremconnsubclo 21605 If a clopen set meets a connected subspace, it must contain the entire subspace. (Contributed by Mario Carneiro, 10-Mar-2015.)
𝑋 = 𝐽    &   (𝜑𝐴𝑋)    &   (𝜑 → (𝐽t 𝐴) ∈ Conn)    &   (𝜑𝐵𝐽)    &   (𝜑 → (𝐵𝐴) ≠ ∅)    &   (𝜑𝐵 ∈ (Clsd‘𝐽))       (𝜑𝐴𝐵)

Theoremconnima 21606 The image of a connected set is connected. (Contributed by Mario Carneiro, 7-Jul-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
𝑋 = 𝐽    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐴𝑋)    &   (𝜑 → (𝐽t 𝐴) ∈ Conn)       (𝜑 → (𝐾t (𝐹𝐴)) ∈ Conn)

Theoremconncn 21607 A continuous function from a connected topology with one point in a clopen set must lie entirely within the set. (Contributed by Mario Carneiro, 16-Feb-2015.)
𝑋 = 𝐽    &   (𝜑𝐽 ∈ Conn)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑈𝐾)    &   (𝜑𝑈 ∈ (Clsd‘𝐾))    &   (𝜑𝐴𝑋)    &   (𝜑 → (𝐹𝐴) ∈ 𝑈)       (𝜑𝐹:𝑋𝑈)

Theoremiunconnlem 21608* Lemma for iunconn 21609. (Contributed by Mario Carneiro, 11-Jun-2014.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   ((𝜑𝑘𝐴) → 𝐵𝑋)    &   ((𝜑𝑘𝐴) → 𝑃𝐵)    &   ((𝜑𝑘𝐴) → (𝐽t 𝐵) ∈ Conn)    &   (𝜑𝑈𝐽)    &   (𝜑𝑉𝐽)    &   (𝜑 → (𝑉 𝑘𝐴 𝐵) ≠ ∅)    &   (𝜑 → (𝑈𝑉) ⊆ (𝑋 𝑘𝐴 𝐵))    &   (𝜑 𝑘𝐴 𝐵 ⊆ (𝑈𝑉))    &   𝑘𝜑       (𝜑 → ¬ 𝑃𝑈)

Theoremiunconn 21609* The indexed union of connected overlapping subspaces sharing a common point is connected. (Contributed by Mario Carneiro, 11-Jun-2014.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   ((𝜑𝑘𝐴) → 𝐵𝑋)    &   ((𝜑𝑘𝐴) → 𝑃𝐵)    &   ((𝜑𝑘𝐴) → (𝐽t 𝐵) ∈ Conn)       (𝜑 → (𝐽t 𝑘𝐴 𝐵) ∈ Conn)

Theoremunconn 21610 The union of two connected overlapping subspaces is connected. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 11-Jun-2014.)
((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ≠ ∅) → (((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn) → (𝐽t (𝐴𝐵)) ∈ Conn))

Theoremclsconn 21611 The closure of a connected set is connected. (Contributed by Mario Carneiro, 19-Mar-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) → (𝐽t ((cls‘𝐽)‘𝐴)) ∈ Conn)

Theoremconncompid 21612* The connected component containing 𝐴 contains 𝐴. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑆)

Theoremconncompconn 21613* The connected component containing 𝐴 is connected. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝑆) ∈ Conn)

Theoremconncompss 21614* The connected component containing 𝐴 is a superset of any other connected set containing 𝐴. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}       ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → 𝑇𝑆)

Theoremconncompcld 21615* The connected component containing 𝐴 is a closed set. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑆 ∈ (Clsd‘𝐽))

Theoremconncompclo 21616* The connected component containing 𝐴 is a subset of any clopen set containing 𝐴. (Contributed by Mario Carneiro, 20-Sep-2015.)
𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → 𝑆𝑇)

Theoremt1connperf 21617 A connected T1 space is perfect, unless it is the topology of a singleton. (Contributed by Mario Carneiro, 26-Dec-2016.)
𝑋 = 𝐽       ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn ∧ ¬ 𝑋 ≈ 1o) → 𝐽 ∈ Perf)

12.1.14  First- and second-countability

Syntaxc1stc 21618 Extend class definition to include the class of all first-countable topologies.
class 1st𝜔

Syntaxc2ndc 21619 Extend class definition to include the class of all second-countable topologies.
class 2nd𝜔

Definitiondf-1stc 21620* Define the class of all first-countable topologies. (Contributed by Jeff Hankins, 22-Aug-2009.)
1st𝜔 = {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)))}

Definitiondf-2ndc 21621* Define the class of all second-countable topologies. (Contributed by Jeff Hankins, 17-Jan-2010.)
2nd𝜔 = {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)}

Theoremis1stc 21622* The predicate "is a first-countable topology." This can be described as "every point has a countable local basis" - that is, every point has a countable collection of open sets containing it such that every open set containing the point has an open set from this collection as a subset. (Contributed by Jeff Hankins, 22-Aug-2009.)
𝑋 = 𝐽       (𝐽 ∈ 1st𝜔 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)))))

Theoremis1stc2 21623* An equivalent way of saying "is a first-countable topology." (Contributed by Jeff Hankins, 22-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.)
𝑋 = 𝐽       (𝐽 ∈ 1st𝜔 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧𝐽 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)))))

Theorem1stctop 21624 A first-countable topology is a topology. (Contributed by Jeff Hankins, 22-Aug-2009.)
(𝐽 ∈ 1st𝜔 → 𝐽 ∈ Top)

Theorem1stcclb 21625* A property of points in a first-countable topology. (Contributed by Jeff Hankins, 22-Aug-2009.)
𝑋 = 𝐽       ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) → ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦))))

Theorem1stcfb 21626* For any point 𝐴 in a first-countable topology, there is a function 𝑓:ℕ⟶𝐽 enumerating neighborhoods of 𝐴 which is decreasing and forms a local base. (Contributed by Mario Carneiro, 21-Mar-2015.)
𝑋 = 𝐽       ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) → ∃𝑓(𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓𝑘)) ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑘 ∈ ℕ (𝑓𝑘) ⊆ 𝑦)))

Theoremis2ndc 21627* The property of being second-countable. (Contributed by Jeff Hankins, 17-Jan-2010.) (Revised by Mario Carneiro, 21-Mar-2015.)
(𝐽 ∈ 2nd𝜔 ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽))

Theorem2ndctop 21628 A second-countable topology is a topology. (Contributed by Jeff Hankins, 17-Jan-2010.) (Revised by Mario Carneiro, 21-Mar-2015.)
(𝐽 ∈ 2nd𝜔 → 𝐽 ∈ Top)

Theorem2ndci 21629 A countable basis generates a second-countable topology. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → (topGen‘𝐵) ∈ 2nd𝜔)

Theorem2ndcsb 21630* Having a countable subbase is a sufficient condition for second-countability. (Contributed by Jeff Hankins, 17-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
(𝐽 ∈ 2nd𝜔 ↔ ∃𝑥(𝑥 ≼ ω ∧ (topGen‘(fi‘𝑥)) = 𝐽))

Theorem2ndcredom 21631 A second-countable space has at most the cardinality of the continuum. (Contributed by Mario Carneiro, 9-Apr-2015.)
(𝐽 ∈ 2nd𝜔 → 𝐽 ≼ ℝ)

Theorem2ndc1stc 21632 A second-countable space is first-countable. (Contributed by Jeff Hankins, 17-Jan-2010.)
(𝐽 ∈ 2nd𝜔 → 𝐽 ∈ 1st𝜔)

Theorem1stcrestlem 21633* Lemma for 1stcrest 21634. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)
(𝐵 ≼ ω → ran (𝑥𝐵𝐶) ≼ ω)

Theorem1stcrest 21634 A subspace of a first-countable space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ 1st𝜔)

Theorem2ndcrest 21635 A subspace of a second-countable space is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝐽 ∈ 2nd𝜔 ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ 2nd𝜔)

Theorem2ndcctbss 21636* 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.)
𝐽 = (topGen‘𝐵)    &   𝑆 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝑐𝑣𝑐 ∧ ∃𝑤𝐵 (𝑢𝑤𝑤𝑣))}       ((𝐵 ∈ TopBases ∧ 𝐽 ∈ 2nd𝜔) → ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ 𝑏𝐵𝐽 = (topGen‘𝑏)))

Theorem2ndcdisj 21637* 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 21638* 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 21639* 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 21640* 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 21641 A discrete space is second-countable iff it is countable. (Contributed by Mario Carneiro, 13-Apr-2015.)
(𝑋 ≼ ω ↔ 𝒫 𝑋 ∈ 2nd𝜔)

Theorem1stcelcls 21642* 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 9579. 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 21643* 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 9579, but only via 1stcelcls 21642, 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 21644* 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 21645 Extend class notation with the "locally 𝐴 " predicate of a topological space.
class Locally 𝐴

Syntaxcnlly 21646 Extend class notation with the "N-locally 𝐴 " predicate of a topological space.
class 𝑛-Locally 𝐴

Definitiondf-lly 21647* 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 21648* 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 21649* 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 21650* 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 21651 Equality theorem for the Locally 𝐴 predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐴 = 𝐵 → Locally 𝐴 = Locally 𝐵)

Theoremnllyeq 21652 Equality theorem for the Locally 𝐴 predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐴 = 𝐵 → 𝑛-Locally 𝐴 = 𝑛-Locally 𝐵)

Theoremllytop 21653 A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐽 ∈ Locally 𝐴𝐽 ∈ Top)

Theoremnllytop 21654 A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐽 ∈ 𝑛-Locally 𝐴𝐽 ∈ Top)

Theoremllyi 21655* The property of a locally 𝐴 topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
((𝐽 ∈ Locally 𝐴𝑈𝐽𝑃𝑈) → ∃𝑢𝐽 (𝑢𝑈𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))

Theoremnllyi 21656* 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 21657* Eliminate the neighborhood symbol from nllyi 21656. (Contributed by Mario Carneiro, 2-Mar-2015.)
((𝐽 ∈ 𝑛-Locally 𝐴𝑈𝐽𝑃𝑈) → ∃𝑠 ∈ 𝒫 𝑈𝑢𝐽 (𝑃𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))

Theoremllynlly 21658 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 21659 A locally 𝐴 space is n-locally 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally 𝐴 ⊆ 𝑛-Locally 𝐴

Theoremllyss 21660 The "locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐴𝐵 → Locally 𝐴 ⊆ Locally 𝐵)

Theoremnllyss 21661 The "n-locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐴𝐵 → 𝑛-Locally 𝐴 ⊆ 𝑛-Locally 𝐵)

Theoremsubislly 21662* 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 21663* 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 21664* 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 21665* 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 21666 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 21667 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 21668 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 21669 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 21670 Idempotence of the "n-locally" predicate, i.e. being "n-locally 𝐴 " is a local property. (Use loclly 21668 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 21671 A topology is locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally Top = Top

Theoremtopnlly 21672 A topology is n-locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛-Locally Top = Top

Theoremhauslly 21673 A Hausdorff space is locally Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐽 ∈ Haus → 𝐽 ∈ Locally Haus)

Theoremhausnlly 21674 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 21675 A compact Hausdorff space is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → 𝐽 ∈ 𝑛-Locally Comp)

Theoremcldllycmp 21676 A closed subspace of a locally compact space is also locally compact. (The analogous result for open subspaces follows from the more general nllyrest 21667.) (Contributed by Mario Carneiro, 2-Mar-2015.)
((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽t 𝐴) ∈ 𝑛-Locally Comp)

Theoremlly1stc 21677 First-countability is a local property (unlike second-countability). (Contributed by Mario Carneiro, 21-Mar-2015.)
Locally 1st𝜔 = 1st𝜔

Theoremdislly 21678* 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 21679 A discrete space is locally compact. (Contributed by Mario Carneiro, 20-Mar-2015.)
(𝑋𝑉 → 𝒫 𝑋 ∈ Locally Comp)

Theoremdis1stc 21680 A discrete space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝑋𝑉 → 𝒫 𝑋 ∈ 1st𝜔)

Theoremhausmapdom 21681 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 22169 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 21682 Simplify the cardinal 𝐴↑ℕ of hausmapdom 21681 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 21683 Extend class definition to include the refinement relation.
class Ref

Syntaxcptfin 21684 Extend class definition to include the class of point-finite covers.
class PtFin

Syntaxclocfin 21685 Extend class definition to include the class of locally finite covers.
class LocFin

Definitiondf-ref 21686* Define the refinement relation. (Contributed by Jeff Hankins, 18-Jan-2010.)
Ref = {⟨𝑥, 𝑦⟩ ∣ ( 𝑦 = 𝑥 ∧ ∀𝑧𝑥𝑤𝑦 𝑧𝑤)}

Definitiondf-ptfin 21687* Define "point-finite." (Contributed by Jeff Hankins, 21-Jan-2010.)
PtFin = {𝑥 ∣ ∀𝑦 𝑥{𝑧𝑥𝑦𝑧} ∈ Fin}

Definitiondf-locfin 21688* 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 21689 Refinement is a relation. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
Rel Ref

Theoremisref 21690* 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 32867. 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 21691 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 21692* 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 21693 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 21694 Reflexivity of refinement. (Contributed by Jeff Hankins, 18-Jan-2010.)
(𝐴𝑉𝐴Ref𝐴)

Theoremreftr 21695 Refinement is transitive. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
((𝐴Ref𝐵𝐵Ref𝐶) → 𝐴Ref𝐶)

Theoremrefun0 21696 Adding the empty set preserves refinements. (Contributed by Thierry Arnoux, 31-Jan-2020.)
((𝐴Ref𝐵𝐵 ≠ ∅) → (𝐴 ∪ {∅})Ref𝐵)

Theoremisptfin 21697* The statement "is a point-finite cover." (Contributed by Jeff Hankins, 21-Jan-2010.)
𝑋 = 𝐴       (𝐴𝐵 → (𝐴 ∈ PtFin ↔ ∀𝑥𝑋 {𝑦𝐴𝑥𝑦} ∈ Fin))

Theoremislocfin 21698* 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 21699 A finite cover is a point-finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.)
(𝐴 ∈ Fin → 𝐴 ∈ PtFin)

Theoremptfinfin 21700* 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)

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