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Theorem List for Metamath Proof Explorer - 20901-21000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremordthaus 20901 The order topology of a total order is Hausdorff. (Contributed by Mario Carneiro, 13-Sep-2015.)
(𝑅 ∈ TosetRel → (ordTop‘𝑅) ∈ Haus)

12.1.11  Compactness

Syntaxccmp 20902 Extend class notation with the class of all compact spaces.
class Comp

Definitiondf-cmp 20903* Definition of a compact topology. A topology is compact iff any open covering of its underlying set contains a finite sub-covering (Heine-Borel property). Definition C''' of [BourbakiTop1] p. I.59. Note: Bourbaki uses the term quasi-compact topology but it is not the modern usage (which we follow). (Contributed by FL, 22-Dec-2008.)
Comp = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥( 𝑥 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝑥 = 𝑧)}

Theoremiscmp 20904* The predicate "is a compact topology". (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 11-Feb-2015.)
𝑋 = 𝐽       (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)))

Theoremcmpcov 20905* An open cover of a compact topology has a finite subcover. (Contributed by Jeff Hankins, 29-Jun-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Comp ∧ 𝑆𝐽𝑋 = 𝑆) → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = 𝑠)

Theoremcmpcov2 20906* Rewrite cmpcov 20905 for the cover {𝑦𝐽𝜑}. (Contributed by Mario Carneiro, 11-Sep-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦𝜑)) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑))

Theoremcmpcovf 20907* Combine cmpcov 20905 with ac6sfi 7965 to show the existence of a function that indexes the elements that are generating the open cover. (Contributed by Mario Carneiro, 14-Sep-2014.)
𝑋 = 𝐽    &   (𝑧 = (𝑓𝑦) → (𝜑𝜓))       ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∃𝑧𝐴 𝜑)) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑠 ∧ ∃𝑓(𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓)))

Theoremcncmp 20908 Compactness is respected by a continuous onto map. (Contributed by Jeff Hankins, 12-Jul-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
𝑌 = 𝐾       ((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Comp)

Theoremfincmp 20909 A finite topology is compact. (Contributed by FL, 22-Dec-2008.)
(𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Comp)

Theorem0cmp 20910 The singleton of the empty set is compact. (Contributed by FL, 2-Aug-2009.)
{∅} ∈ Comp

Theoremcmptop 20911 A compact topology is a topology. (Contributed by Jeff Hankins, 29-Jun-2009.)
(𝐽 ∈ Comp → 𝐽 ∈ Top)

Theoremrncmp 20912 The image of a compact set under a continuous function is compact. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐾t ran 𝐹) ∈ Comp)

Theoremimacmp 20913 The image of a compact set under a continuous function is compact. (Contributed by Mario Carneiro, 18-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝐴) ∈ Comp) → (𝐾t (𝐹𝐴)) ∈ Comp)

Theoremdiscmp 20914 A discrete topology is compact iff the base set is finite. (Contributed by Mario Carneiro, 19-Mar-2015.)
(𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Comp)

Theoremcmpsublem 20915* Lemma for cmpsub 20916. (Contributed by Jeff Hankins, 28-Jun-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → (∀𝑐 ∈ 𝒫 𝐽(𝑆 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑑) → ∀𝑠 ∈ 𝒫 (𝐽t 𝑆)( (𝐽t 𝑆) = 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin) (𝐽t 𝑆) = 𝑡)))

Theoremcmpsub 20916* Two equivalent ways of describing a compact subset of a topological space. Inspired by Sue E. Goodman's Beginning Topology. (Contributed by Jeff Hankins, 22-Jun-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐽t 𝑆) ∈ Comp ↔ ∀𝑐 ∈ 𝒫 𝐽(𝑆 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑑)))

Theoremtgcmp 20917* A topology generated by a basis is compact iff open covers drawn from the basis have finite subcovers. (See also alexsub 21562, which further specializes to subbases, assuming the ultrafilter lemma.) (Contributed by Mario Carneiro, 26-Aug-2015.)
((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → ((topGen‘𝐵) ∈ Comp ↔ ∀𝑦 ∈ 𝒫 𝐵(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)))

Theoremcmpcld 20918 A closed subset of a compact space is compact. (Contributed by Jeff Hankins, 29-Jun-2009.)
((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝐽t 𝑆) ∈ Comp)

Theoremuncmp 20919 The union of two compact sets is compact. (Contributed by Jeff Hankins, 30-Jan-2010.)
𝑋 = 𝐽       (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ ((𝐽t 𝑆) ∈ Comp ∧ (𝐽t 𝑇) ∈ Comp)) → 𝐽 ∈ Comp)

Theoremfiuncmp 20920* A finite union of compact sets is compact. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝐽t 𝐵) ∈ Comp) → (𝐽t 𝑥𝐴 𝐵) ∈ Comp)

Theoremsscmp 20921 A subset of a compact topology (i.e. a coarser topology) is compact. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝑋 = 𝐾       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) → 𝐽 ∈ Comp)

Theoremhauscmplem 20922* Lemma for hauscmp 20923. (Contributed by Mario Carneiro, 27-Nov-2013.)
𝑋 = 𝐽    &   𝑂 = {𝑦𝐽 ∣ ∃𝑤𝐽 (𝐴𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋𝑦))}    &   (𝜑𝐽 ∈ Haus)    &   (𝜑𝑆𝑋)    &   (𝜑 → (𝐽t 𝑆) ∈ Comp)    &   (𝜑𝐴 ∈ (𝑋𝑆))       (𝜑 → ∃𝑧𝐽 (𝐴𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋𝑆)))

Theoremhauscmp 20923 A compact subspace of a T2 space is closed. (Contributed by Jeff Hankins, 16-Jan-2010.) (Proof shortened by Mario Carneiro, 14-Dec-2013.)
𝑋 = 𝐽       ((𝐽 ∈ Haus ∧ 𝑆𝑋 ∧ (𝐽t 𝑆) ∈ Comp) → 𝑆 ∈ (Clsd‘𝐽))

Theoremcmpfi 20924* If a topology is compact and a collection of closed sets has the finite intersection property, its intersection is nonempty. (Contributed by Jeff Hankins, 25-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
(𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅)))

Theoremcmpfii 20925 In a compact topology, a system of closed sets with nonempty finite intersections has a nonempty intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝐽 ∈ Comp ∧ 𝑋 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑋)) → 𝑋 ≠ ∅)

12.1.12  Bolzano-Weierstrass theorem

Theorembwth 20926* The glorious Bolzano-Weierstrass theorem. The first general topology theorem ever proved. The first mention of this theorem can be found in a course by Weierstrass from 1865. In his course Weierstrass called it a lemma. He didn't know how famous this theorem would be. He used a Euclidean space instead of a general compact space. And he was not aware of the Heine-Borel property. But the concepts of neighborhood and limit point were already there although not precisely defined. Cantor was one of his students. He published and used the theorem in an article from 1872. The rest of the general topology followed from that. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) Revised by BL to significantly shorten the proof and avoid infinity, regularity, and choice. (Revised by Brendan Leahy, 26-Dec-2018.)
𝑋 = 𝐽       ((𝐽 ∈ Comp ∧ 𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → ∃𝑥𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝐴))

12.1.13  Connectedness

Syntaxccon 20927 Extend class notation with the class of all connected topologies.
class Con

Definitiondf-con 20928 Topologies are connected when only and 𝑗 are both open and closed. (Contributed by FL, 17-Nov-2008.)
Con = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, 𝑗}}

Theoremiscon 20929 The predicate 𝐽 is a connected topology . (Contributed by FL, 17-Nov-2008.)
𝑋 = 𝐽       (𝐽 ∈ Con ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))

Theoremiscon2 20930 The predicate 𝐽 is a connected topology . (Contributed by Mario Carneiro, 10-Mar-2015.)
𝑋 = 𝐽       (𝐽 ∈ Con ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋}))

Theoremconclo 20931 The only nonempty clopen set of a connected topology is the whole space. (Contributed by Mario Carneiro, 10-Mar-2015.)
𝑋 = 𝐽    &   (𝜑𝐽 ∈ Con)    &   (𝜑𝐴𝐽)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑𝐴 ∈ (Clsd‘𝐽))       (𝜑𝐴 = 𝑋)

Theoremconndisj 20932 If a topology is connected, its underlying set can't be partitioned into two nonempty non-overlapping open sets. (Contributed by FL, 16-Nov-2008.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
𝑋 = 𝐽    &   (𝜑𝐽 ∈ Con)    &   (𝜑𝐴𝐽)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑𝐵𝐽)    &   (𝜑𝐵 ≠ ∅)    &   (𝜑 → (𝐴𝐵) = ∅)       (𝜑 → (𝐴𝐵) ≠ 𝑋)

Theoremcontop 20933 A connected topology is a topology. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 14-Dec-2013.)
(𝐽 ∈ Con → 𝐽 ∈ Top)

Theoremindiscon 20934 The indiscrete topology (or trivial topology) on any set is connected. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 14-Aug-2015.)
{∅, 𝐴} ∈ Con

Theoremdfcon2 20935* An alternate definition of connectedness. (Contributed by Jeff Hankins, 9-Jul-2009.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
(𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Con ↔ ∀𝑥𝐽𝑦𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥𝑦) = ∅) → (𝑥𝑦) ≠ 𝑋)))

Theoremconsuba 20936* Connectedness for a subspace. See connsub 20937. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → ((𝐽t 𝐴) ∈ Con ↔ ∀𝑥𝐽𝑦𝐽 (((𝑥𝐴) ≠ ∅ ∧ (𝑦𝐴) ≠ ∅ ∧ ((𝑥𝑦) ∩ 𝐴) = ∅) → ((𝑥𝑦) ∩ 𝐴) ≠ 𝐴)))

Theoremconnsub 20937* 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) ∈ Con ↔ ∀𝑥𝐽𝑦𝐽 (((𝑥𝑆) ≠ ∅ ∧ (𝑦𝑆) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋𝑆)) → ¬ 𝑆 ⊆ (𝑥𝑦))))

Theoremcnconn 20938 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 ∈ Con ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Con)

Theoremnconsubb 20939 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) ∈ Con)

Theoremconsubclo 20940 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) ∈ Con)    &   (𝜑𝐵𝐽)    &   (𝜑 → (𝐵𝐴) ≠ ∅)    &   (𝜑𝐵 ∈ (Clsd‘𝐽))       (𝜑𝐴𝐵)

Theoremconima 20941 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) ∈ Con)       (𝜑 → (𝐾t (𝐹𝐴)) ∈ Con)

Theoremconcn 20942 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 ∈ Con)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑈𝐾)    &   (𝜑𝑈 ∈ (Clsd‘𝐾))    &   (𝜑𝐴𝑋)    &   (𝜑 → (𝐹𝐴) ∈ 𝑈)       (𝜑𝐹:𝑋𝑈)

Theoremiunconlem 20943* Lemma for iuncon 20944. (Contributed by Mario Carneiro, 11-Jun-2014.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   ((𝜑𝑘𝐴) → 𝐵𝑋)    &   ((𝜑𝑘𝐴) → 𝑃𝐵)    &   ((𝜑𝑘𝐴) → (𝐽t 𝐵) ∈ Con)    &   (𝜑𝑈𝐽)    &   (𝜑𝑉𝐽)    &   (𝜑 → (𝑉 𝑘𝐴 𝐵) ≠ ∅)    &   (𝜑 → (𝑈𝑉) ⊆ (𝑋 𝑘𝐴 𝐵))    &   (𝜑 𝑘𝐴 𝐵 ⊆ (𝑈𝑉))    &   𝑘𝜑       (𝜑 → ¬ 𝑃𝑈)

Theoremiuncon 20944* 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) ∈ Con)       (𝜑 → (𝐽t 𝑘𝐴 𝐵) ∈ Con)

Theoremuncon 20945 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) ∈ Con ∧ (𝐽t 𝐵) ∈ Con) → (𝐽t (𝐴𝐵)) ∈ Con))

Theoremclscon 20946 The closure of a connected set is connected. (Contributed by Mario Carneiro, 19-Mar-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Con) → (𝐽t ((cls‘𝐽)‘𝐴)) ∈ Con)

Theoremconcompid 20947* The connected component containing 𝐴 contains 𝐴. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Con)}       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑆)

Theoremconcompcon 20948* The connected component containing 𝐴 is connected. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Con)}       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝑆) ∈ Con)

Theoremconcompss 20949* 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) ∈ Con)}       ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Con) → 𝑇𝑆)

Theoremconcompcld 20950* The connected component containing 𝐴 is a closed set. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Con)}       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑆 ∈ (Clsd‘𝐽))

Theoremconcompclo 20951* 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) ∈ Con)}       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴𝑇) → 𝑆𝑇)

Theoremt1conperf 20952 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 ∈ Con ∧ ¬ 𝑋 ≈ 1𝑜) → 𝐽 ∈ Perf)

12.1.14  First- and second-countability

Syntaxc1stc 20953 Extend class definition to include the class of all first-countable topologies.
class 1st𝜔

Syntaxc2ndc 20954 Extend class definition to include the class of all second-countable topologies.
class 2nd𝜔

Definitiondf-1stc 20955* 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 20956* 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 20957* 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 20958* 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 20959 A first-countable topology is a topology. (Contributed by Jeff Hankins, 22-Aug-2009.)
(𝐽 ∈ 1st𝜔 → 𝐽 ∈ Top)

Theorem1stcclb 20960* 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 20961* 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 20962* 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 20963 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 20964 A countable basis generates a second-countable topology. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → (topGen‘𝐵) ∈ 2nd𝜔)

Theorem2ndcsb 20965* 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 20966 A second-countable space has at most the cardinality of the continuum. (Contributed by Mario Carneiro, 9-Apr-2015.)
(𝐽 ∈ 2nd𝜔 → 𝐽 ≼ ℝ)

Theorem2ndc1stc 20967 A second-countable space is first-countable. (Contributed by Jeff Hankins, 17-Jan-2010.)
(𝐽 ∈ 2nd𝜔 → 𝐽 ∈ 1st𝜔)

Theorem1stcrestlem 20968* Lemma for 1stcrest 20969. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)
(𝐵 ≼ ω → ran (𝑥𝐵𝐶) ≼ ω)

Theorem1stcrest 20969 A subspace of a first-countable space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ 1st𝜔)

Theorem2ndcrest 20970 A subspace of a second-countable space is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝐽 ∈ 2nd𝜔 ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ 2nd𝜔)

Theorem2ndcctbss 20971* 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 20972* 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 20973* 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 20974* 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 20975* 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 20976 A discrete space is second-countable iff it is countable. (Contributed by Mario Carneiro, 13-Apr-2015.)
(𝑋 ≼ ω ↔ 𝒫 𝑋 ∈ 2nd𝜔)

Theorem1stcelcls 20977* 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 9016. 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 20978* 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 9016, but only via 1stcelcls 20977, 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 20979* 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 20980 Extend class notation with the "locally 𝐴 " predicate of a topological space.
class Locally 𝐴

Syntaxcnlly 20981 Extend class notation with the "N-locally 𝐴 " predicate of a topological space.
class 𝑛-Locally 𝐴

Definitiondf-lly 20982* 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 20983* 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 20984* 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 20985* 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 20986 Equality theorem for the Locally 𝐴 predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐴 = 𝐵 → Locally 𝐴 = Locally 𝐵)

Theoremnllyeq 20987 Equality theorem for the Locally 𝐴 predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐴 = 𝐵 → 𝑛-Locally 𝐴 = 𝑛-Locally 𝐵)

Theoremllytop 20988 A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐽 ∈ Locally 𝐴𝐽 ∈ Top)

Theoremnllytop 20989 A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐽 ∈ 𝑛-Locally 𝐴𝐽 ∈ Top)

Theoremllyi 20990* The property of a locally 𝐴 topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
((𝐽 ∈ Locally 𝐴𝑈𝐽𝑃𝑈) → ∃𝑢𝐽 (𝑢𝑈𝑃𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))

Theoremnllyi 20991* 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 20992* Eliminate the neighborhood symbol from nllyi 20991. (Contributed by Mario Carneiro, 2-Mar-2015.)
((𝐽 ∈ 𝑛-Locally 𝐴𝑈𝐽𝑃𝑈) → ∃𝑠 ∈ 𝒫 𝑈𝑢𝐽 (𝑃𝑢𝑢𝑠 ∧ (𝐽t 𝑠) ∈ 𝐴))

Theoremllynlly 20993 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 20994 A locally 𝐴 space is n-locally 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally 𝐴 ⊆ 𝑛-Locally 𝐴

Theoremllyss 20995 The "locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐴𝐵 → Locally 𝐴 ⊆ Locally 𝐵)

Theoremnllyss 20996 The "n-locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)
(𝐴𝐵 → 𝑛-Locally 𝐴 ⊆ 𝑛-Locally 𝐵)

Theoremsubislly 20997* 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 20998* 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 20999* 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 21000* 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))))

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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
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