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Theorem List for Metamath Proof Explorer - 21901-22000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremresthauslem 21901 Lemma for resthaus 21906 and similar theorems. If the topological property 𝐴 is preserved under injective preimages, then property 𝐴 passes to subspaces. (Contributed by Mario Carneiro, 25-Aug-2015.)
(𝐽𝐴𝐽 ∈ Top)    &   ((𝐽𝐴 ∧ ( I ↾ (𝑆 𝐽)):(𝑆 𝐽)–1-1→(𝑆 𝐽) ∧ ( I ↾ (𝑆 𝐽)) ∈ ((𝐽t 𝑆) Cn 𝐽)) → (𝐽t 𝑆) ∈ 𝐴)       ((𝐽𝐴𝑆𝑉) → (𝐽t 𝑆) ∈ 𝐴)
 
Theoremlpcls 21902 The limit points of the closure of a subset are the same as the limit points of the set in a T1 space. (Contributed by Mario Carneiro, 26-Dec-2016.)
𝑋 = 𝐽       ((𝐽 ∈ Fre ∧ 𝑆𝑋) → ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆))
 
Theoremperfcls 21903 A subset of a perfect space is perfect iff its closure is perfect (and the closure is an actual perfect set, since it is both closed and perfect in the subspace topology). (Contributed by Mario Carneiro, 26-Dec-2016.)
𝑋 = 𝐽       ((𝐽 ∈ Fre ∧ 𝑆𝑋) → ((𝐽t 𝑆) ∈ Perf ↔ (𝐽t ((cls‘𝐽)‘𝑆)) ∈ Perf))
 
Theoremrestt0 21904 A subspace of a T0 topology is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
((𝐽 ∈ Kol2 ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Kol2)
 
Theoremrestt1 21905 A subspace of a T1 topology is T1. (Contributed by Mario Carneiro, 25-Aug-2015.)
((𝐽 ∈ Fre ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Fre)
 
Theoremresthaus 21906 A subspace of a Hausdorff topology is Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
((𝐽 ∈ Haus ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Haus)
 
Theoremt1sep2 21907* Any two points in a T1 space which have no separation are equal. (Contributed by Jeff Hankins, 1-Feb-2010.)
𝑋 = 𝐽       ((𝐽 ∈ Fre ∧ 𝐴𝑋𝐵𝑋) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵))
 
Theoremt1sep 21908* Any two distinct points in a T1 space are separated by an open set. (Contributed by Jeff Hankins, 1-Feb-2010.)
𝑋 = 𝐽       ((𝐽 ∈ Fre ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → ∃𝑜𝐽 (𝐴𝑜 ∧ ¬ 𝐵𝑜))
 
Theoremsncld 21909 A singleton is closed in a Hausdorff space. (Contributed by NM, 5-Mar-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Haus ∧ 𝑃𝑋) → {𝑃} ∈ (Clsd‘𝐽))
 
Theoremsshauslem 21910 Lemma for sshaus 21913 and similar theorems. If the topological property 𝐴 is preserved under injective preimages, then a topology finer than one with property 𝐴 also has property 𝐴. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝑋 = 𝐽    &   (𝐽𝐴𝐽 ∈ Top)    &   ((𝐽𝐴 ∧ ( I ↾ 𝑋):𝑋1-1𝑋 ∧ ( I ↾ 𝑋) ∈ (𝐾 Cn 𝐽)) → 𝐾𝐴)       ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐾𝐴)
 
Theoremsst0 21911 A topology finer than a T0 topology is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Kol2 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐾 ∈ Kol2)
 
Theoremsst1 21912 A topology finer than a T1 topology is T1. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Fre ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐾 ∈ Fre)
 
Theoremsshaus 21913 A topology finer than a Hausdorff topology is Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Haus ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐾 ∈ Haus)
 
Theoremregsep2 21914* In a regular space, a closed set is separated by open sets from a point not in it. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 25-Aug-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → ∃𝑥𝐽𝑦𝐽 (𝐶𝑥𝐴𝑦 ∧ (𝑥𝑦) = ∅))
 
Theoremisreg2 21915* A topological space is regular if any closed set is separated from any point not in it by neighborhoods. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 25-Aug-2015.)
(𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Reg ↔ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))))
 
Theoremdnsconst 21916 If a continuous mapping to a T1 space is constant on a dense subset, it is constant on the entire space. Note that ((cls‘𝐽)‘𝐴) = 𝑋 means "𝐴 is dense in 𝑋 " and 𝐴 ⊆ (𝐹 “ {𝑃}) means "𝐹 is constant on 𝐴 " (see funconstss 6819). (Contributed by NM, 15-Mar-2007.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
𝑋 = 𝐽    &   𝑌 = 𝐾       (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃𝑌𝐴 ⊆ (𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐹:𝑋⟶{𝑃})
 
Theoremordtt1 21917 The order topology is T1 for any poset. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑅 ∈ PosetRel → (ordTop‘𝑅) ∈ Fre)
 
Theoremlmmo 21918 A sequence in a Hausdorff space converges to at most one limit. Part of Lemma 1.4-2(a) of [Kreyszig] p. 26. (Contributed by NM, 31-Jan-2008.) (Proof shortened by Mario Carneiro, 1-May-2014.)
(𝜑𝐽 ∈ Haus)    &   (𝜑𝐹(⇝𝑡𝐽)𝐴)    &   (𝜑𝐹(⇝𝑡𝐽)𝐵)       (𝜑𝐴 = 𝐵)
 
Theoremlmfun 21919 The convergence relation is function-like in a Hausdorff space. (Contributed by Mario Carneiro, 26-Dec-2013.)
(𝐽 ∈ Haus → Fun (⇝𝑡𝐽))
 
Theoremdishaus 21920 A discrete topology is Hausdorff. Morris, Topology without tears, p.72, ex. 13. (Contributed by FL, 24-Jun-2007.) (Proof shortened by Mario Carneiro, 8-Apr-2015.)
(𝐴𝑉 → 𝒫 𝐴 ∈ Haus)
 
Theoremordthauslem 21921* Lemma for ordthaus 21922. (Contributed by Mario Carneiro, 13-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵 → (𝐴𝐵 → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅))))
 
Theoremordthaus 21922 The order topology of a total order is Hausdorff. (Contributed by Mario Carneiro, 13-Sep-2015.)
(𝑅 ∈ TosetRel → (ordTop‘𝑅) ∈ Haus)
 
Theoremxrhaus 21923 The topology of the extended reals is Hausdorff. (Contributed by Thierry Arnoux, 24-Mar-2017.)
(ordTop‘ ≤ ) ∈ Haus
 
12.1.11  Compactness
 
Syntaxccmp 21924 Extend class notation with the class of all compact spaces.
class Comp
 
Definitiondf-cmp 21925* Definition of a compact topology. A topology is compact iff any open covering of its underlying set contains a finite subcovering (Heine-Borel property). Definition C''' of [BourbakiTop1] p. I.59. Note: Bourbaki uses the term "quasi-compact" 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 21926* 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 21927* 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 21928* Rewrite cmpcov 21927 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 21929* Combine cmpcov 21927 with ac6sfi 8751 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 21930 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 21931 A finite topology is compact. (Contributed by FL, 22-Dec-2008.)
(𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Comp)
 
Theorem0cmp 21932 The singleton of the empty set is compact. (Contributed by FL, 2-Aug-2009.)
{∅} ∈ Comp
 
Theoremcmptop 21933 A compact topology is a topology. (Contributed by Jeff Hankins, 29-Jun-2009.)
(𝐽 ∈ Comp → 𝐽 ∈ Top)
 
Theoremrncmp 21934 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 21935 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 21936 A discrete topology is compact iff the base set is finite. (Contributed by Mario Carneiro, 19-Mar-2015.)
(𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Comp)
 
Theoremcmpsublem 21937* Lemma for cmpsub 21938. (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 21938* 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 21939* A topology generated by a basis is compact iff open covers drawn from the basis have finite subcovers. (See also alexsub 22583, 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 21940 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 21941 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 21942* 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 21943 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 21944* Lemma for hauscmp 21945. (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 21945 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 21946* 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 21947 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 21948* 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
 
Syntaxcconn 21949 Extend class notation with the class of all connected topologies.
class Conn
 
Definitiondf-conn 21950 Topologies are connected when only and 𝑗 are both open and closed. (Contributed by FL, 17-Nov-2008.)
Conn = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, 𝑗}}
 
Theoremisconn 21951 The predicate 𝐽 is a connected topology . (Contributed by FL, 17-Nov-2008.)
𝑋 = 𝐽       (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
 
Theoremisconn2 21952 The predicate 𝐽 is a connected topology . (Contributed by Mario Carneiro, 10-Mar-2015.)
𝑋 = 𝐽       (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋}))
 
Theoremconnclo 21953 The only nonempty clopen set of a connected topology is the whole space. (Contributed by Mario Carneiro, 10-Mar-2015.)
𝑋 = 𝐽    &   (𝜑𝐽 ∈ Conn)    &   (𝜑𝐴𝐽)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑𝐴 ∈ (Clsd‘𝐽))       (𝜑𝐴 = 𝑋)
 
Theoremconndisj 21954 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 ∈ Conn)    &   (𝜑𝐴𝐽)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑𝐵𝐽)    &   (𝜑𝐵 ≠ ∅)    &   (𝜑 → (𝐴𝐵) = ∅)       (𝜑 → (𝐴𝐵) ≠ 𝑋)
 
Theoremconntop 21955 A connected topology is a topology. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 14-Dec-2013.)
(𝐽 ∈ Conn → 𝐽 ∈ Top)
 
Theoremindisconn 21956 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} ∈ Conn
 
Theoremdfconn2 21957* An alternate definition of connectedness. (Contributed by Jeff Hankins, 9-Jul-2009.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
(𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Conn ↔ ∀𝑥𝐽𝑦𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥𝑦) = ∅) → (𝑥𝑦) ≠ 𝑋)))
 
Theoremconnsuba 21958* Connectedness for a subspace. See connsub 21959. (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 21959* 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 21960 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 21961 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 21962 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 21963 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 21964 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 21965* Lemma for iunconn 21966. (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 21966* 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 21967 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 21968 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 21969* 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 21970* 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 21971* 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 21972* 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 21973* 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 21974 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 21975 Extend class definition to include the class of all first-countable topologies.
class 1stω
 
Syntaxc2ndc 21976 Extend class definition to include the class of all second-countable topologies.
class 2ndω
 
Definitiondf-1stc 21977* 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 21978* 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 21979* 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 21980* 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 21981 A first-countable topology is a topology. (Contributed by Jeff Hankins, 22-Aug-2009.)
(𝐽 ∈ 1stω → 𝐽 ∈ Top)
 
Theorem1stcclb 21982* 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 21983* 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 21984* 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 21985 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 21986 A countable basis generates a second-countable topology. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → (topGen‘𝐵) ∈ 2ndω)
 
Theorem2ndcsb 21987* 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 21988 A second-countable space has at most the cardinality of the continuum. (Contributed by Mario Carneiro, 9-Apr-2015.)
(𝐽 ∈ 2ndω → 𝐽 ≼ ℝ)
 
Theorem2ndc1stc 21989 A second-countable space is first-countable. (Contributed by Jeff Hankins, 17-Jan-2010.)
(𝐽 ∈ 2ndω → 𝐽 ∈ 1stω)
 
Theorem1stcrestlem 21990* Lemma for 1stcrest 21991. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)
(𝐵 ≼ ω → ran (𝑥𝐵𝐶) ≼ ω)
 
Theorem1stcrest 21991 A subspace of a first-countable space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝐽 ∈ 1stω ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ 1stω)
 
Theorem2ndcrest 21992 A subspace of a second-countable space is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝐽 ∈ 2ndω ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ 2ndω)
 
Theorem2ndcctbss 21993* 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 21994* 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 21995* 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 21996* 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 21997* 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 21998 A discrete space is second-countable iff it is countable. (Contributed by Mario Carneiro, 13-Apr-2015.)
(𝑋 ≼ ω ↔ 𝒫 𝑋 ∈ 2ndω)
 
Theorem1stcelcls 21999* 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 9846. 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 22000* 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 9846, but only via 1stcelcls 21999, 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)))))
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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 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
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