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Theorem List for Metamath Proof Explorer - 21201-21300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremist1-3 21201* A space is T1 iff every point is the only point in the intersection of all open sets containing that point. (Contributed by Jeff Hankins, 31-Jan-2010.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
(𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥𝑋 {𝑜𝐽𝑥𝑜} = {𝑥}))

Theoremcnt1 21202 The preimage of a T1 topology under an injective map is T1. (Contributed by Mario Carneiro, 26-Aug-2015.)
((𝐾 ∈ Fre ∧ 𝐹:𝑋1-1𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Fre)

Theoremishaus2 21203* Express the predicate "𝐽 is a Hausdorff space." (Contributed by NM, 8-Mar-2007.)
(𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Haus ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))))

Theoremhaust1 21204 A Hausdorff space is a T1 space. (Contributed by FL, 11-Jun-2007.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
(𝐽 ∈ Haus → 𝐽 ∈ Fre)

Theoremhausnei2 21205* The Hausdorff condition still holds if one considers general neighborhoods instead of open sets. (Contributed by Jeff Hankins, 5-Sep-2009.)
(𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Haus ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝑦 → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢𝑣) = ∅)))

Theoremcnhaus 21206 The preimage of a Hausdorff topology under an injective map is Hausdorff. (Contributed by Mario Carneiro, 25-Aug-2015.)
((𝐾 ∈ Haus ∧ 𝐹:𝑋1-1𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Haus)

Theoremnrmsep3 21207* In a normal space, given a closed set 𝐵 inside an open set 𝐴, there is an open set 𝑥 such that 𝐵𝑥 ⊆ cls(𝑥) ⊆ 𝐴. (Contributed by Mario Carneiro, 24-Aug-2015.)
((𝐽 ∈ Nrm ∧ (𝐴𝐽𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴)) → ∃𝑥𝐽 (𝐵𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴))

Theoremnrmsep2 21208* In a normal space, any two disjoint closed sets have the property that each one is a subset of an open set whose closure is disjoint from the other. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 24-Aug-2015.)
((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → ∃𝑥𝐽 (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))

Theoremnrmsep 21209* In a normal space, disjoint closed sets are separated by open sets. (Contributed by Jeff Hankins, 1-Feb-2010.)
((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → ∃𝑥𝐽𝑦𝐽 (𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅))

Theoremisnrm2 21210* An alternate characterization of normality. This is the important property in the proof of Urysohn's lemma. (Contributed by Jeff Hankins, 1-Feb-2010.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
(𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))))

Theoremisnrm3 21211* A topological space is normal iff any two disjoint closed sets are separated by open sets. (Contributed by Mario Carneiro, 24-Aug-2015.)
(𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑥𝐽𝑦𝐽 (𝑐𝑥𝑑𝑦 ∧ (𝑥𝑦) = ∅))))

Theoremcnrmi 21212 A subspace of a completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Nrm)

Theoremcnrmnrm 21213 A completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝐽 ∈ CNrm → 𝐽 ∈ Nrm)

Theoremrestcnrm 21214 A subspace of a completely normal space is completely normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ CNrm)

Theoremresthauslem 21215 Lemma for resthaus 21220 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 21216 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 21217 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 21218 A subspace of a T0 topology is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
((𝐽 ∈ Kol2 ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Kol2)

Theoremrestt1 21219 A subspace of a T1 topology is T1. (Contributed by Mario Carneiro, 25-Aug-2015.)
((𝐽 ∈ Fre ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Fre)

Theoremresthaus 21220 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 21221* 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 21222* 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 21223 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 21224 Lemma for sshaus 21227 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 21225 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 21226 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 21227 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 21228* 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 21229* 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 21230 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 6375). (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 21231 The order topology is T1 for any poset. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑅 ∈ PosetRel → (ordTop‘𝑅) ∈ Fre)

Theoremlmmo 21232 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 21233 The convergence relation is function-like in a Hausdorff space. (Contributed by Mario Carneiro, 26-Dec-2013.)
(𝐽 ∈ Haus → Fun (⇝𝑡𝐽))

Theoremdishaus 21234 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 21235* Lemma for ordthaus 21236. (Contributed by Mario Carneiro, 13-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵 → (𝐴𝐵 → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅))))

Theoremordthaus 21236 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 21237 Extend class notation with the class of all compact spaces.
class Comp

Definitiondf-cmp 21238* 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 21239* 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 21240* 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 21241* Rewrite cmpcov 21240 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 21242* Combine cmpcov 21240 with ac6sfi 8245 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 21243 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 21244 A finite topology is compact. (Contributed by FL, 22-Dec-2008.)
(𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Comp)

Theorem0cmp 21245 The singleton of the empty set is compact. (Contributed by FL, 2-Aug-2009.)
{∅} ∈ Comp

Theoremcmptop 21246 A compact topology is a topology. (Contributed by Jeff Hankins, 29-Jun-2009.)
(𝐽 ∈ Comp → 𝐽 ∈ Top)

Theoremrncmp 21247 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 21248 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 21249 A discrete topology is compact iff the base set is finite. (Contributed by Mario Carneiro, 19-Mar-2015.)
(𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Comp)

Theoremcmpsublem 21250* Lemma for cmpsub 21251. (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 21251* 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 21252* A topology generated by a basis is compact iff open covers drawn from the basis have finite subcovers. (See also alexsub 21896, 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 21253 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 21254 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 21255* 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 21256 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 21257* Lemma for hauscmp 21258. (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 21258 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 21259* 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 21260 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 21261* 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 21262 Extend class notation with the class of all connected topologies.
class Conn

Definitiondf-conn 21263 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 21264 The predicate 𝐽 is a connected topology . (Contributed by FL, 17-Nov-2008.)
𝑋 = 𝐽       (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))

Theoremisconn2 21265 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 21266 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 21267 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 21268 A connected topology is a topology. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 14-Dec-2013.)
(𝐽 ∈ Conn → 𝐽 ∈ Top)

Theoremindisconn 21269 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 21270* 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 21271* Connectedness for a subspace. See connsub 21272. (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 21272* 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 21273 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 21274 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 21275 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 21276 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 21277 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 21278* Lemma for iunconn 21279. (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 21279* 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 21280 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 21281 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 21282* 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 21283* 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 21284* 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 21285* 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 21286* 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 21287 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 21288 Extend class definition to include the class of all first-countable topologies.
class 1st𝜔

Syntaxc2ndc 21289 Extend class definition to include the class of all second-countable topologies.
class 2nd𝜔

Definitiondf-1stc 21290* 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 21291* 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 21292* 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 21293* 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 21294 A first-countable topology is a topology. (Contributed by Jeff Hankins, 22-Aug-2009.)
(𝐽 ∈ 1st𝜔 → 𝐽 ∈ Top)

Theorem1stcclb 21295* 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 21296* 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 21297* 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 21298 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 21299 A countable basis generates a second-countable topology. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝐵 ∈ TopBases ∧ 𝐵 ≼ ω) → (topGen‘𝐵) ∈ 2nd𝜔)

Theorem2ndcsb 21300* 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))

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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 42801-42879
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