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Theorem List for Metamath Proof Explorer - 21101-21200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-reg 21101* Define regular spaces. A space is regular if a point and a closed set can be separated by neighborhoods. (Contributed by Jeff Hankins, 1-Feb-2010.)
Reg = {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑧𝑗 (𝑦𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}

Definitiondf-nrm 21102* Define normal spaces. A space is normal if disjoint closed sets can be separated by neighborhoods. (Contributed by Jeff Hankins, 1-Feb-2010.)
Nrm = {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦 ∈ ((Clsd‘𝑗) ∩ 𝒫 𝑥)∃𝑧𝑗 (𝑦𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}

Definitiondf-cnrm 21103* Define completely normal spaces. A space is completely normal if all its subspaces are normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
CNrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝒫 𝑗(𝑗t 𝑥) ∈ Nrm}

Definitiondf-pnrm 21104* Define perfectly normal spaces. A space is perfectly normal if it is normal and every closed set is a G&delta; set, meaning that it is a countable intersection of open sets. (Contributed by Mario Carneiro, 26-Aug-2015.)
PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗𝑚 ℕ) ↦ ran 𝑓)}

Theoremist0 21105* The predicate "is a T0 space." Every pair of distinct points is topologically distinguishable. For the way this definition is usually encountered, see ist0-3 21130. (Contributed by Jeff Hankins, 1-Feb-2010.)
𝑋 = 𝐽       (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))

Theoremist1 21106* The predicate 𝐽 is T1. (Contributed by FL, 18-Jun-2007.)
𝑋 = 𝐽       (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑎𝑋 {𝑎} ∈ (Clsd‘𝐽)))

Theoremishaus 21107* Express the predicate "𝐽 is a Hausdorff space." (Contributed by NM, 8-Mar-2007.)
𝑋 = 𝐽       (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))))

Theoremiscnrm 21108* The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝑋 = 𝐽       (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐽t 𝑥) ∈ Nrm))

Theoremt0sep 21109* Any two topologically indistinguishable points in a T0 space are identical. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Kol2 ∧ (𝐴𝑋𝐵𝑋)) → (∀𝑥𝐽 (𝐴𝑥𝐵𝑥) → 𝐴 = 𝐵))

Theoremt0dist 21110* Any two distinct points in a T0 space are topologically distinguishable. (Contributed by Jeff Hankins, 1-Feb-2010.)
𝑋 = 𝐽       ((𝐽 ∈ Kol2 ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → ∃𝑜𝐽 ¬ (𝐴𝑜𝐵𝑜))

Theoremt1sncld 21111 In a T1 space, one-point sets are closed. (Contributed by Jeff Hankins, 1-Feb-2010.)
𝑋 = 𝐽       ((𝐽 ∈ Fre ∧ 𝐴𝑋) → {𝐴} ∈ (Clsd‘𝐽))

Theoremt1ficld 21112 In a T1 space, finite sets are closed. (Contributed by Mario Carneiro, 25-Dec-2016.)
𝑋 = 𝐽       ((𝐽 ∈ Fre ∧ 𝐴𝑋𝐴 ∈ Fin) → 𝐴 ∈ (Clsd‘𝐽))

Theoremhausnei 21113* Neighborhood property of a Hausdorff space. (Contributed by NM, 8-Mar-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Haus ∧ (𝑃𝑋𝑄𝑋𝑃𝑄)) → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅))

Theoremt0top 21114 A T0 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
(𝐽 ∈ Kol2 → 𝐽 ∈ Top)

Theoremt1top 21115 A T1 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
(𝐽 ∈ Fre → 𝐽 ∈ Top)

Theoremhaustop 21116 A Hausdorff space is a topology. (Contributed by NM, 5-Mar-2007.)
(𝐽 ∈ Haus → 𝐽 ∈ Top)

Theoremisreg 21117* The predicate "is a regular space." In a regular space, any open neighborhood has a closed subneighborhood. Note that some authors require the space to be Hausdorff (which would make it the same as T3), but we reserve the phrase "regular Hausdorff" for that as many topologists do. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 25-Aug-2015.)
(𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))

Theoremregtop 21118 A regular space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
(𝐽 ∈ Reg → 𝐽 ∈ Top)

Theoremregsep 21119* In a regular space, every neighborhood of a point contains a closed subneighborhood. (Contributed by Mario Carneiro, 25-Aug-2015.)
((𝐽 ∈ Reg ∧ 𝑈𝐽𝐴𝑈) → ∃𝑥𝐽 (𝐴𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈))

Theoremisnrm 21120* The predicate "is a normal space." Much like the case for regular spaces, normal does not imply Hausdorff or even regular. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 24-Aug-2015.)
(𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))

Theoremnrmtop 21121 A normal space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
(𝐽 ∈ Nrm → 𝐽 ∈ Top)

Theoremcnrmtop 21122 A completely normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝐽 ∈ CNrm → 𝐽 ∈ Top)

Theoremiscnrm2 21123* The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ CNrm ↔ ∀𝑥 ∈ 𝒫 𝑋(𝐽t 𝑥) ∈ Nrm))

Theoremispnrm 21124* The property of being perfectly normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓)))

Theorempnrmnrm 21125 A perfectly normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝐽 ∈ PNrm → 𝐽 ∈ Nrm)

Theorempnrmtop 21126 A perfectly normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝐽 ∈ PNrm → 𝐽 ∈ Top)

Theorempnrmcld 21127* A closed set in a perfectly normal space is a countable intersection of open sets. (Contributed by Mario Carneiro, 26-Aug-2015.)
((𝐽 ∈ PNrm ∧ 𝐴 ∈ (Clsd‘𝐽)) → ∃𝑓 ∈ (𝐽𝑚 ℕ)𝐴 = ran 𝑓)

Theorempnrmopn 21128* An open set in a perfectly normal space is a countable union of closed sets. (Contributed by Mario Carneiro, 26-Aug-2015.)
((𝐽 ∈ PNrm ∧ 𝐴𝐽) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑𝑚 ℕ)𝐴 = ran 𝑓)

Theoremist0-2 21129* The predicate "is a T0 space". (Contributed by Mario Carneiro, 24-Aug-2015.)
(𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))

Theoremist0-3 21130* The predicate "is a T0 space," expressed in more familiar terms. (Contributed by Jeff Hankins, 1-Feb-2010.)
(𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝑦 → ∃𝑜𝐽 ((𝑥𝑜 ∧ ¬ 𝑦𝑜) ∨ (¬ 𝑥𝑜𝑦𝑜)))))

Theoremcnt0 21131 The preimage of a T0 topology under an injective map is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
((𝐾 ∈ Kol2 ∧ 𝐹:𝑋1-1𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Kol2)

Theoremist1-2 21132* An alternate characterization of T1 spaces. (Contributed by Jeff Hankins, 31-Jan-2010.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
(𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))

Theoremt1t0 21133 A T1 space is a T0 space. (Contributed by Jeff Hankins, 1-Feb-2010.)
(𝐽 ∈ Fre → 𝐽 ∈ Kol2)

Theoremist1-3 21134* 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 21135 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 21136* 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 21137 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 21138* 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 21139 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 21140* 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 21141* 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 21142* 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 21143* 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 21144* 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 21145 A subspace of a completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Nrm)

Theoremcnrmnrm 21146 A completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝐽 ∈ CNrm → 𝐽 ∈ Nrm)

Theoremrestcnrm 21147 A subspace of a completely normal space is completely normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ CNrm)

Theoremresthauslem 21148 Lemma for resthaus 21153 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 21149 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 21150 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 21151 A subspace of a T0 topology is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
((𝐽 ∈ Kol2 ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Kol2)

Theoremrestt1 21152 A subspace of a T1 topology is T1. (Contributed by Mario Carneiro, 25-Aug-2015.)
((𝐽 ∈ Fre ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Fre)

Theoremresthaus 21153 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 21154* 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 21155* 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 21156 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 21157 Lemma for sshaus 21160 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 21158 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 21159 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 21160 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 21161* 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 21162* 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 21163 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 6321). (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 21164 The order topology is T1 for any poset. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑅 ∈ PosetRel → (ordTop‘𝑅) ∈ Fre)

Theoremlmmo 21165 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 21166 The convergence relation is function-like in a Hausdorff space. (Contributed by Mario Carneiro, 26-Dec-2013.)
(𝐽 ∈ Haus → Fun (⇝𝑡𝐽))

Theoremdishaus 21167 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 21168* Lemma for ordthaus 21169. (Contributed by Mario Carneiro, 13-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵 → (𝐴𝐵 → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅))))

Theoremordthaus 21169 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 21170 Extend class notation with the class of all compact spaces.
class Comp

Definitiondf-cmp 21171* 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 21172* 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 21173* 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 21174* Rewrite cmpcov 21173 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 21175* Combine cmpcov 21173 with ac6sfi 8189 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 21176 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 21177 A finite topology is compact. (Contributed by FL, 22-Dec-2008.)
(𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Comp)

Theorem0cmp 21178 The singleton of the empty set is compact. (Contributed by FL, 2-Aug-2009.)
{∅} ∈ Comp

Theoremcmptop 21179 A compact topology is a topology. (Contributed by Jeff Hankins, 29-Jun-2009.)
(𝐽 ∈ Comp → 𝐽 ∈ Top)

Theoremrncmp 21180 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 21181 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 21182 A discrete topology is compact iff the base set is finite. (Contributed by Mario Carneiro, 19-Mar-2015.)
(𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Comp)

Theoremcmpsublem 21183* Lemma for cmpsub 21184. (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 21184* 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 21185* A topology generated by a basis is compact iff open covers drawn from the basis have finite subcovers. (See also alexsub 21830, 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 21186 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 21187 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 21188* 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 21189 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 21190* Lemma for hauscmp 21191. (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 21191 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 21192* 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 21193 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 21194* 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 21195 Extend class notation with the class of all connected topologies.
class Conn

Definitiondf-conn 21196 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 21197 The predicate 𝐽 is a connected topology . (Contributed by FL, 17-Nov-2008.)
𝑋 = 𝐽       (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))

Theoremisconn2 21198 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 21199 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 21200 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)

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