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Theorem List for Metamath Proof Explorer - 21901-22000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcnindis 21901 Every function is continuous when the codomain is indiscrete (trivial). (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝐽 Cn {∅, 𝐴}) = (𝐴m 𝑋))
 
Theoremcnpdis 21902 If 𝐴 is an isolated point in 𝑋 (or equivalently, the singleton {𝐴} is open in 𝑋), then every function is continuous at 𝐴. (Contributed by Mario Carneiro, 9-Sep-2015.)
(((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → ((𝐽 CnP 𝐾)‘𝐴) = (𝑌m 𝑋))
 
Theorempaste 21903 Pasting lemma. If 𝐴 and 𝐵 are closed sets in 𝑋 with 𝐴𝐵 = 𝑋, then any function whose restrictions to 𝐴 and 𝐵 are continuous is continuous on all of 𝑋. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
𝑋 = 𝐽    &   𝑌 = 𝐾    &   (𝜑𝐴 ∈ (Clsd‘𝐽))    &   (𝜑𝐵 ∈ (Clsd‘𝐽))    &   (𝜑 → (𝐴𝐵) = 𝑋)    &   (𝜑𝐹:𝑋𝑌)    &   (𝜑 → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾))    &   (𝜑 → (𝐹𝐵) ∈ ((𝐽t 𝐵) Cn 𝐾))       (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
 
Theoremlmfpm 21904 If 𝐹 converges, then 𝐹 is a partial function. (Contributed by Mario Carneiro, 23-Dec-2013.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡𝐽)𝑃) → 𝐹 ∈ (𝑋pm ℂ))
 
Theoremlmfss 21905 Inclusion of a function having a limit (used to ensure the limit relation is a set, under our definition). (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡𝐽)𝑃) → 𝐹 ⊆ (ℂ × 𝑋))
 
Theoremlmcl 21906 Closure of a limit. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡𝐽)𝑃) → 𝑃𝑋)
 
Theoremlmss 21907 Limit on a subspace. (Contributed by NM, 30-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)
𝐾 = (𝐽t 𝑌)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑌𝑉)    &   (𝜑𝐽 ∈ Top)    &   (𝜑𝑃𝑌)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍𝑌)       (𝜑 → (𝐹(⇝𝑡𝐽)𝑃𝐹(⇝𝑡𝐾)𝑃))
 
Theoremsslm 21908 A finer topology has fewer convergent sequences (but the sequences that do converge, converge to the same value). (Contributed by Mario Carneiro, 15-Sep-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (⇝𝑡𝐾) ⊆ (⇝𝑡𝐽))
 
Theoremlmres 21909 A function converges iff its restriction to an upper integers set converges. (Contributed by Mario Carneiro, 31-Dec-2013.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝑋pm ℂ))    &   (𝜑𝑀 ∈ ℤ)       (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ↾ (ℤ𝑀))(⇝𝑡𝐽)𝑃))
 
Theoremlmff 21910* If 𝐹 converges, there is some upper integer set on which 𝐹 is a total function. (Contributed by Mario Carneiro, 31-Dec-2013.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹 ∈ dom (⇝𝑡𝐽))       (𝜑 → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋)
 
Theoremlmcls 21911* Any convergent sequence of points in a subset of a topological space converges to a point in the closure of the subset. (Contributed by Mario Carneiro, 30-Dec-2013.) (Revised by Mario Carneiro, 1-May-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹(⇝𝑡𝐽)𝑃)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ 𝑆)    &   (𝜑𝑆𝑋)       (𝜑𝑃 ∈ ((cls‘𝐽)‘𝑆))
 
Theoremlmcld 21912* Any convergent sequence of points in a closed subset of a topological space converges to a point in the set. (Contributed by Mario Carneiro, 30-Dec-2013.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹(⇝𝑡𝐽)𝑃)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ 𝑆)    &   (𝜑𝑆 ∈ (Clsd‘𝐽))       (𝜑𝑃𝑆)
 
Theoremlmcnp 21913 The image of a convergent sequence under a continuous map is convergent to the image of the original point. (Contributed by Mario Carneiro, 3-May-2014.)
(𝜑𝐹(⇝𝑡𝐽)𝑃)    &   (𝜑𝐺 ∈ ((𝐽 CnP 𝐾)‘𝑃))       (𝜑 → (𝐺𝐹)(⇝𝑡𝐾)(𝐺𝑃))
 
Theoremlmcn 21914 The image of a convergent sequence under a continuous map is convergent to the image of the original point. (Contributed by Mario Carneiro, 3-May-2014.)
(𝜑𝐹(⇝𝑡𝐽)𝑃)    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝐺𝐹)(⇝𝑡𝐾)(𝐺𝑃))
 
12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...
 
Syntaxct0 21915 Extend class notation with the class of all T0 spaces.
class Kol2
 
Syntaxct1 21916 Extend class notation to include T1 spaces (also called Fréchet spaces).
class Fre
 
Syntaxcha 21917 Extend class notation with the class of all Hausdorff spaces.
class Haus
 
Syntaxcreg 21918 Extend class notation with the class of all regular topologies.
class Reg
 
Syntaxcnrm 21919 Extend class notation with the class of all normal topologies.
class Nrm
 
Syntaxccnrm 21920 Extend class notation with the class of all completely normal topologies.
class CNrm
 
Syntaxcpnrm 21921 Extend class notation with the class of all perfectly normal topologies.
class PNrm
 
Definitiondf-t0 21922* Define T0 or Kolmogorov spaces. A T0 space satisfies a kind of "topological extensionality" principle (compare ax-ext 2773): any two points which are members of the same open sets are equal, or in contraposition, for any two distinct points there is an open set which contains one point but not the other. This differs from T1 spaces (see ist1-2 21956) in that in a T1 space you can choose which point will be in the open set and which outside; in a T0 space you only know that one of the two points is in the set. (Contributed by Jeff Hankins, 1-Feb-2010.)
Kol2 = {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 𝑗(∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)}
 
Definitiondf-t1 21923* The class of all T1 spaces, also called Fréchet spaces. Morris, Topology without tears, p. 30 ex. 3. (Contributed by FL, 18-Jun-2007.)
Fre = {𝑥 ∈ Top ∣ ∀𝑎 𝑥{𝑎} ∈ (Clsd‘𝑥)}
 
Definitiondf-haus 21924* Define the class of all Hausdorff (or T2) spaces. A Hausdorff space is a topology in which distinct points have disjoint open neighborhoods. Definition of Hausdorff space in [Munkres] p. 98. (Contributed by NM, 8-Mar-2007.)
Haus = {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 𝑗(𝑥𝑦 → ∃𝑛𝑗𝑚𝑗 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))}
 
Definitiondf-reg 21925* 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 21926* 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 21927* 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 21928* 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 21929* 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 21954. (Contributed by Jeff Hankins, 1-Feb-2010.)
𝑋 = 𝐽       (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
 
Theoremist1 21930* The predicate "is a T1 space". (Contributed by FL, 18-Jun-2007.)
𝑋 = 𝐽       (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑎𝑋 {𝑎} ∈ (Clsd‘𝐽)))
 
Theoremishaus 21931* The predicate "is a Hausdorff space". (Contributed by NM, 8-Mar-2007.)
𝑋 = 𝐽       (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))))
 
Theoremiscnrm 21932* 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 21933* 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 21934* 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 21935 In a T1 space, singletons are closed. (Contributed by Jeff Hankins, 1-Feb-2010.)
𝑋 = 𝐽       ((𝐽 ∈ Fre ∧ 𝐴𝑋) → {𝐴} ∈ (Clsd‘𝐽))
 
Theoremt1ficld 21936 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 21937* Neighborhood property of a Hausdorff space. (Contributed by NM, 8-Mar-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Haus ∧ (𝑃𝑋𝑄𝑋𝑃𝑄)) → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅))
 
Theoremt0top 21938 A T0 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
(𝐽 ∈ Kol2 → 𝐽 ∈ Top)
 
Theoremt1top 21939 A T1 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
(𝐽 ∈ Fre → 𝐽 ∈ Top)
 
Theoremhaustop 21940 A Hausdorff space is a topology. (Contributed by NM, 5-Mar-2007.)
(𝐽 ∈ Haus → 𝐽 ∈ Top)
 
Theoremisreg 21941* 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 21942 A regular space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
(𝐽 ∈ Reg → 𝐽 ∈ Top)
 
Theoremregsep 21943* 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 21944* 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 21945 A normal space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
(𝐽 ∈ Nrm → 𝐽 ∈ Top)
 
Theoremcnrmtop 21946 A completely normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝐽 ∈ CNrm → 𝐽 ∈ Top)
 
Theoremiscnrm2 21947* 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 21948* 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 21949 A perfectly normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝐽 ∈ PNrm → 𝐽 ∈ Nrm)
 
Theorempnrmtop 21950 A perfectly normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝐽 ∈ PNrm → 𝐽 ∈ Top)
 
Theorempnrmcld 21951* 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 21952* 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 21953* 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 21954* 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 21955 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 21956* 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 21957 A T1 space is a T0 space. (Contributed by Jeff Hankins, 1-Feb-2010.)
(𝐽 ∈ Fre → 𝐽 ∈ Kol2)
 
Theoremist1-3 21958* 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 21959 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 21960* 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 21961 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 21962* 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 21963 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 21964* 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 21965* 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 21966* 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 21967* 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 21968* 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 21969 A subspace of a completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Nrm)
 
Theoremcnrmnrm 21970 A completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝐽 ∈ CNrm → 𝐽 ∈ Nrm)
 
Theoremrestcnrm 21971 A subspace of a completely normal space is completely normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ CNrm)
 
Theoremresthauslem 21972 Lemma for resthaus 21977 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 21973 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 21974 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 21975 A subspace of a T0 topology is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
((𝐽 ∈ Kol2 ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Kol2)
 
Theoremrestt1 21976 A subspace of a T1 topology is T1. (Contributed by Mario Carneiro, 25-Aug-2015.)
((𝐽 ∈ Fre ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Fre)
 
Theoremresthaus 21977 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 21978* 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 21979* 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 21980 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 21981 Lemma for sshaus 21984 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 21982 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 21983 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 21984 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 21985* 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 21986* 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 21987 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 6807). (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 21988 The order topology is T1 for any poset. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑅 ∈ PosetRel → (ordTop‘𝑅) ∈ Fre)
 
Theoremlmmo 21989 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 21990 The convergence relation is function-like in a Hausdorff space. (Contributed by Mario Carneiro, 26-Dec-2013.)
(𝐽 ∈ Haus → Fun (⇝𝑡𝐽))
 
Theoremdishaus 21991 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 21992* Lemma for ordthaus 21993. (Contributed by Mario Carneiro, 13-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵 → (𝐴𝐵 → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅))))
 
Theoremordthaus 21993 The order topology of a total order is Hausdorff. (Contributed by Mario Carneiro, 13-Sep-2015.)
(𝑅 ∈ TosetRel → (ordTop‘𝑅) ∈ Haus)
 
Theoremxrhaus 21994 The topology of the extended reals is Hausdorff. (Contributed by Thierry Arnoux, 24-Mar-2017.)
(ordTop‘ ≤ ) ∈ Haus
 
12.1.11  Compactness
 
Syntaxccmp 21995 Extend class notation with the class of all compact spaces.
class Comp
 
Definitiondf-cmp 21996* 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" (saving "compact" for "compact Hausdorff"), 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 21997* 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 21998* 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 21999* Rewrite cmpcov 21998 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 22000* Combine cmpcov 21998 with ac6sfi 8750 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)))
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