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Theorem List for Metamath Proof Explorer - 20801-20900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcnconst 20801 A constant function is continuous. (Contributed by FL, 15-Jan-2007.) (Proof shortened by Mario Carneiro, 19-Mar-2015.)
(((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐵𝑌𝐹:𝑋⟶{𝐵})) → 𝐹 ∈ (𝐽 Cn 𝐾))

Theoremcnrest 20802 Continuity of a restriction from a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 21-Aug-2015.)
𝑋 = 𝐽       ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾))

Theoremcnrest2 20803 Equivalence of continuity in the parent topology and continuity in a subspace. (Contributed by Jeff Hankins, 10-Jul-2009.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾t 𝐵))))

Theoremcnrest2r 20804 Equivalence of continuity in the parent topology and continuity in a subspace. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Jun-2014.)
(𝐾 ∈ Top → (𝐽 Cn (𝐾t 𝐵)) ⊆ (𝐽 Cn 𝐾))

Theoremcnpresti 20805 One direction of cnprest 20806 under the weaker condition that the point is in the subset rather than the interior of the subset. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 1-May-2015.)
𝑋 = 𝐽       ((𝐴𝑋𝑃𝐴𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃))

Theoremcnprest 20806 Equivalence of continuity at a point and continuity of the restricted function at a point. (Contributed by Mario Carneiro, 8-Aug-2014.)
𝑋 = 𝐽    &   𝑌 = 𝐾       (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝑃 ∈ ((int‘𝐽)‘𝐴) ∧ 𝐹:𝑋𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹𝐴) ∈ (((𝐽t 𝐴) CnP 𝐾)‘𝑃)))

Theoremcnprest2 20807 Equivalence of point-continuity in the parent topology and point-continuity in a subspace. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 21-Aug-2015.)
𝑋 = 𝐽    &   𝑌 = 𝐾       ((𝐾 ∈ Top ∧ 𝐹:𝑋𝐵𝐵𝑌) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ 𝐹 ∈ ((𝐽 CnP (𝐾t 𝐵))‘𝑃)))

Theoremcndis 20808 Every function is continuous when the domain is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
((𝐴𝑉𝐽 ∈ (TopOn‘𝑋)) → (𝒫 𝐴 Cn 𝐽) = (𝑋𝑚 𝐴))

Theoremcnindis 20809 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 20810 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 20811 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 20812 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 20813 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 20814 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 20815 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 20816 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 20817 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 20818* 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 20819* 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 20820* 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 20821 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 20822 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 20823 Extend class notation with the class of all T0 spaces.
class Kol2

Syntaxct1 20824 Extend class notation to include T1 spaces (also called Fréchet spaces).
class Fre

Syntaxcha 20825 Extend class notation with the class of all Hausdorff spaces.
class Haus

Syntaxcreg 20826 Extend class notation with the class of all regular topologies.
class Reg

Syntaxcnrm 20827 Extend class notation with the class of all normal topologies.
class Nrm

Syntaxccnrm 20828 Extend class notation with the class of all completely normal topologies.
class CNrm

Syntaxcpnrm 20829 Extend class notation with the class of all perfectly normal topologies.
class PNrm

Definitiondf-t0 20830* Define T0 or Kolmogorov spaces. A T0 space satisfies a kind of "topological extensionality" principle (compare ax-ext 2494): 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 20864) 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 20831* 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 20832* Define the class of all Hausdorff 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 20833* 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 20834* 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 20835* 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 20836* 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 20837* 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 20862. (Contributed by Jeff Hankins, 1-Feb-2010.)
𝑋 = 𝐽       (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))

Theoremist1 20838* The predicate 𝐽 is T1. (Contributed by FL, 18-Jun-2007.)
𝑋 = 𝐽       (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑎𝑋 {𝑎} ∈ (Clsd‘𝐽)))

Theoremishaus 20839* 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 20840* 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 20841* 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 20842* 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 20843 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 20844 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 20845* Neighborhood property of a Hausdorff space. (Contributed by NM, 8-Mar-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Haus ∧ (𝑃𝑋𝑄𝑋𝑃𝑄)) → ∃𝑛𝐽𝑚𝐽 (𝑃𝑛𝑄𝑚 ∧ (𝑛𝑚) = ∅))

Theoremt0top 20846 A T0 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
(𝐽 ∈ Kol2 → 𝐽 ∈ Top)

Theoremt1top 20847 A T1 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
(𝐽 ∈ Fre → 𝐽 ∈ Top)

Theoremhaustop 20848 A Hausdorff space is a topology. (Contributed by NM, 5-Mar-2007.)
(𝐽 ∈ Haus → 𝐽 ∈ Top)

Theoremisreg 20849* 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 20850 A regular space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
(𝐽 ∈ Reg → 𝐽 ∈ Top)

Theoremregsep 20851* 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 20852* 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 20853 A normal space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
(𝐽 ∈ Nrm → 𝐽 ∈ Top)

Theoremcnrmtop 20854 A completely normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝐽 ∈ CNrm → 𝐽 ∈ Top)

Theoremiscnrm2 20855* 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 20856* 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 20857 A perfectly normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝐽 ∈ PNrm → 𝐽 ∈ Nrm)

Theorempnrmtop 20858 A perfectly normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝐽 ∈ PNrm → 𝐽 ∈ Top)

Theorempnrmcld 20859* 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 20860* 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 20861* 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 20862* 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 20863 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 20864* 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 20865 A T1 space is a T0 space. (Contributed by Jeff Hankins, 1-Feb-2010.)
(𝐽 ∈ Fre → 𝐽 ∈ Kol2)

Theoremist1-3 20866* 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 20867 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 20868* 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 20869 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 20870* 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 20871 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 20872* 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 20873* 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 20874* 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 20875* 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 20876* 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 20877 A subspace of a completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Nrm)

Theoremcnrmnrm 20878 A completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝐽 ∈ CNrm → 𝐽 ∈ Nrm)

Theoremrestcnrm 20879 A subspace of a completely normal space is completely normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ CNrm)

Theoremresthauslem 20880 Lemma for resthaus 20885 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 20881 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 20882 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 20883 A subspace of a T0 topology is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
((𝐽 ∈ Kol2 ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Kol2)

Theoremrestt1 20884 A subspace of a T1 topology is T1. (Contributed by Mario Carneiro, 25-Aug-2015.)
((𝐽 ∈ Fre ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Fre)

Theoremresthaus 20885 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 20886* 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 20887* 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 20888 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 20889 Lemma for sshaus 20892 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 20890 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 20891 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 20892 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 20893* 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 20894* 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 20895 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 6127). (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 20896 The order topology is T1 for any poset. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑅 ∈ PosetRel → (ordTop‘𝑅) ∈ Fre)

Theoremlmmo 20897 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 20898 The convergence relation is function-like in a Hausdorff space. (Contributed by Mario Carneiro, 26-Dec-2013.)
(𝐽 ∈ Haus → Fun (⇝𝑡𝐽))

Theoremdishaus 20899 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 20900* Lemma for ordthaus 20901. (Contributed by Mario Carneiro, 13-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵 → (𝐴𝐵 → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴𝑚𝐵𝑛 ∧ (𝑚𝑛) = ∅))))

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