P. 233 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 23201-23300   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-t1 23201*	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)}
Definition	df-haus 23202*	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 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝑗 ∃𝑚 ∈ 𝑗 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))}
Definition	df-reg 23203*	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 ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑗 (𝑦 ∈ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}
Definition	df-nrm 23204*	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 ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ ((Clsd‘𝑗) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝑗 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}
Definition	df-cnrm 23205*	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(𝑗 ↾t 𝑥) ∈ Nrm}
Definition	df-pnrm 23206*	Define perfectly normal spaces. A space is perfectly normal if it is normal and every closed set is a Gδ 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 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓)}
Theorem	ist0 23207*	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 23232. (Contributed by Jeff Hankins, 1-Feb-2010.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
Theorem	ist1 23208*	The predicate "is a T1 space". (Contributed by FL, 18-Jun-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝑋 {𝑎} ∈ (Clsd‘𝐽)))
Theorem	ishaus 23209*	The predicate "is a Hausdorff space". (Contributed by NM, 8-Mar-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
Theorem	iscnrm 23210*	The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐽 ↾t 𝑥) ∈ Nrm))
Theorem	t0sep 23211*	Any two topologically indistinguishable points in a T0 space are identical. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Kol2 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ↔ 𝐵 ∈ 𝑥) → 𝐴 = 𝐵))
Theorem	t0dist 23212*	Any two distinct points in a T0 space are topologically distinguishable. (Contributed by Jeff Hankins, 1-Feb-2010.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Kol2 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ≠ 𝐵)) → ∃𝑜 ∈ 𝐽 ¬ (𝐴 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜))
Theorem	t1sncld 23213	In a T1 space, singletons are closed. (Contributed by Jeff Hankins, 1-Feb-2010.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Fre ∧ 𝐴 ∈ 𝑋) → {𝐴} ∈ (Clsd‘𝐽))
Theorem	t1ficld 23214	In a T1 space, finite sets are closed. (Contributed by Mario Carneiro, 25-Dec-2016.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Fre ∧ 𝐴 ⊆ 𝑋 ∧ 𝐴 ∈ Fin) → 𝐴 ∈ (Clsd‘𝐽))
Theorem	hausnei 23215*	Neighborhood property of a Hausdorff space. (Contributed by NM, 8-Mar-2007.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Haus ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋 ∧ 𝑃 ≠ 𝑄)) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ 𝑄 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))
Theorem	t0top 23216	A T0 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
⊢ (𝐽 ∈ Kol2 → 𝐽 ∈ Top)
Theorem	t1top 23217	A T1 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top)
Theorem	haustop 23218	A Hausdorff space is a topology. (Contributed by NM, 5-Mar-2007.)
⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
Theorem	isreg 23219*	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 ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
Theorem	regtop 23220	A regular space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
⊢ (𝐽 ∈ Reg → 𝐽 ∈ Top)
Theorem	regsep 23221*	In a regular space, every neighborhood of a point contains a closed subneighborhood. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ ((𝐽 ∈ Reg ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈) → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈))
Theorem	isnrm 23222*	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 ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝐽 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
Theorem	nrmtop 23223	A normal space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
Theorem	cnrmtop 23224	A completely normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ (𝐽 ∈ CNrm → 𝐽 ∈ Top)
Theorem	iscnrm2 23225*	The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ CNrm ↔ ∀𝑥 ∈ 𝒫 𝑋(𝐽 ↾t 𝑥) ∈ Nrm))
Theorem	ispnrm 23226*	The property of being perfectly normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓)))
Theorem	pnrmnrm 23227	A perfectly normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Nrm)
Theorem	pnrmtop 23228	A perfectly normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Top)
Theorem	pnrmcld 23229*	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 ∈ (𝐽 ↑m ℕ)𝐴 = ∩ ran 𝑓)
Theorem	pnrmopn 23230*	An open set in a perfectly normal space is a countable union of closed sets. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ 𝐽) → ∃𝑓 ∈ ((Clsd‘𝐽) ↑m ℕ)𝐴 = ∪ ran 𝑓)
Theorem	ist0-2 23231*	The predicate "is a T0 space". (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
Theorem	ist0-3 23232*	The predicate "is a T0 space" expressed in more familiar terms. (Contributed by Jeff Hankins, 1-Feb-2010.)
⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 ((𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ∨ (¬ 𝑥 ∈ 𝑜 ∧ 𝑦 ∈ 𝑜)))))
Theorem	cnt0 23233	The preimage of a T0 topology under an injective map is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ ((𝐾 ∈ Kol2 ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Kol2)
Theorem	ist1-2 23234*	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 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
Theorem	t1t0 23235	A T1 space is a T0 space. (Contributed by Jeff Hankins, 1-Feb-2010.)
⊢ (𝐽 ∈ Fre → 𝐽 ∈ Kol2)
Theorem	ist1-3 23236*	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 ↔ ∀𝑥 ∈ 𝑋 ∩ {𝑜 ∈ 𝐽 ∣ 𝑥 ∈ 𝑜} = {𝑥}))
Theorem	cnt1 23237	The preimage of a T1 topology under an injective map is T1. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ ((𝐾 ∈ Fre ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Fre)
Theorem	ishaus2 23238*	Express the predicate "𝐽 is a Hausdorff space." (Contributed by NM, 8-Mar-2007.)
⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Haus ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
Theorem	haust1 23239	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)
Theorem	hausnei2 23240*	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 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅)))
Theorem	cnhaus 23241	The preimage of a Hausdorff topology under an injective map is Hausdorff. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ ((𝐾 ∈ Haus ∧ 𝐹:𝑋–1-1→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Haus)
Theorem	nrmsep3 23242*	In a normal space, given a closed set 𝐵 inside an open set 𝐴, there is an open set 𝑥 such that 𝐵 ⊆ 𝑥 ⊆ cls(𝑥) ⊆ 𝐴. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ ((𝐽 ∈ Nrm ∧ (𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴)) → ∃𝑥 ∈ 𝐽 (𝐵 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝐴))
Theorem	nrmsep2 23243*	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) ∧ (𝐶 ∩ 𝐷) = ∅)) → ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))
Theorem	nrmsep 23244*	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) ∧ (𝐶 ∩ 𝐷) = ∅)) → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))
Theorem	isnrm2 23245*	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)((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))))
Theorem	isnrm3 23246*	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)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))))
Theorem	cnrmi 23247	A subspace of a completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ Nrm)
Theorem	cnrmnrm 23248	A completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ (𝐽 ∈ CNrm → 𝐽 ∈ Nrm)
Theorem	restcnrm 23249	A subspace of a completely normal space is completely normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ CNrm)
Theorem	resthauslem 23250	Lemma for resthaus 23255 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.)
⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Top)    &   ⊢ ((𝐽 ∈ 𝐴 ∧ ( I ↾ (𝑆 ∩ ∪ 𝐽)):(𝑆 ∩ ∪ 𝐽)–1-1→(𝑆 ∩ ∪ 𝐽) ∧ ( I ↾ (𝑆 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t 𝑆) Cn 𝐽)) → (𝐽 ↾t 𝑆) ∈ 𝐴)    ⇒   ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → (𝐽 ↾t 𝑆) ∈ 𝐴)
Theorem	lpcls 23251	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 ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆))
Theorem	perfcls 23252	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 ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t 𝑆) ∈ Perf ↔ (𝐽 ↾t ((cls‘𝐽)‘𝑆)) ∈ Perf))
Theorem	restt0 23253	A subspace of a T0 topology is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ ((𝐽 ∈ Kol2 ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ Kol2)
Theorem	restt1 23254	A subspace of a T1 topology is T1. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ ((𝐽 ∈ Fre ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ Fre)
Theorem	resthaus 23255	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 ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ Haus)
Theorem	t1sep2 23256*	Any two points in a T1 space which have no separation are equal. (Contributed by Jeff Hankins, 1-Feb-2010.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Fre ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜) → 𝐴 = 𝐵))
Theorem	t1sep 23257*	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 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ≠ 𝐵)) → ∃𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ∧ ¬ 𝐵 ∈ 𝑜))
Theorem	sncld 23258	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 ∧ 𝑃 ∈ 𝑋) → {𝑃} ∈ (Clsd‘𝐽))
Theorem	sshauslem 23259	Lemma for sshaus 23262 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    &   ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Top)    &   ⊢ ((𝐽 ∈ 𝐴 ∧ ( I ↾ 𝑋):𝑋–1-1→𝑋 ∧ ( I ↾ 𝑋) ∈ (𝐾 Cn 𝐽)) → 𝐾 ∈ 𝐴)    ⇒   ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐾 ∈ 𝐴)
Theorem	sst0 23260	A topology finer than a T0 topology is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Kol2 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐾 ∈ Kol2)
Theorem	sst1 23261	A topology finer than a T1 topology is T1. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Fre ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐾 ∈ Fre)
Theorem	sshaus 23262	A topology finer than a Hausdorff topology is Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Haus ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐾 ∈ Haus)
Theorem	regsep2 23263*	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) ∧ 𝐴 ∈ 𝑋 ∧ ¬ 𝐴 ∈ 𝐶)) → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐴 ∈ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))
Theorem	isreg2 23264*	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)∀𝑥 ∈ 𝑋 (¬ 𝑥 ∈ 𝑐 → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))))
Theorem	dnsconst 23265	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 7028). (Contributed by NM, 15-Mar-2007.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ (((𝐾 ∈ Fre ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑃 ∈ 𝑌 ∧ 𝐴 ⊆ (◡𝐹 “ {𝑃}) ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐹:𝑋⟶{𝑃})
Theorem	ordtt1 23266	The order topology is T1 for any poset. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (𝑅 ∈ PosetRel → (ordTop‘𝑅) ∈ Fre)
Theorem	lmmo 23267	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.)
⊢ (𝜑 → 𝐽 ∈ Haus)    &   ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝐴)    &   ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝐵)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	lmfun 23268	The convergence relation is function-like in a Hausdorff space. (Contributed by Mario Carneiro, 26-Dec-2013.)
⊢ (𝐽 ∈ Haus → Fun (⇝𝑡‘𝐽))
Theorem	dishaus 23269	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.)
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Haus)
Theorem	ordthauslem 23270*	Lemma for ordthaus 23271. (Contributed by Mario Carneiro, 13-Sep-2015.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ ((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 → (𝐴 ≠ 𝐵 → ∃𝑚 ∈ (ordTop‘𝑅)∃𝑛 ∈ (ordTop‘𝑅)(𝐴 ∈ 𝑚 ∧ 𝐵 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
Theorem	ordthaus 23271	The order topology of a total order is Hausdorff. (Contributed by Mario Carneiro, 13-Sep-2015.)
⊢ (𝑅 ∈ TosetRel → (ordTop‘𝑅) ∈ Haus)
Theorem	xrhaus 23272	The topology of the extended reals is Hausdorff. (Contributed by Thierry Arnoux, 24-Mar-2017.)
⊢ (ordTop‘ ≤ ) ∈ Haus
12.1.11  Compactness
Syntax	ccmp 23273	Extend class notation with the class of all compact spaces.
class Comp
Definition	df-cmp 23274*	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)}
Theorem	iscmp 23275*	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)))
Theorem	cmpcov 23276*	An open cover of a compact topology has a finite subcover. (Contributed by Jeff Hankins, 29-Jun-2009.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑆) → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = ∪ 𝑠)
Theorem	cmpcov2 23277*	Rewrite cmpcov 23276 for the cover {𝑦 ∈ 𝐽 ∣ 𝜑}. (Contributed by Mario Carneiro, 11-Sep-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Comp ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝜑)) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑠 ∧ ∀𝑦 ∈ 𝑠 𝜑))
Theorem	cmpcovf 23278*	Combine cmpcov 23276 with ac6sfi 9231 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 = (𝑓‘𝑦) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐽 ∈ Comp ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝐴 𝜑)) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑠 ∧ ∃𝑓(𝑓:𝑠⟶𝐴 ∧ ∀𝑦 ∈ 𝑠 𝜓)))
Theorem	cncmp 23279	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:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Comp)
Theorem	fincmp 23280	A finite topology is compact. (Contributed by FL, 22-Dec-2008.)
⊢ (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Comp)
Theorem	0cmp 23281	The singleton of the empty set is compact. (Contributed by FL, 2-Aug-2009.)
⊢ {∅} ∈ Comp
Theorem	cmptop 23282	A compact topology is a topology. (Contributed by Jeff Hankins, 29-Jun-2009.)
⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
Theorem	rncmp 23283	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)) → (𝐾 ↾t ran 𝐹) ∈ Comp)
Theorem	imacmp 23284	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) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐾 ↾t (𝐹 “ 𝐴)) ∈ Comp)
Theorem	discmp 23285	A discrete topology is compact iff the base set is finite. (Contributed by Mario Carneiro, 19-Mar-2015.)
⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Comp)
Theorem	cmpsublem 23286*	Lemma for cmpsub 23287. (Contributed by Jeff Hankins, 28-Jun-2009.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∀𝑐 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑) → ∀𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)(∪ (𝐽 ↾t 𝑆) = ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡)))
Theorem	cmpsub 23287*	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 ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t 𝑆) ∈ Comp ↔ ∀𝑐 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑)))
Theorem	tgcmp 23288*	A topology generated by a basis is compact iff open covers drawn from the basis have finite subcovers. (See also alexsub 23932, 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)))
Theorem	cmpcld 23289	A closed subset of a compact space is compact. (Contributed by Jeff Hankins, 29-Jun-2009.)
⊢ ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝐽 ↾t 𝑆) ∈ Comp)
Theorem	uncmp 23290	The union of two compact sets is compact. (Contributed by Jeff Hankins, 30-Jan-2010.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (((𝐽 ∈ Top ∧ 𝑋 = (𝑆 ∪ 𝑇)) ∧ ((𝐽 ↾t 𝑆) ∈ Comp ∧ (𝐽 ↾t 𝑇) ∈ Comp)) → 𝐽 ∈ Comp)
Theorem	fiuncmp 23291*	A finite union of compact sets is compact. (Contributed by Mario Carneiro, 19-Mar-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾t 𝐵) ∈ Comp) → (𝐽 ↾t ∪ 𝑥 ∈ 𝐴 𝐵) ∈ Comp)
Theorem	sscmp 23292	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 ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ Comp)
Theorem	hauscmplem 23293*	Lemma for hauscmp 23294. (Contributed by Mario Carneiro, 27-Nov-2013.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑂 = {𝑦 ∈ 𝐽 ∣ ∃𝑤 ∈ 𝐽 (𝐴 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑋 ∖ 𝑦))}    &   ⊢ (𝜑 → 𝐽 ∈ Haus)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    &   ⊢ (𝜑 → (𝐽 ↾t 𝑆) ∈ Comp)    &   ⊢ (𝜑 → 𝐴 ∈ (𝑋 ∖ 𝑆))    ⇒   ⊢ (𝜑 → ∃𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ (𝑋 ∖ 𝑆)))
Theorem	hauscmp 23294	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 ∧ 𝑆 ⊆ 𝑋 ∧ (𝐽 ↾t 𝑆) ∈ Comp) → 𝑆 ∈ (Clsd‘𝐽))
Theorem	cmpfi 23295*	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 ≠ ∅)))
Theorem	cmpfii 23296	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
Theorem	bwth 23297*	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 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → ∃𝑥 ∈ 𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝐴))
12.1.13  Connectedness
Syntax	cconn 23298	Extend class notation with the class of all connected topologies.
class Conn
Definition	df-conn 23299	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}}
Theorem	isconn 23300	The predicate 𝐽 is a connected topology . (Contributed by FL, 17-Nov-2008.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))