P. 238 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 23701-23800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cnmptcom 23701*	The argument converse of a continuous function is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))    ⇒   ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
Theorem	cnmptkc 23702*	The curried first projection function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝑥)) ∈ (𝐽 Cn (𝐽 ↑ko 𝐾)))
Theorem	cnmptkp 23703*	The evaluation of the inner function in a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))    &   ⊢ (𝜑 → 𝐵 ∈ 𝑌)    &   ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) ∈ (𝐽 Cn 𝐿))
Theorem	cnmptk1 23704*	The composition of a curried function with a one-arg function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))    &   ⊢ (𝜑 → (𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀))    &   ⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)) ∈ (𝐽 Cn (𝑀 ↑ko 𝐾)))
Theorem	cnmpt1k 23705*	The composition of a one-arg function with a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))    &   ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑊))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐿))    &   ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑧 ∈ 𝑍 ↦ 𝐵)) ∈ (𝐾 Cn (𝑀 ↑ko 𝐿)))    &   ⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑥 ∈ 𝑋 ↦ 𝐶)) ∈ (𝐾 Cn (𝑀 ↑ko 𝐽)))
Theorem	cnmptkk 23706*	The composition of two curried functions is jointly continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))    &   ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑊))    &   ⊢ (𝜑 → 𝐿 ∈ 𝑛-Locally Comp)    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑧 ∈ 𝑍 ↦ 𝐵)) ∈ (𝐽 Cn (𝑀 ↑ko 𝐿)))    &   ⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)) ∈ (𝐽 Cn (𝑀 ↑ko 𝐾)))
Theorem	xkofvcn 23707*	Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 23679.) (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ 𝑋 = ∪ 𝑅    &   ⊢ 𝐹 = (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥 ∈ 𝑋 ↦ (𝑓‘𝑥))    ⇒   ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝐹 ∈ (((𝑆 ↑ko 𝑅) ×t 𝑅) Cn 𝑆))
Theorem	cnmptk1p 23708*	The evaluation of a curried function by a one-arg function is jointly continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Comp)    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) ∈ (𝐽 Cn 𝐿))
Theorem	cnmptk2 23709*	The uncurrying of a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Comp)    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
Theorem	xkoinjcn 23710*	Continuity of "injection", i.e. currying, as a function on continuous function spaces. (Contributed by Mario Carneiro, 23-Mar-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩))    ⇒   ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝐹 ∈ (𝑅 Cn ((𝑆 ×t 𝑅) ↑ko 𝑆)))
Theorem	cnmpt2k 23711*	The currying of a two-argument function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
Theorem	txconn 23712	The topological product of two connected spaces is connected. (Contributed by Mario Carneiro, 29-Mar-2015.)
⊢ ((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) → (𝑅 ×t 𝑆) ∈ Conn)
Theorem	imasnopn 23713	If a relation graph is open, then an image set of a singleton is also open. Corollary of Proposition 4 of [BourbakiTop1] p. I.26. (Contributed by Thierry Arnoux, 14-Jan-2018.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) ∈ 𝐾)
Theorem	imasncld 23714	If a relation graph is closed, then an image set of a singleton is also closed. Corollary of Proposition 4 of [BourbakiTop1] p. I.26. (Contributed by Thierry Arnoux, 14-Jan-2018.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsd‘(𝐽 ×t 𝐾)) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) ∈ (Clsd‘𝐾))
Theorem	imasncls 23715	If a relation graph is closed, then an image set of a singleton is also closed. Corollary of Proposition 4 of [BourbakiTop1] p. I.26. (Contributed by Thierry Arnoux, 14-Jan-2018.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → ((cls‘𝐾)‘(𝑅 “ {𝐴})) ⊆ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}))
12.1.20  Quotient maps and quotient topology
Syntax	ckq 23716	Extend class notation with the Kolmogorov quotient function.
class KQ
Definition	df-kq 23717*	Define the Kolmogorov quotient. This is a function on topologies which maps a topology to its quotient under the topological distinguishability map, which takes a point to the set of open sets that contain it. Two points are mapped to the same image under this function iff they are topologically indistinguishable. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})))
Theorem	qtopval 23718*	Value of the quotient topology function. (Contributed by Mario Carneiro, 23-Mar-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽})
Theorem	qtopval2 23719*	Value of the quotient topology function when 𝐹 is a function on the base set. (Contributed by Mario Carneiro, 23-Mar-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 𝑌 ∣ (◡𝐹 “ 𝑠) ∈ 𝐽})
Theorem	elqtop 23720	Value of the quotient topology function. (Contributed by Mario Carneiro, 23-Mar-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐴 ∈ (𝐽 qTop 𝐹) ↔ (𝐴 ⊆ 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ 𝐽)))
Theorem	qtopres 23721	The quotient topology is unaffected by restriction to the base set. This property makes it slightly more convenient to use, since we don't have to require that 𝐹 be a function with domain 𝑋. (Contributed by Mario Carneiro, 23-Mar-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐹 ∈ 𝑉 → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 ↾ 𝑋)))
Theorem	qtoptop2 23722	The quotient topology is a topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝐽 qTop 𝐹) ∈ Top)
Theorem	qtoptop 23723	The quotient topology is a topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top)
Theorem	elqtop2 23724	Value of the quotient topology function. (Contributed by Mario Carneiro, 9-Apr-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑋–onto→𝑌) → (𝐴 ∈ (𝐽 qTop 𝐹) ↔ (𝐴 ⊆ 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ 𝐽)))
Theorem	qtopuni 23725	The base set of the quotient topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→𝑌) → 𝑌 = ∪ (𝐽 qTop 𝐹))
Theorem	elqtop3 23726	Value of the quotient topology function. (Contributed by Mario Carneiro, 9-Apr-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐴 ∈ (𝐽 qTop 𝐹) ↔ (𝐴 ⊆ 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ 𝐽)))
Theorem	qtoptopon 23727	The base set of the quotient topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
Theorem	qtopid 23728	A quotient map is a continuous function into its quotient topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
Theorem	idqtop 23729	The quotient topology induced by the identity function is the original topology. (Contributed by Mario Carneiro, 30-Aug-2015.)
⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 qTop ( I ↾ 𝑋)) = 𝐽)
Theorem	qtopcmplem 23730	Lemma for qtopcmp 23731 and qtopconn 23732. (Contributed by Mario Carneiro, 24-Mar-2015.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Top)    &   ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹:𝑋–onto→∪ (𝐽 qTop 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))) → (𝐽 qTop 𝐹) ∈ 𝐴)    ⇒   ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ 𝐴)
Theorem	qtopcmp 23731	A quotient of a compact space is compact. (Contributed by Mario Carneiro, 24-Mar-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Comp ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Comp)
Theorem	qtopconn 23732	A quotient of a connected space is connected. (Contributed by Mario Carneiro, 24-Mar-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Conn ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Conn)
Theorem	qtopkgen 23733	A quotient of a compactly generated space is compactly generated. (Contributed by Mario Carneiro, 9-Apr-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ ran 𝑘Gen)
Theorem	basqtop 23734	An injection maps bases to bases. (Contributed by Mario Carneiro, 27-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝐽 qTop 𝐹) ∈ TopBases)
Theorem	tgqtop 23735	An injection maps generated topologies to each other. (Contributed by Mario Carneiro, 27-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → ((topGen‘𝐽) qTop 𝐹) = (topGen‘(𝐽 qTop 𝐹)))
Theorem	qtopcld 23736	The property of being a closed set in the quotient topology. (Contributed by Mario Carneiro, 24-Mar-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐴 ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ (𝐴 ⊆ 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽))))
Theorem	qtopcn 23737	Universal property of a quotient map. (Contributed by Mario Carneiro, 23-Mar-2015.)
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → (𝐺 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ↔ (𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐾)))
Theorem	qtopss 23738	A surjective continuous function from 𝐽 to 𝐾 induces a topology 𝐽 qTop 𝐹 on the base set of 𝐾. This topology is in general finer than 𝐾. Together with qtopid 23728, this implies that 𝐽 qTop 𝐹 is the finest topology making 𝐹 continuous, i.e. the final topology with respect to the family {𝐹}. (Contributed by Mario Carneiro, 24-Mar-2015.)
⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → 𝐾 ⊆ (𝐽 qTop 𝐹))
Theorem	qtopeu 23739*	Universal property of the quotient topology. If 𝐺 is a function from 𝐽 to 𝐾 which is equal on all equivalent elements under 𝐹, then there is a unique continuous map 𝑓:(𝐽 / 𝐹)⟶𝐾 such that 𝐺 = 𝑓 ∘ 𝐹, and we say that 𝐺 "passes to the quotient". (Contributed by Mario Carneiro, 24-Mar-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐺‘𝑥) = (𝐺‘𝑦))    ⇒   ⊢ (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)𝐺 = (𝑓 ∘ 𝐹))
Theorem	qtoprest 23740	If 𝐴 is a saturated open or closed set (where saturated means that 𝐴 = (◡𝐹 “ 𝑈) for some 𝑈), then the restriction of the quotient map 𝐹 to 𝐴 is a quotient map. (Contributed by Mario Carneiro, 24-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝑌)    &   ⊢ (𝜑 → 𝐴 = (◡𝐹 “ 𝑈))    &   ⊢ (𝜑 → (𝐴 ∈ 𝐽 ∨ 𝐴 ∈ (Clsd‘𝐽)))    ⇒   ⊢ (𝜑 → ((𝐽 qTop 𝐹) ↾t 𝑈) = ((𝐽 ↾t 𝐴) qTop (𝐹 ↾ 𝐴)))
Theorem	qtopomap 23741*	If 𝐹 is a surjective continuous open map, then it is a quotient map. (An open map is a function that maps open sets to open sets.) (Contributed by Mario Carneiro, 24-Mar-2015.)
⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → ran 𝐹 = 𝑌)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐾)    ⇒   ⊢ (𝜑 → 𝐾 = (𝐽 qTop 𝐹))
Theorem	qtopcmap 23742*	If 𝐹 is a surjective continuous closed map, then it is a quotient map. (A closed map is a function that maps closed sets to closed sets.) (Contributed by Mario Carneiro, 24-Mar-2015.)
⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → ran 𝐹 = 𝑌)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝐹 “ 𝑥) ∈ (Clsd‘𝐾))    ⇒   ⊢ (𝜑 → 𝐾 = (𝐽 qTop 𝐹))
Theorem	imastopn 23743	The topology of an image structure. (Contributed by Mario Carneiro, 27-Aug-2015.)
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑅)    &   ⊢ 𝑂 = (TopOpen‘𝑈)    ⇒   ⊢ (𝜑 → 𝑂 = (𝐽 qTop 𝐹))
Theorem	imastps 23744	The image of a topological space under a function is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ TopSp)    ⇒   ⊢ (𝜑 → 𝑈 ∈ TopSp)
Theorem	qustps 23745	A quotient structure is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
⊢ (𝜑 → 𝑈 = (𝑅 /s 𝐸))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ TopSp)    ⇒   ⊢ (𝜑 → 𝑈 ∈ TopSp)
Theorem	kqfval 23746*	Value of the function appearing in df-kq 23717. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})    ⇒   ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) = {𝑦 ∈ 𝐽 ∣ 𝐴 ∈ 𝑦})
Theorem	kqfeq 23747*	Two points in the Kolmogorov quotient are equal iff the original points are topologically indistinguishable. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})    ⇒   ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐹‘𝐴) = (𝐹‘𝐵) ↔ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ↔ 𝐵 ∈ 𝑦)))
Theorem	kqffn 23748*	The topological indistinguishability map is a function on the base. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})    ⇒   ⊢ (𝐽 ∈ 𝑉 → 𝐹 Fn 𝑋)
Theorem	kqval 23749*	Value of the quotient topology function. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})    ⇒   ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
Theorem	kqtopon 23750*	The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})    ⇒   ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
Theorem	kqid 23751*	The topological indistinguishability map is a continuous function into the Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})    ⇒   ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
Theorem	ist0-4 23752*	The topological indistinguishability map is injective iff the space is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})    ⇒   ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹:𝑋–1-1→V))
Theorem	kqfvima 23753*	When the image set is open, the quotient map satisfies a partial converse to fnfvima 7252, which is normally only true for injective functions. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})    ⇒   ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ 𝑈 ↔ (𝐹‘𝐴) ∈ (𝐹 “ 𝑈)))
Theorem	kqsat 23754*	Any open set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 23740). (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})    ⇒   ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (◡𝐹 “ (𝐹 “ 𝑈)) = 𝑈)
Theorem	kqdisj 23755*	A version of imain 6652 for the topological indistinguishability map. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})    ⇒   ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝐴 ∖ 𝑈))) = ∅)
Theorem	kqcldsat 23756*	Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 23740). (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})    ⇒   ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (◡𝐹 “ (𝐹 “ 𝑈)) = 𝑈)
Theorem	kqopn 23757*	The topological indistinguishability map is an open map. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})    ⇒   ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝐹 “ 𝑈) ∈ (KQ‘𝐽))
Theorem	kqcld 23758*	The topological indistinguishability map is a closed map. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})    ⇒   ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹 “ 𝑈) ∈ (Clsd‘(KQ‘𝐽)))
Theorem	kqt0lem 23759*	Lemma for kqt0 23769. (Contributed by Mario Carneiro, 23-Mar-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})    ⇒   ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ Kol2)
Theorem	isr0 23760*	The property "𝐽 is an R0 space". A space is R0 if any two topologically distinguishable points are separated (there is an open set containing each one and disjoint from the other). Or in contraposition, if every open set which contains 𝑥 also contains 𝑦, so there is no separation, then 𝑥 and 𝑦 are members of the same open sets. We have chosen not to give this definition a name, because it turns out that a space is R0 if and only if its Kolmogorov quotient is T1, so that is what we prove here. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})    ⇒   ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((KQ‘𝐽) ∈ Fre ↔ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜))))
Theorem	r0cld 23761*	The analogue of the T1 axiom (singletons are closed) for an R0 space. In an R0 space the set of all points topologically indistinguishable from 𝐴 is closed. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})    ⇒   ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → {𝑧 ∈ 𝑋 ∣ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)} ∈ (Clsd‘𝐽))
Theorem	regr1lem 23762*	Lemma for regr1 23773. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐽 ∈ Reg)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐽)    &   ⊢ (𝜑 → ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝐴) ∈ 𝑚 ∧ (𝐹‘𝐵) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))    ⇒   ⊢ (𝜑 → (𝐴 ∈ 𝑈 → 𝐵 ∈ 𝑈))
Theorem	regr1lem2 23763*	A Kolmogorov quotient of a regular space is Hausdorff. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})    ⇒   ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Haus)
Theorem	kqreglem1 23764*	A Kolmogorov quotient of a regular space is regular. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})    ⇒   ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Reg)
Theorem	kqreglem2 23765*	If the Kolmogorov quotient of a space is regular then so is the original space. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})    ⇒   ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Reg)
Theorem	kqnrmlem1 23766*	A Kolmogorov quotient of a normal space is normal. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})    ⇒   ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Nrm)
Theorem	kqnrmlem2 23767*	If the Kolmogorov quotient of a space is normal then so is the original space. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})    ⇒   ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Nrm)
Theorem	kqtop 23768	The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top)
Theorem	kqt0 23769	The Kolmogorov quotient is T0 even if the original topology is not. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Kol2)
Theorem	kqf 23770	The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ KQ:Top⟶Kol2
Theorem	r0sep 23771*	The separation property of an R0 space. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜)))
Theorem	nrmr0reg 23772	A normal R0 space is also regular. These spaces are usually referred to as normal regular spaces. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → 𝐽 ∈ Reg)
Theorem	regr1 23773	A regular space is R1, which means that any two topologically distinct points can be separated by neighborhoods. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ (𝐽 ∈ Reg → (KQ‘𝐽) ∈ Haus)
Theorem	kqreg 23774	The Kolmogorov quotient of a regular space is regular. By regr1 23773 it is also Hausdorff, so we can also say that a space is regular iff the Kolmogorov quotient is regular Hausdorff (T3). (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ (𝐽 ∈ Reg ↔ (KQ‘𝐽) ∈ Reg)
Theorem	kqnrm 23775	The Kolmogorov quotient of a normal space is normal. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ (𝐽 ∈ Nrm ↔ (KQ‘𝐽) ∈ Nrm)
12.1.21  Homeomorphisms
Syntax	chmeo 23776	Extend class notation with the class of all homeomorphisms.
class Homeo
Syntax	chmph 23777	Extend class notation with the relation "is homeomorphic to.".
class ≃
Definition	df-hmeo 23778*	Function returning all the homeomorphisms from topology 𝑗 to topology 𝑘. (Contributed by FL, 14-Feb-2007.)
⊢ Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)})
Definition	df-hmph 23779	Definition of the relation 𝑥 is homeomorphic to 𝑦. (Contributed by FL, 14-Feb-2007.)
⊢ ≃ = (◡Homeo “ (V ∖ 1o))
Theorem	hmeofn 23780	The set of homeomorphisms is a function on topologies. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ Homeo Fn (Top × Top)
Theorem	hmeofval 23781*	The set of all the homeomorphisms between two topologies. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)}
Theorem	ishmeo 23782	The predicate F is a homeomorphism between topology 𝐽 and topology 𝐾. Criterion of [BourbakiTop1] p. I.2. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn 𝐽)))
Theorem	hmeocn 23783	A homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.)
⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
Theorem	hmeocnvcn 23784	The converse of a homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.)
⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
Theorem	hmeocnv 23785	The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾Homeo𝐽))
Theorem	hmeof1o2 23786	A homeomorphism is a 1-1-onto mapping. (Contributed by Mario Carneiro, 22-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽Homeo𝐾)) → 𝐹:𝑋–1-1-onto→𝑌)
Theorem	hmeof1o 23787	A homeomorphism is a 1-1-onto mapping. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 30-May-2014.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1-onto→𝑌)
Theorem	hmeoima 23788	The image of an open set by a homeomorphism is an open set. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ∈ 𝐽) → (𝐹 “ 𝐴) ∈ 𝐾)
Theorem	hmeoopn 23789	Homeomorphisms preserve openness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ 𝐽 ↔ (𝐹 “ 𝐴) ∈ 𝐾))
Theorem	hmeocld 23790	Homeomorphisms preserve closedness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (𝐹 “ 𝐴) ∈ (Clsd‘𝐾)))
Theorem	hmeocls 23791	Homeomorphisms preserve closures. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐾)‘(𝐹 “ 𝐴)) = (𝐹 “ ((cls‘𝐽)‘𝐴)))
Theorem	hmeontr 23792	Homeomorphisms preserve interiors. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐾)‘(𝐹 “ 𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴)))
Theorem	hmeoimaf1o 23793*	The function mapping open sets to their images under a homeomorphism is a bijection of topologies. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ 𝐺 = (𝑥 ∈ 𝐽 ↦ (𝐹 “ 𝑥))    ⇒   ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐺:𝐽–1-1-onto→𝐾)
Theorem	hmeores 23794	The restriction of a homeomorphism is a homeomorphism. (Contributed by Mario Carneiro, 14-Sep-2014.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌)Homeo(𝐾 ↾t (𝐹 “ 𝑌))))
Theorem	hmeoco 23795	The composite of two homeomorphisms is a homeomorphism. (Contributed by FL, 9-Mar-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (𝐺 ∘ 𝐹) ∈ (𝐽Homeo𝐿))
Theorem	idhmeo 23796	The identity function is a homeomorphism. (Contributed by FL, 14-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
⊢ (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽Homeo𝐽))
Theorem	hmeocnvb 23797	The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
⊢ (Rel 𝐹 → (◡𝐹 ∈ (𝐽Homeo𝐾) ↔ 𝐹 ∈ (𝐾Homeo𝐽)))
Theorem	hmeoqtop 23798	A homeomorphism is a quotient map. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐾 = (𝐽 qTop 𝐹))
Theorem	hmph 23799	Express the predicate 𝐽 is homeomorphic to 𝐾. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
Theorem	hmphi 23800	If there is a homeomorphism between spaces, then the spaces are homeomorphic. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐽 ≃ 𝐾)