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Theorem List for Metamath Proof Explorer - 22201-22300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcnmpt11 22201* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑 → (𝑦𝑌𝐵) ∈ (𝐾 Cn 𝐿))    &   (𝑦 = 𝐴𝐵 = 𝐶)       (𝜑 → (𝑥𝑋𝐶) ∈ (𝐽 Cn 𝐿))
 
Theoremcnmpt11f 22202* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐹 ∈ (𝐾 Cn 𝐿))       (𝜑 → (𝑥𝑋 ↦ (𝐹𝐴)) ∈ (𝐽 Cn 𝐿))
 
Theoremcnmpt1t 22203* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿))       (𝜑 → (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
 
Theoremcnmpt12f 22204* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿))    &   (𝜑𝐹 ∈ ((𝐾 ×t 𝐿) Cn 𝑀))       (𝜑 → (𝑥𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝐽 Cn 𝑀))
 
Theoremcnmpt12 22205* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 12-Jun-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐿 ∈ (TopOn‘𝑍))    &   (𝜑 → (𝑦𝑌, 𝑧𝑍𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀))    &   ((𝑦 = 𝐴𝑧 = 𝐵) → 𝐶 = 𝐷)       (𝜑 → (𝑥𝑋𝐷) ∈ (𝐽 Cn 𝑀))
 
Theoremcnmpt1st 22206* The projection onto the first coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))       (𝜑 → (𝑥𝑋, 𝑦𝑌𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
 
Theoremcnmpt2nd 22207* The projection onto the second coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))       (𝜑 → (𝑥𝑋, 𝑦𝑌𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
 
Theoremcnmpt2c 22208* A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐿 ∈ (TopOn‘𝑍))    &   (𝜑𝑃𝑍)       (𝜑 → (𝑥𝑋, 𝑦𝑌𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
 
Theoremcnmpt21 22209* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))    &   (𝜑𝐿 ∈ (TopOn‘𝑍))    &   (𝜑 → (𝑧𝑍𝐵) ∈ (𝐿 Cn 𝑀))    &   (𝑧 = 𝐴𝐵 = 𝐶)       (𝜑 → (𝑥𝑋, 𝑦𝑌𝐶) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
 
Theoremcnmpt21f 22210* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))    &   (𝜑𝐹 ∈ (𝐿 Cn 𝑀))       (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐹𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
 
Theoremcnmpt2t 22211* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))       (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))
 
Theoremcnmpt22 22212* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))    &   (𝜑𝐿 ∈ (TopOn‘𝑍))    &   (𝜑𝑀 ∈ (TopOn‘𝑊))    &   (𝜑 → (𝑧𝑍, 𝑤𝑊𝐶) ∈ ((𝐿 ×t 𝑀) Cn 𝑁))    &   ((𝑧 = 𝐴𝑤 = 𝐵) → 𝐶 = 𝐷)       (𝜑 → (𝑥𝑋, 𝑦𝑌𝐷) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))
 
Theoremcnmpt22f 22213* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))    &   (𝜑𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁))       (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴𝐹𝐵)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))
 
Theoremcnmpt1res 22214* The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 5-Jun-2014.)
𝐾 = (𝐽t 𝑌)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑌𝑋)    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐿))       (𝜑 → (𝑥𝑌𝐴) ∈ (𝐾 Cn 𝐿))
 
Theoremcnmpt2res 22215* The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
𝐾 = (𝐽t 𝑌)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑌𝑋)    &   𝑁 = (𝑀t 𝑊)    &   (𝜑𝑀 ∈ (TopOn‘𝑍))    &   (𝜑𝑊𝑍)    &   (𝜑 → (𝑥𝑋, 𝑦𝑍𝐴) ∈ ((𝐽 ×t 𝑀) Cn 𝐿))       (𝜑 → (𝑥𝑌, 𝑦𝑊𝐴) ∈ ((𝐾 ×t 𝑁) Cn 𝐿))
 
Theoremcnmptcom 22216* The argument converse of a continuous function is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))       (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
 
Theoremcnmptkc 22217* The curried first projection function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))       (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝑥)) ∈ (𝐽 Cn (𝐽ko 𝐾)))
 
Theoremcnmptkp 22218* 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.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐿 ∈ (TopOn‘𝑍))    &   (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))    &   (𝜑𝐵𝑌)    &   (𝑦 = 𝐵𝐴 = 𝐶)       (𝜑 → (𝑥𝑋𝐶) ∈ (𝐽 Cn 𝐿))
 
Theoremcnmptk1 22219* The composition of a curried function with a one-arg function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐿 ∈ (TopOn‘𝑍))    &   (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))    &   (𝜑 → (𝑧𝑍𝐵) ∈ (𝐿 Cn 𝑀))    &   (𝑧 = 𝐴𝐵 = 𝐶)       (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐶)) ∈ (𝐽 Cn (𝑀ko 𝐾)))
 
Theoremcnmpt1k 22220* 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.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐿 ∈ (TopOn‘𝑍))    &   (𝜑𝑀 ∈ (TopOn‘𝑊))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐿))    &   (𝜑 → (𝑦𝑌 ↦ (𝑧𝑍𝐵)) ∈ (𝐾 Cn (𝑀ko 𝐿)))    &   (𝑧 = 𝐴𝐵 = 𝐶)       (𝜑 → (𝑦𝑌 ↦ (𝑥𝑋𝐶)) ∈ (𝐾 Cn (𝑀ko 𝐽)))
 
Theoremcnmptkk 22221* The composition of two curried functions is jointly continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐿 ∈ (TopOn‘𝑍))    &   (𝜑𝑀 ∈ (TopOn‘𝑊))    &   (𝜑𝐿 ∈ 𝑛-Locally Comp)    &   (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))    &   (𝜑 → (𝑥𝑋 ↦ (𝑧𝑍𝐵)) ∈ (𝐽 Cn (𝑀ko 𝐿)))    &   (𝑧 = 𝐴𝐵 = 𝐶)       (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐶)) ∈ (𝐽 Cn (𝑀ko 𝐾)))
 
Theoremxkofvcn 22222* Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 22194.) (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
𝑋 = 𝑅    &   𝐹 = (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋 ↦ (𝑓𝑥))       ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝐹 ∈ (((𝑆ko 𝑅) ×t 𝑅) Cn 𝑆))
 
Theoremcnmptk1p 22223* 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.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐿 ∈ (TopOn‘𝑍))    &   (𝜑𝐾 ∈ 𝑛-Locally Comp)    &   (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐾))    &   (𝑦 = 𝐵𝐴 = 𝐶)       (𝜑 → (𝑥𝑋𝐶) ∈ (𝐽 Cn 𝐿))
 
Theoremcnmptk2 22224* The uncurrying of a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐿 ∈ (TopOn‘𝑍))    &   (𝜑𝐾 ∈ 𝑛-Locally Comp)    &   (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))       (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
 
Theoremxkoinjcn 22225* Continuity of "injection", i.e. currying, as a function on continuous function spaces. (Contributed by Mario Carneiro, 23-Mar-2015.)
𝐹 = (𝑥𝑋 ↦ (𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩))       ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝐹 ∈ (𝑅 Cn ((𝑆 ×t 𝑅) ↑ko 𝑆)))
 
Theoremcnmpt2k 22226* The currying of a two-argument function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))       (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))
 
Theoremtxconn 22227 The topological product of two connected spaces is connected. (Contributed by Mario Carneiro, 29-Mar-2015.)
((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) → (𝑅 ×t 𝑆) ∈ Conn)
 
Theoremimasnopn 22228 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) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) ∈ 𝐾)
 
Theoremimasncld 22229 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)) ∧ 𝐴𝑋)) → (𝑅 “ {𝐴}) ∈ (Clsd‘𝐾))
 
Theoremimasncls 22230 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) ∧ 𝐴𝑋)) → ((cls‘𝐾)‘(𝑅 “ {𝐴})) ⊆ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}))
 
12.1.20  Quotient maps and quotient topology
 
Syntaxckq 22231 Extend class notation with the Kolmogorov quotient function.
class KQ
 
Definitiondf-kq 22232* 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 ↦ {𝑦𝑗𝑥𝑦})))
 
Theoremqtopval 22233* Value of the quotient topology function. (Contributed by Mario Carneiro, 23-Mar-2015.)
𝑋 = 𝐽       ((𝐽𝑉𝐹𝑊) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹𝑋) ∣ ((𝐹𝑠) ∩ 𝑋) ∈ 𝐽})
 
Theoremqtopval2 22234* Value of the quotient topology function when 𝐹 is a function on the base set. (Contributed by Mario Carneiro, 23-Mar-2015.)
𝑋 = 𝐽       ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 𝑌 ∣ (𝐹𝑠) ∈ 𝐽})
 
Theoremelqtop 22235 Value of the quotient topology function. (Contributed by Mario Carneiro, 23-Mar-2015.)
𝑋 = 𝐽       ((𝐽𝑉𝐹:𝑍onto𝑌𝑍𝑋) → (𝐴 ∈ (𝐽 qTop 𝐹) ↔ (𝐴𝑌 ∧ (𝐹𝐴) ∈ 𝐽)))
 
Theoremqtopres 22236 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       (𝐹𝑉 → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹𝑋)))
 
Theoremqtoptop2 22237 The quotient topology is a topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐽 qTop 𝐹) ∈ Top)
 
Theoremqtoptop 22238 The quotient topology is a topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top)
 
Theoremelqtop2 22239 Value of the quotient topology function. (Contributed by Mario Carneiro, 9-Apr-2015.)
𝑋 = 𝐽       ((𝐽𝑉𝐹:𝑋onto𝑌) → (𝐴 ∈ (𝐽 qTop 𝐹) ↔ (𝐴𝑌 ∧ (𝐹𝐴) ∈ 𝐽)))
 
Theoremqtopuni 22240 The base set of the quotient topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐹:𝑋onto𝑌) → 𝑌 = (𝐽 qTop 𝐹))
 
Theoremelqtop3 22241 Value of the quotient topology function. (Contributed by Mario Carneiro, 9-Apr-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐴 ∈ (𝐽 qTop 𝐹) ↔ (𝐴𝑌 ∧ (𝐹𝐴) ∈ 𝐽)))
 
Theoremqtoptopon 22242 The base set of the quotient topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
 
Theoremqtopid 22243 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)))
 
Theoremidqtop 22244 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)
 
Theoremqtopcmplem 22245 Lemma for qtopcmp 22246 and qtopconn 22247. (Contributed by Mario Carneiro, 24-Mar-2015.)
𝑋 = 𝐽    &   (𝐽𝐴𝐽 ∈ Top)    &   ((𝐽𝐴𝐹:𝑋onto (𝐽 qTop 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))) → (𝐽 qTop 𝐹) ∈ 𝐴)       ((𝐽𝐴𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ 𝐴)
 
Theoremqtopcmp 22246 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)
 
Theoremqtopconn 22247 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)
 
Theoremqtopkgen 22248 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)
 
Theorembasqtop 22249 An injection maps bases to bases. (Contributed by Mario Carneiro, 27-Aug-2015.)
𝑋 = 𝐽       ((𝐽 ∈ TopBases ∧ 𝐹:𝑋1-1-onto𝑌) → (𝐽 qTop 𝐹) ∈ TopBases)
 
Theoremtgqtop 22250 An injection maps generated topologies to each other. (Contributed by Mario Carneiro, 27-Aug-2015.)
𝑋 = 𝐽       ((𝐽 ∈ TopBases ∧ 𝐹:𝑋1-1-onto𝑌) → ((topGen‘𝐽) qTop 𝐹) = (topGen‘(𝐽 qTop 𝐹)))
 
Theoremqtopcld 22251 The property of being a closed set in the quotient topology. (Contributed by Mario Carneiro, 24-Mar-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐴 ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ (𝐴𝑌 ∧ (𝐹𝐴) ∈ (Clsd‘𝐽))))
 
Theoremqtopcn 22252 Universal property of a quotient map. (Contributed by Mario Carneiro, 23-Mar-2015.)
(((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋onto𝑌𝐺:𝑌𝑍)) → (𝐺 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ↔ (𝐺𝐹) ∈ (𝐽 Cn 𝐾)))
 
Theoremqtopss 22253 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 22243, 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))
 
Theoremqtopeu 22254* 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 = 𝑓𝐹, and we say that 𝐺 "passes to the quotient". (Contributed by Mario Carneiro, 24-Mar-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹:𝑋onto𝑌)    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋 ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐺𝑥) = (𝐺𝑦))       (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)𝐺 = (𝑓𝐹))
 
Theoremqtoprest 22255 If 𝐴 is a saturated open or closed set (where saturated means that 𝐴 = (𝐹𝑈) 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.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹:𝑋onto𝑌)    &   (𝜑𝑈𝑌)    &   (𝜑𝐴 = (𝐹𝑈))    &   (𝜑 → (𝐴𝐽𝐴 ∈ (Clsd‘𝐽)))       (𝜑 → ((𝐽 qTop 𝐹) ↾t 𝑈) = ((𝐽t 𝐴) qTop (𝐹𝐴)))
 
Theoremqtopomap 22256* 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.)
(𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑 → ran 𝐹 = 𝑌)    &   ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ 𝐾)       (𝜑𝐾 = (𝐽 qTop 𝐹))
 
Theoremqtopcmap 22257* 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.)
(𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑 → ran 𝐹 = 𝑌)    &   ((𝜑𝑥 ∈ (Clsd‘𝐽)) → (𝐹𝑥) ∈ (Clsd‘𝐾))       (𝜑𝐾 = (𝐽 qTop 𝐹))
 
Theoremimastopn 22258 The topology of an image structure. (Contributed by Mario Carneiro, 27-Aug-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   (𝜑𝑅𝑊)    &   𝐽 = (TopOpen‘𝑅)    &   𝑂 = (TopOpen‘𝑈)       (𝜑𝑂 = (𝐽 qTop 𝐹))
 
Theoremimastps 22259 The image of a topological space under a function is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   (𝜑𝑅 ∈ TopSp)       (𝜑𝑈 ∈ TopSp)
 
Theoremqustps 22260 A quotient structure is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
(𝜑𝑈 = (𝑅 /s 𝐸))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐸𝑊)    &   (𝜑𝑅 ∈ TopSp)       (𝜑𝑈 ∈ TopSp)
 
Theoremkqfval 22261* Value of the function appearing in df-kq 22232. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽𝑉𝐴𝑋) → (𝐹𝐴) = {𝑦𝐽𝐴𝑦})
 
Theoremkqfeq 22262* Two points in the Kolmogorov quotient are equal iff the original points are topologically indistinguishable. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽𝑉𝐴𝑋𝐵𝑋) → ((𝐹𝐴) = (𝐹𝐵) ↔ ∀𝑦𝐽 (𝐴𝑦𝐵𝑦)))
 
Theoremkqffn 22263* The topological indistinguishability map is a function on the base. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       (𝐽𝑉𝐹 Fn 𝑋)
 
Theoremkqval 22264* Value of the quotient topology function. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
 
Theoremkqtopon 22265* The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
 
Theoremkqid 22266* The topological indistinguishability map is a continuous function into the Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
 
Theoremist0-4 22267* The topological indistinguishability map is injective iff the space is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹:𝑋1-1→V))
 
Theoremkqfvima 22268* When the image set is open, the quotient map satisfies a partial converse to fnfvima 6986, which is normally only true for injective functions. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → (𝐴𝑈 ↔ (𝐹𝐴) ∈ (𝐹𝑈)))
 
Theoremkqsat 22269* Any open set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 22255). (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ (𝐹𝑈)) = 𝑈)
 
Theoremkqdisj 22270* A version of imain 6433 for the topological indistinguishability map. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ((𝐹𝑈) ∩ (𝐹 “ (𝐴𝑈))) = ∅)
 
Theoremkqcldsat 22271* Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 22255). (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹 “ (𝐹𝑈)) = 𝑈)
 
Theoremkqopn 22272* The topological indistinguishability map is an open map. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹𝑈) ∈ (KQ‘𝐽))
 
Theoremkqcld 22273* The topological indistinguishability map is a closed map. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹𝑈) ∈ (Clsd‘(KQ‘𝐽)))
 
Theoremkqt0lem 22274* Lemma for kqt0 22284. (Contributed by Mario Carneiro, 23-Mar-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ Kol2)
 
Theoremisr0 22275* 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 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       (𝐽 ∈ (TopOn‘𝑋) → ((KQ‘𝐽) ∈ Fre ↔ ∀𝑧𝑋𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜))))
 
Theoremr0cld 22276* 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 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → {𝑧𝑋 ∣ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)} ∈ (Clsd‘𝐽))
 
Theoremregr1lem 22277* Lemma for regr1 22288. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐽 ∈ Reg)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝑈𝐽)    &   (𝜑 → ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝐴) ∈ 𝑚 ∧ (𝐹𝐵) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))       (𝜑 → (𝐴𝑈𝐵𝑈))
 
Theoremregr1lem2 22278* A Kolmogorov quotient of a regular space is Hausdorff. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Haus)
 
Theoremkqreglem1 22279* A Kolmogorov quotient of a regular space is regular. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Reg)
 
Theoremkqreglem2 22280* If the Kolmogorov quotient of a space is regular then so is the original space. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Reg)
 
Theoremkqnrmlem1 22281* A Kolmogorov quotient of a normal space is normal. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Nrm)
 
Theoremkqnrmlem2 22282* If the Kolmogorov quotient of a space is normal then so is the original space. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Nrm)
 
Theoremkqtop 22283 The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
(𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top)
 
Theoremkqt0 22284 The Kolmogorov quotient is T0 even if the original topology is not. (Contributed by Mario Carneiro, 25-Aug-2015.)
(𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Kol2)
 
Theoremkqf 22285 The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ:Top⟶Kol2
 
Theoremr0sep 22286* The separation property of an R0 space. (Contributed by Mario Carneiro, 25-Aug-2015.)
(((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝐴𝑋𝐵𝑋)) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → ∀𝑜𝐽 (𝐴𝑜𝐵𝑜)))
 
Theoremnrmr0reg 22287 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)
 
Theoremregr1 22288 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)
 
Theoremkqreg 22289 The Kolmogorov quotient of a regular space is regular. By regr1 22288 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)
 
Theoremkqnrm 22290 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
 
Syntaxchmeo 22291 Extend class notation with the class of all homeomorphisms.
class Homeo
 
Syntaxchmph 22292 Extend class notation with the relation "is homeomorphic to.".
class
 
Definitiondf-hmeo 22293* 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)})
 
Definitiondf-hmph 22294 Definition of the relation 𝑥 is homeomorphic to 𝑦. (Contributed by FL, 14-Feb-2007.)
≃ = (Homeo “ (V ∖ 1o))
 
Theoremhmeofn 22295 The set of homeomorphisms is a function on topologies. (Contributed by Mario Carneiro, 23-Aug-2015.)
Homeo Fn (Top × Top)
 
Theoremhmeofval 22296* 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)}
 
Theoremishmeo 22297 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)))
 
Theoremhmeocn 22298 A homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.)
(𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
 
Theoremhmeocnvcn 22299 The converse of a homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.)
(𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
 
Theoremhmeocnv 22300 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))
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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