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Theorem List for Metamath Proof Explorer - 22401-22500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempthaus 22401 The product of a collection of Hausdorff spaces is Hausdorff. (Contributed by Mario Carneiro, 2-Sep-2015.)
((𝐴𝑉𝐹:𝐴⟶Haus) → (∏t𝐹) ∈ Haus)
 
Theoremptrescn 22402* Restriction is a continuous function on product topologies. (Contributed by Mario Carneiro, 7-Feb-2015.)
𝑋 = 𝐽    &   𝐽 = (∏t𝐹)    &   𝐾 = (∏t‘(𝐹𝐵))       ((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐵𝐴) → (𝑥𝑋 ↦ (𝑥𝐵)) ∈ (𝐽 Cn 𝐾))
 
Theoremtxtube 22403* The "tube lemma". If 𝑋 is compact and there is an open set 𝑈 containing the line 𝑋 × {𝐴}, then there is a "tube" 𝑋 × 𝑢 for some neighborhood 𝑢 of 𝐴 which is entirely contained within 𝑈. (Contributed by Mario Carneiro, 21-Mar-2015.)
𝑋 = 𝑅    &   𝑌 = 𝑆    &   (𝜑𝑅 ∈ Comp)    &   (𝜑𝑆 ∈ Top)    &   (𝜑𝑈 ∈ (𝑅 ×t 𝑆))    &   (𝜑 → (𝑋 × {𝐴}) ⊆ 𝑈)    &   (𝜑𝐴𝑌)       (𝜑 → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈))
 
Theoremtxcmplem1 22404* Lemma for txcmp 22406. (Contributed by Mario Carneiro, 14-Sep-2014.)
𝑋 = 𝑅    &   𝑌 = 𝑆    &   (𝜑𝑅 ∈ Comp)    &   (𝜑𝑆 ∈ Comp)    &   (𝜑𝑊 ⊆ (𝑅 ×t 𝑆))    &   (𝜑 → (𝑋 × 𝑌) = 𝑊)    &   (𝜑𝐴𝑌)       (𝜑 → ∃𝑢𝑆 (𝐴𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣))
 
Theoremtxcmplem2 22405* Lemma for txcmp 22406. (Contributed by Mario Carneiro, 14-Sep-2014.)
𝑋 = 𝑅    &   𝑌 = 𝑆    &   (𝜑𝑅 ∈ Comp)    &   (𝜑𝑆 ∈ Comp)    &   (𝜑𝑊 ⊆ (𝑅 ×t 𝑆))    &   (𝜑 → (𝑋 × 𝑌) = 𝑊)       (𝜑 → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣)
 
Theoremtxcmp 22406 The topological product of two compact spaces is compact. (Contributed by Mario Carneiro, 14-Sep-2014.) (Proof shortened 21-Mar-2015.)
((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑅 ×t 𝑆) ∈ Comp)
 
Theoremtxcmpb 22407 The topological product of two nonempty topologies is compact iff the component topologies are both compact. (Contributed by Mario Carneiro, 14-Sep-2014.)
𝑋 = 𝑅    &   𝑌 = 𝑆       (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → ((𝑅 ×t 𝑆) ∈ Comp ↔ (𝑅 ∈ Comp ∧ 𝑆 ∈ Comp)))
 
Theoremhausdiag 22408 A topology is Hausdorff iff the diagonal set is closed in the topology's product with itself. EDITORIAL: very clumsy proof, can probably be shortened substantially. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
𝑋 = 𝐽       (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×t 𝐽))))
 
Theoremhauseqlcld 22409 In a Hausdorff topology, the equalizer of two continuous functions is closed (thus, two continuous functions which agree on a dense set agree everywhere). (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐾 ∈ Haus)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))       (𝜑 → dom (𝐹𝐺) ∈ (Clsd‘𝐽))
 
Theoremtxhaus 22410 The topological product of two Hausdorff spaces is Hausdorff. (Contributed by Mario Carneiro, 23-Mar-2015.)
((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) → (𝑅 ×t 𝑆) ∈ Haus)
 
Theoremtxlm 22411* Two sequences converge iff the sequence of their ordered pairs converges. Proposition 14-2.6 of [Gleason] p. 230. (Contributed by NM, 16-Jul-2007.) (Revised by Mario Carneiro, 5-May-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐹:𝑍𝑋)    &   (𝜑𝐺:𝑍𝑌)    &   𝐻 = (𝑛𝑍 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩)       (𝜑 → ((𝐹(⇝𝑡𝐽)𝑅𝐺(⇝𝑡𝐾)𝑆) ↔ 𝐻(⇝𝑡‘(𝐽 ×t 𝐾))⟨𝑅, 𝑆⟩))
 
Theoremlmcn2 22412* The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 15-May-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐹:𝑍𝑋)    &   (𝜑𝐺:𝑍𝑌)    &   (𝜑𝐹(⇝𝑡𝐽)𝑅)    &   (𝜑𝐺(⇝𝑡𝐾)𝑆)    &   (𝜑𝑂 ∈ ((𝐽 ×t 𝐾) Cn 𝑁))    &   𝐻 = (𝑛𝑍 ↦ ((𝐹𝑛)𝑂(𝐺𝑛)))       (𝜑𝐻(⇝𝑡𝑁)(𝑅𝑂𝑆))
 
Theoremtx1stc 22413 The topological product of two first-countable spaces is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) → (𝑅 ×t 𝑆) ∈ 1stω)
 
Theoremtx2ndc 22414 The topological product of two second-countable spaces is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝑅 ∈ 2ndω ∧ 𝑆 ∈ 2ndω) → (𝑅 ×t 𝑆) ∈ 2ndω)
 
Theoremtxkgen 22415 The topological product of a locally compact space and a compactly generated Hausdorff space is compactly generated. (The condition on 𝑆 can also be replaced with either "compactly generated weak Hausdorff (CGWH)" or "compact Hausdorff-ly generated (CHG)", where WH means that all images of compact Hausdorff spaces are closed and CHG means that a set is open iff it is open in all compact Hausdorff spaces.) (Contributed by Mario Carneiro, 23-Mar-2015.)
((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) → (𝑅 ×t 𝑆) ∈ ran 𝑘Gen)
 
Theoremxkohaus 22416 If the codomain space is Hausdorff, then the compact-open topology of continuous functions is also Hausdorff. (Contributed by Mario Carneiro, 19-Mar-2015.)
((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) → (𝑆ko 𝑅) ∈ Haus)
 
Theoremxkoptsub 22417 The compact-open topology is finer than the product topology restricted to continuous functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑋 = 𝑅    &   𝐽 = (∏t‘(𝑋 × {𝑆}))       ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝐽t (𝑅 Cn 𝑆)) ⊆ (𝑆ko 𝑅))
 
Theoremxkopt 22418 The compact-open topology on a discrete set coincides with the product topology where all the factors are the same. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 12-Sep-2015.)
((𝑅 ∈ Top ∧ 𝐴𝑉) → (𝑅ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝑅})))
 
Theoremxkopjcn 22419* Continuity of a projection map from the space of continuous functions. (This theorem can be strengthened, to joint continuity in both 𝑓 and 𝐴 as a function on (𝑆ko 𝑅) ×t 𝑅, but not without stronger assumptions on 𝑅; see xkofvcn 22447.) (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
𝑋 = 𝑅       ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝐴)) ∈ ((𝑆ko 𝑅) Cn 𝑆))
 
Theoremxkoco1cn 22420* If 𝐹 is a continuous function, then 𝑔𝑔𝐹 is a continuous function on function spaces. (The reason we prove this and xkoco2cn 22421 independently of the more general xkococn 22423 is because that requires some inconvenient extra assumptions on 𝑆.) (Contributed by Mario Carneiro, 20-Mar-2015.)
(𝜑𝑇 ∈ Top)    &   (𝜑𝐹 ∈ (𝑅 Cn 𝑆))       (𝜑 → (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) ∈ ((𝑇ko 𝑆) Cn (𝑇ko 𝑅)))
 
Theoremxkoco2cn 22421* If 𝐹 is a continuous function, then 𝑔𝐹𝑔 is a continuous function on function spaces. (Contributed by Mario Carneiro, 23-Mar-2015.)
(𝜑𝑅 ∈ Top)    &   (𝜑𝐹 ∈ (𝑆 Cn 𝑇))       (𝜑 → (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) ∈ ((𝑆ko 𝑅) Cn (𝑇ko 𝑅)))
 
Theoremxkococnlem 22422* Continuity of the composition operation as a function on continuous function spaces. (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
𝐹 = (𝑓 ∈ (𝑆 Cn 𝑇), 𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝑔))    &   (𝜑𝑆 ∈ 𝑛-Locally Comp)    &   (𝜑𝐾 𝑅)    &   (𝜑 → (𝑅t 𝐾) ∈ Comp)    &   (𝜑𝑉𝑇)    &   (𝜑𝐴 ∈ (𝑆 Cn 𝑇))    &   (𝜑𝐵 ∈ (𝑅 Cn 𝑆))    &   (𝜑 → ((𝐴𝐵) “ 𝐾) ⊆ 𝑉)       (𝜑 → ∃𝑧 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉})))
 
Theoremxkococn 22423* Continuity of the composition operation as a function on continuous function spaces. (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
𝐹 = (𝑓 ∈ (𝑆 Cn 𝑇), 𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝑔))       ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝐹 ∈ (((𝑇ko 𝑆) ×t (𝑆ko 𝑅)) Cn (𝑇ko 𝑅)))
 
12.1.19  Continuous function-builders
 
Theoremcnmptid 22424* The identity function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))       (𝜑 → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))
 
Theoremcnmptc 22425* 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))    &   (𝜑𝑃𝑌)       (𝜑 → (𝑥𝑋𝑃) ∈ (𝐽 Cn 𝐾))
 
Theoremcnmpt11 22426* 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 22427* 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 22428* 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 22429* 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 22430* 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 22431* 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 22432* 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 22433* 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 22434* 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 22435* 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 22436* 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 22437* 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 22438* 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 22439* 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 22440* 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 22441* 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 22442* 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 22443* 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 22444* 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 22445* 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 22446* 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 22447* Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 22419.) (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 22448* 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 22449* 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 22450* 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 22451* 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 22452 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 22453 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 22454 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 22455 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 22456 Extend class notation with the Kolmogorov quotient function.
class KQ
 
Definitiondf-kq 22457* 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 22458* 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 22459* 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 22460 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 22461 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 22462 The quotient topology is a topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐽 qTop 𝐹) ∈ Top)
 
Theoremqtoptop 22463 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 22464 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 22465 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 22466 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 22467 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 22468 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 22469 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 22470 Lemma for qtopcmp 22471 and qtopconn 22472. (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 22471 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 22472 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 22473 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 22474 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 22475 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 22476 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 22477 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 22478 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 22468, 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 22479* 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 22480 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 22481* 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 22482* 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 22483 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 22484 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 22485 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 22486* Value of the function appearing in df-kq 22457. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽𝑉𝐴𝑋) → (𝐹𝐴) = {𝑦𝐽𝐴𝑦})
 
Theoremkqfeq 22487* 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 22488* The topological indistinguishability map is a function on the base. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       (𝐽𝑉𝐹 Fn 𝑋)
 
Theoremkqval 22489* 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 22490* 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 22491* 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 22492* 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 22493* When the image set is open, the quotient map satisfies a partial converse to fnfvima 7018, 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 22494* Any open set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 22480). (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ (𝐹𝑈)) = 𝑈)
 
Theoremkqdisj 22495* A version of imain 6434 for the topological indistinguishability map. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ((𝐹𝑈) ∩ (𝐹 “ (𝐴𝑈))) = ∅)
 
Theoremkqcldsat 22496* Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 22480). (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹 “ (𝐹𝑈)) = 𝑈)
 
Theoremkqopn 22497* 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 22498* 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 22499* Lemma for kqt0 22509. (Contributed by Mario Carneiro, 23-Mar-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ Kol2)
 
Theoremisr0 22500* 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))))
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