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Theorem List for Metamath Proof Explorer - 21401-21500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdfac14lem 21401* Lemma for dfac14 21402. By equipping 𝑆 ∪ {𝑃} for some 𝑃𝑆 with the particular point topology, we can show that 𝑃 is in the closure of 𝑆; hence the sequence 𝑃(𝑥) is in the product of the closures, and we can utilize this instance of ptcls 21400 to extract an element of the closure of X𝑘𝐼𝑆. (Contributed by Mario Carneiro, 2-Sep-2015.)
(𝜑𝐼𝑉)    &   ((𝜑𝑥𝐼) → 𝑆𝑊)    &   ((𝜑𝑥𝐼) → 𝑆 ≠ ∅)    &   𝑃 = 𝒫 𝑆    &   𝑅 = {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃𝑦𝑦 = (𝑆 ∪ {𝑃}))}    &   𝐽 = (∏t‘(𝑥𝐼𝑅))    &   (𝜑 → ((cls‘𝐽)‘X𝑥𝐼 𝑆) = X𝑥𝐼 ((cls‘𝑅)‘𝑆))       (𝜑X𝑥𝐼 𝑆 ≠ ∅)

Theoremdfac14 21402* Theorem ptcls 21400 is an equivalent of the axiom of choice. (Contributed by Mario Carneiro, 3-Sep-2015.)
(CHOICE ↔ ∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))))

Theoremxkoccn 21403* The "constant function" function which maps 𝑥𝑌 to the constant function 𝑧𝑋𝑥 is a continuous function from 𝑋 into the space of continuous functions from 𝑌 to 𝑋. This can also be understood as the currying of the first projection function. (The currying of the second projection function is 𝑥𝑌 ↦ (𝑧𝑋𝑧), which we already know is continuous because it is a constant function.) (Contributed by Mario Carneiro, 19-Mar-2015.)
((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑥𝑌 ↦ (𝑋 × {𝑥})) ∈ (𝑆 Cn (𝑆 ^ko 𝑅)))

Theoremtxcnp 21404* If two functions are continuous at 𝐷, then the ordered pair of them is continuous at 𝐷 into the product topology. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐿 ∈ (TopOn‘𝑍))    &   (𝜑𝐷𝑋)    &   (𝜑 → (𝑥𝑋𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷))    &   (𝜑 → (𝑥𝑋𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷))       (𝜑 → (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 CnP (𝐾 ×t 𝐿))‘𝐷))

Theoremptcnplem 21405* Lemma for ptcnp 21406. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
𝐾 = (∏t𝐹)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼⟶Top)    &   (𝜑𝐷𝑋)    &   ((𝜑𝑘𝐼) → (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝐷))    &   𝑘𝜓    &   ((𝜑𝜓) → 𝐺 Fn 𝐼)    &   (((𝜑𝜓) ∧ 𝑘𝐼) → (𝐺𝑘) ∈ (𝐹𝑘))    &   ((𝜑𝜓) → 𝑊 ∈ Fin)    &   (((𝜑𝜓) ∧ 𝑘 ∈ (𝐼𝑊)) → (𝐺𝑘) = (𝐹𝑘))    &   ((𝜑𝜓) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑘𝐼 (𝐺𝑘))       ((𝜑𝜓) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝐺𝑘)))

Theoremptcnp 21406* If every projection of a function is continuous at 𝐷, then the function itself is continuous at 𝐷 into the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
𝐾 = (∏t𝐹)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼⟶Top)    &   (𝜑𝐷𝑋)    &   ((𝜑𝑘𝐼) → (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝐷))       (𝜑 → (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝐷))

Theoremupxp 21407* Universal property of the Cartesian product considered as a categorical product in the category of sets. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)
𝑃 = (1st ↾ (𝐵 × 𝐶))    &   𝑄 = (2nd ↾ (𝐵 × 𝐶))       ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → ∃!(:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))

Theoremtxcnmpt 21408* A map into the product of two topological spaces is continuous if both of its projections are continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Aug-2015.)
𝑊 = 𝑈    &   𝐻 = (𝑥𝑊 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)       ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐻 ∈ (𝑈 Cn (𝑅 ×t 𝑆)))

Theoremuptx 21409* Universal property of the binary topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
𝑇 = (𝑅 ×t 𝑆)    &   𝑋 = 𝑅    &   𝑌 = 𝑆    &   𝑍 = (𝑋 × 𝑌)    &   𝑃 = (1st𝑍)    &   𝑄 = (2nd𝑍)       ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃! ∈ (𝑈 Cn 𝑇)(𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))

Theoremtxcn 21410 A map into the product of two topological spaces is continuous iff both of its projections are continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
𝑋 = 𝑅    &   𝑌 = 𝑆    &   𝑍 = (𝑋 × 𝑌)    &   𝑊 = 𝑈    &   𝑃 = (1st𝑍)    &   𝑄 = (2nd𝑍)       ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) → (𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ↔ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))))

Theoremptcn 21411* If every projection of a function is continuous, then the function itself is continuous into the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐾 = (∏t𝐹)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼⟶Top)    &   ((𝜑𝑘𝐼) → (𝑥𝑋𝐴) ∈ (𝐽 Cn (𝐹𝑘)))       (𝜑 → (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ (𝐽 Cn 𝐾))

Theoremprdstopn 21412 Topology of a structure product. (Contributed by Mario Carneiro, 27-Aug-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)    &   𝑂 = (TopOpen‘𝑌)       (𝜑𝑂 = (∏t‘(TopOpen ∘ 𝑅)))

Theoremprdstps 21413 A structure product of topologies is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅:𝐼⟶TopSp)       (𝜑𝑌 ∈ TopSp)

Theorempwstps 21414 A structure product of topologies is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ TopSp ∧ 𝐼𝑉) → 𝑌 ∈ TopSp)

Theoremtxrest 21415 The subspace of a topological product space induced by a subset with a Cartesian product representation is a topological product of the subspaces induced by the subspaces of the terms of the products. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
(((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = ((𝑅t 𝐴) ×t (𝑆t 𝐵)))

Theoremtxdis 21416 The topological product of discrete spaces is discrete. (Contributed by Mario Carneiro, 14-Aug-2015.)
((𝐴𝑉𝐵𝑊) → (𝒫 𝐴 ×t 𝒫 𝐵) = 𝒫 (𝐴 × 𝐵))

Theoremtxindislem 21417 Lemma for txindis 21418. (Contributed by Mario Carneiro, 14-Aug-2015.)
(( I ‘𝐴) × ( I ‘𝐵)) = ( I ‘(𝐴 × 𝐵))

Theoremtxindis 21418 The topological product of indiscrete spaces is indiscrete. (Contributed by Mario Carneiro, 14-Aug-2015.)
({∅, 𝐴} ×t {∅, 𝐵}) = {∅, (𝐴 × 𝐵)}

Theoremtxdis1cn 21419* A function is jointly continuous on a discrete left topology iff it is continuous as a function of its right argument, for each fixed left value. (Contributed by Mario Carneiro, 19-Sep-2015.)
(𝜑𝑋𝑉)    &   (𝜑𝐽 ∈ (TopOn‘𝑌))    &   (𝜑𝐾 ∈ Top)    &   (𝜑𝐹 Fn (𝑋 × 𝑌))    &   ((𝜑𝑥𝑋) → (𝑦𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾))       (𝜑𝐹 ∈ ((𝒫 𝑋 ×t 𝐽) Cn 𝐾))

Theoremtxlly 21420* If the property 𝐴 is preserved under topological products, then so is the property of being locally 𝐴. (Contributed by Mario Carneiro, 10-Mar-2015.)
((𝑗𝐴𝑘𝐴) → (𝑗 ×t 𝑘) ∈ 𝐴)       ((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) → (𝑅 ×t 𝑆) ∈ Locally 𝐴)

Theoremtxnlly 21421* If the property 𝐴 is preserved under topological products, then so is the property of being n-locally 𝐴. (Contributed by Mario Carneiro, 13-Apr-2015.)
((𝑗𝐴𝑘𝐴) → (𝑗 ×t 𝑘) ∈ 𝐴)       ((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) → (𝑅 ×t 𝑆) ∈ 𝑛-Locally 𝐴)

Theorempthaus 21422 The product of a collection of Hausdorff spaces is Hausdorff. (Contributed by Mario Carneiro, 2-Sep-2015.)
((𝐴𝑉𝐹:𝐴⟶Haus) → (∏t𝐹) ∈ Haus)

Theoremptrescn 21423* 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 21424* 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 21425* Lemma for txcmp 21427. (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 21426* Lemma for txcmp 21427. (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 21427 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 21428 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 21429 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.)
𝑋 = 𝐽       (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×t 𝐽))))

Theoremhauseqlcld 21430 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 21431 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 21432* 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 21433* 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 21434 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 21435 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 21436 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 21437 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) → (𝑆 ^ko 𝑅) ∈ Haus)

Theoremxkoptsub 21438 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)) ⊆ (𝑆 ^ko 𝑅))

Theoremxkopt 21439 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) → (𝑅 ^ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝑅})))

Theoremxkopjcn 21440* 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 (𝑆 ^ko 𝑅) ×t 𝑅, but not without stronger assumptions on 𝑅; see xkofvcn 21468.) (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)) ∈ ((𝑆 ^ko 𝑅) Cn 𝑆))

Theoremxkoco1cn 21441* If 𝐹 is a continuous function, then 𝑔𝑔𝐹 is a continuous function on function spaces. (The reason we prove this and xkoco2cn 21442 independently of the more general xkococn 21444 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)) ∈ ((𝑇 ^ko 𝑆) Cn (𝑇 ^ko 𝑅)))

Theoremxkoco2cn 21442* 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)) ∈ ((𝑆 ^ko 𝑅) Cn (𝑇 ^ko 𝑅)))

Theoremxkococnlem 21443* 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 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉})))

Theoremxkococn 21444* 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 ∈ (((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅)) Cn (𝑇 ^ko 𝑅)))

12.1.19  Continuous function-builders

Theoremcnmptid 21445* 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 21446* 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 21447* 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 21448* 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 21449* 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 21450* 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 21451* 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 21452* 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 21453* 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 21454* 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 21455* 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 21456* 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 21457* 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 21458* 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 21459* 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 21460* 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 21461* 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 21462* 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 21463* 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 (𝐽 ^ko 𝐾)))

Theoremcnmptkp 21464* 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 (𝐿 ^ko 𝐾)))    &   (𝜑𝐵𝑌)    &   (𝑦 = 𝐵𝐴 = 𝐶)       (𝜑 → (𝑥𝑋𝐶) ∈ (𝐽 Cn 𝐿))

Theoremcnmptk1 21465* 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 (𝐿 ^ko 𝐾)))    &   (𝜑 → (𝑧𝑍𝐵) ∈ (𝐿 Cn 𝑀))    &   (𝑧 = 𝐴𝐵 = 𝐶)       (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐶)) ∈ (𝐽 Cn (𝑀 ^ko 𝐾)))

Theoremcnmpt1k 21466* 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 (𝑀 ^ko 𝐿)))    &   (𝑧 = 𝐴𝐵 = 𝐶)       (𝜑 → (𝑦𝑌 ↦ (𝑥𝑋𝐶)) ∈ (𝐾 Cn (𝑀 ^ko 𝐽)))

Theoremcnmptkk 21467* 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 (𝐿 ^ko 𝐾)))    &   (𝜑 → (𝑥𝑋 ↦ (𝑧𝑍𝐵)) ∈ (𝐽 Cn (𝑀 ^ko 𝐿)))    &   (𝑧 = 𝐴𝐵 = 𝐶)       (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐶)) ∈ (𝐽 Cn (𝑀 ^ko 𝐾)))

Theoremxkofvcn 21468* Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 21440.) (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 ∈ (((𝑆 ^ko 𝑅) ×t 𝑅) Cn 𝑆))

Theoremcnmptk1p 21469* 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 (𝐿 ^ko 𝐾)))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐾))    &   (𝑦 = 𝐵𝐴 = 𝐶)       (𝜑 → (𝑥𝑋𝐶) ∈ (𝐽 Cn 𝐿))

Theoremcnmptk2 21470* 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 (𝐿 ^ko 𝐾)))       (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))

Theoremxkoinjcn 21471* 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 21472* 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 (𝐿 ^ko 𝐾)))

Theoremtxconn 21473 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 21474 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 21475 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 21476 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 21477 Extend class notation with the Kolmogorov quotient function.
class KQ

Definitiondf-kq 21478* 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 21479* 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 21480* 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 21481 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 21482 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 21483 The quotient topology is a topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
((𝐽 ∈ Top ∧ 𝐹𝑉 ∧ Fun 𝐹) → (𝐽 qTop 𝐹) ∈ Top)

Theoremqtoptop 21484 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 21485 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 21486 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 21487 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 21488 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 21489 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 21490 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 21491 Lemma for qtopcmp 21492 and qtopconn 21493. (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 21492 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 21493 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 21494 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 21495 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 21496 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 21497 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 21498 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 21499 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 21489, 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 21500* 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))

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