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Theorem List for Metamath Proof Explorer - 22901-23000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremprdstopn 22901 Topology of a structure product. (Contributed by Mario Carneiro, 27-Aug-2015.)
π‘Œ = (𝑆Xs𝑅)    &   (πœ‘ β†’ 𝑆 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ π‘Š)    &   (πœ‘ β†’ 𝑅 Fn 𝐼)    &   π‘‚ = (TopOpenβ€˜π‘Œ)    β‡’   (πœ‘ β†’ 𝑂 = (∏tβ€˜(TopOpen ∘ 𝑅)))
 
Theoremprdstps 22902 A structure product of topological spaces is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
π‘Œ = (𝑆Xs𝑅)    &   (πœ‘ β†’ 𝑆 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ π‘Š)    &   (πœ‘ β†’ 𝑅:𝐼⟢TopSp)    β‡’   (πœ‘ β†’ π‘Œ ∈ TopSp)
 
Theorempwstps 22903 A structure power of a topological space is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
π‘Œ = (𝑅 ↑s 𝐼)    β‡’   ((𝑅 ∈ TopSp ∧ 𝐼 ∈ 𝑉) β†’ π‘Œ ∈ TopSp)
 
Theoremtxrest 22904 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.)
(((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ π‘Š) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ)) β†’ ((𝑅 Γ—t 𝑆) β†Ύt (𝐴 Γ— 𝐡)) = ((𝑅 β†Ύt 𝐴) Γ—t (𝑆 β†Ύt 𝐡)))
 
Theoremtxdis 22905 The topological product of discrete spaces is discrete. (Contributed by Mario Carneiro, 14-Aug-2015.)
((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝒫 𝐴 Γ—t 𝒫 𝐡) = 𝒫 (𝐴 Γ— 𝐡))
 
Theoremtxindislem 22906 Lemma for txindis 22907. (Contributed by Mario Carneiro, 14-Aug-2015.)
(( I β€˜π΄) Γ— ( I β€˜π΅)) = ( I β€˜(𝐴 Γ— 𝐡))
 
Theoremtxindis 22907 The topological product of indiscrete spaces is indiscrete. (Contributed by Mario Carneiro, 14-Aug-2015.)
({βˆ…, 𝐴} Γ—t {βˆ…, 𝐡}) = {βˆ…, (𝐴 Γ— 𝐡)}
 
Theoremtxdis1cn 22908* 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.)
(πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘Œ))    &   (πœ‘ β†’ 𝐾 ∈ Top)    &   (πœ‘ β†’ 𝐹 Fn (𝑋 Γ— π‘Œ))    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (𝑦 ∈ π‘Œ ↦ (π‘₯𝐹𝑦)) ∈ (𝐽 Cn 𝐾))    β‡’   (πœ‘ β†’ 𝐹 ∈ ((𝒫 𝑋 Γ—t 𝐽) Cn 𝐾))
 
Theoremtxlly 22909* 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.)
((𝑗 ∈ 𝐴 ∧ π‘˜ ∈ 𝐴) β†’ (𝑗 Γ—t π‘˜) ∈ 𝐴)    β‡’   ((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) β†’ (𝑅 Γ—t 𝑆) ∈ Locally 𝐴)
 
Theoremtxnlly 22910* 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.)
((𝑗 ∈ 𝐴 ∧ π‘˜ ∈ 𝐴) β†’ (𝑗 Γ—t π‘˜) ∈ 𝐴)    β‡’   ((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) β†’ (𝑅 Γ—t 𝑆) ∈ 𝑛-Locally 𝐴)
 
Theorempthaus 22911 The product of a collection of Hausdorff spaces is Hausdorff. (Contributed by Mario Carneiro, 2-Sep-2015.)
((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢Haus) β†’ (∏tβ€˜πΉ) ∈ Haus)
 
Theoremptrescn 22912* Restriction is a continuous function on product topologies. (Contributed by Mario Carneiro, 7-Feb-2015.)
𝑋 = βˆͺ 𝐽    &   π½ = (∏tβ€˜πΉ)    &   πΎ = (∏tβ€˜(𝐹 β†Ύ 𝐡))    β‡’   ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢Top ∧ 𝐡 βŠ† 𝐴) β†’ (π‘₯ ∈ 𝑋 ↦ (π‘₯ β†Ύ 𝐡)) ∈ (𝐽 Cn 𝐾))
 
Theoremtxtube 22913* The "tube lemma". If 𝑋 is compact and there is an open set π‘ˆ containing the line 𝑋 Γ— {𝐴}, then there is a "tube" 𝑋 Γ— 𝑒 for some neighborhood 𝑒 of 𝐴 which is entirely contained within π‘ˆ. (Contributed by Mario Carneiro, 21-Mar-2015.)
𝑋 = βˆͺ 𝑅    &   π‘Œ = βˆͺ 𝑆    &   (πœ‘ β†’ 𝑅 ∈ Comp)    &   (πœ‘ β†’ 𝑆 ∈ Top)    &   (πœ‘ β†’ π‘ˆ ∈ (𝑅 Γ—t 𝑆))    &   (πœ‘ β†’ (𝑋 Γ— {𝐴}) βŠ† π‘ˆ)    &   (πœ‘ β†’ 𝐴 ∈ π‘Œ)    β‡’   (πœ‘ β†’ βˆƒπ‘’ ∈ 𝑆 (𝐴 ∈ 𝑒 ∧ (𝑋 Γ— 𝑒) βŠ† π‘ˆ))
 
Theoremtxcmplem1 22914* Lemma for txcmp 22916. (Contributed by Mario Carneiro, 14-Sep-2014.)
𝑋 = βˆͺ 𝑅    &   π‘Œ = βˆͺ 𝑆    &   (πœ‘ β†’ 𝑅 ∈ Comp)    &   (πœ‘ β†’ 𝑆 ∈ Comp)    &   (πœ‘ β†’ π‘Š βŠ† (𝑅 Γ—t 𝑆))    &   (πœ‘ β†’ (𝑋 Γ— π‘Œ) = βˆͺ π‘Š)    &   (πœ‘ β†’ 𝐴 ∈ π‘Œ)    β‡’   (πœ‘ β†’ βˆƒπ‘’ ∈ 𝑆 (𝐴 ∈ 𝑒 ∧ βˆƒπ‘£ ∈ (𝒫 π‘Š ∩ Fin)(𝑋 Γ— 𝑒) βŠ† βˆͺ 𝑣))
 
Theoremtxcmplem2 22915* Lemma for txcmp 22916. (Contributed by Mario Carneiro, 14-Sep-2014.)
𝑋 = βˆͺ 𝑅    &   π‘Œ = βˆͺ 𝑆    &   (πœ‘ β†’ 𝑅 ∈ Comp)    &   (πœ‘ β†’ 𝑆 ∈ Comp)    &   (πœ‘ β†’ π‘Š βŠ† (𝑅 Γ—t 𝑆))    &   (πœ‘ β†’ (𝑋 Γ— π‘Œ) = βˆͺ π‘Š)    β‡’   (πœ‘ β†’ βˆƒπ‘£ ∈ (𝒫 π‘Š ∩ Fin)(𝑋 Γ— π‘Œ) = βˆͺ 𝑣)
 
Theoremtxcmp 22916 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) β†’ (𝑅 Γ—t 𝑆) ∈ Comp)
 
Theoremtxcmpb 22917 The topological product of two nonempty topologies is compact iff the component topologies are both compact. (Contributed by Mario Carneiro, 14-Sep-2014.)
𝑋 = βˆͺ 𝑅    &   π‘Œ = βˆͺ 𝑆    β‡’   (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 β‰  βˆ… ∧ π‘Œ β‰  βˆ…)) β†’ ((𝑅 Γ—t 𝑆) ∈ Comp ↔ (𝑅 ∈ Comp ∧ 𝑆 ∈ Comp)))
 
Theoremhausdiag 22918 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 = βˆͺ 𝐽    β‡’   (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ( I β†Ύ 𝑋) ∈ (Clsdβ€˜(𝐽 Γ—t 𝐽))))
 
Theoremhauseqlcld 22919 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.)
(πœ‘ β†’ 𝐾 ∈ Haus)    &   (πœ‘ β†’ 𝐹 ∈ (𝐽 Cn 𝐾))    &   (πœ‘ β†’ 𝐺 ∈ (𝐽 Cn 𝐾))    β‡’   (πœ‘ β†’ dom (𝐹 ∩ 𝐺) ∈ (Clsdβ€˜π½))
 
Theoremtxhaus 22920 The topological product of two Hausdorff spaces is Hausdorff. (Contributed by Mario Carneiro, 23-Mar-2015.)
((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) β†’ (𝑅 Γ—t 𝑆) ∈ Haus)
 
Theoremtxlm 22921* 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 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))    &   (πœ‘ β†’ 𝐹:π‘βŸΆπ‘‹)    &   (πœ‘ β†’ 𝐺:π‘βŸΆπ‘Œ)    &   π» = (𝑛 ∈ 𝑍 ↦ ⟨(πΉβ€˜π‘›), (πΊβ€˜π‘›)⟩)    β‡’   (πœ‘ β†’ ((𝐹(β‡π‘‘β€˜π½)𝑅 ∧ 𝐺(β‡π‘‘β€˜πΎ)𝑆) ↔ 𝐻(β‡π‘‘β€˜(𝐽 Γ—t 𝐾))βŸ¨π‘…, π‘†βŸ©))
 
Theoremlmcn2 22922* 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 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))    &   (πœ‘ β†’ 𝐹:π‘βŸΆπ‘‹)    &   (πœ‘ β†’ 𝐺:π‘βŸΆπ‘Œ)    &   (πœ‘ β†’ 𝐹(β‡π‘‘β€˜π½)𝑅)    &   (πœ‘ β†’ 𝐺(β‡π‘‘β€˜πΎ)𝑆)    &   (πœ‘ β†’ 𝑂 ∈ ((𝐽 Γ—t 𝐾) Cn 𝑁))    &   π» = (𝑛 ∈ 𝑍 ↦ ((πΉβ€˜π‘›)𝑂(πΊβ€˜π‘›)))    β‡’   (πœ‘ β†’ 𝐻(β‡π‘‘β€˜π‘)(𝑅𝑂𝑆))
 
Theoremtx1stc 22923 The topological product of two first-countable spaces is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝑅 ∈ 1stΟ‰ ∧ 𝑆 ∈ 1stΟ‰) β†’ (𝑅 Γ—t 𝑆) ∈ 1stΟ‰)
 
Theoremtx2ndc 22924 The topological product of two second-countable spaces is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝑅 ∈ 2ndΟ‰ ∧ 𝑆 ∈ 2ndΟ‰) β†’ (𝑅 Γ—t 𝑆) ∈ 2ndΟ‰)
 
Theoremtxkgen 22925 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 π‘˜Gen ∩ Haus)) β†’ (𝑅 Γ—t 𝑆) ∈ ran π‘˜Gen)
 
Theoremxkohaus 22926 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) β†’ (𝑆 ↑ko 𝑅) ∈ Haus)
 
Theoremxkoptsub 22927 The compact-open topology is finer than the product topology restricted to continuous functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑋 = βˆͺ 𝑅    &   π½ = (∏tβ€˜(𝑋 Γ— {𝑆}))    β‡’   ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) β†’ (𝐽 β†Ύt (𝑅 Cn 𝑆)) βŠ† (𝑆 ↑ko 𝑅))
 
Theoremxkopt 22928 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 ∧ 𝐴 ∈ 𝑉) β†’ (𝑅 ↑ko 𝒫 𝐴) = (∏tβ€˜(𝐴 Γ— {𝑅})))
 
Theoremxkopjcn 22929* 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) Γ—t 𝑅, but not without stronger assumptions on 𝑅; see xkofvcn 22957.) (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
𝑋 = βˆͺ 𝑅    β‡’   ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) β†’ (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (π‘“β€˜π΄)) ∈ ((𝑆 ↑ko 𝑅) Cn 𝑆))
 
Theoremxkoco1cn 22930* If 𝐹 is a continuous function, then 𝑔 ↦ 𝑔 ∘ 𝐹 is a continuous function on function spaces. (The reason we prove this and xkoco2cn 22931 independently of the more general xkococn 22933 is because that requires some inconvenient extra assumptions on 𝑆.) (Contributed by Mario Carneiro, 20-Mar-2015.)
(πœ‘ β†’ 𝑇 ∈ Top)    &   (πœ‘ β†’ 𝐹 ∈ (𝑅 Cn 𝑆))    β‡’   (πœ‘ β†’ (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) ∈ ((𝑇 ↑ko 𝑆) Cn (𝑇 ↑ko 𝑅)))
 
Theoremxkoco2cn 22931* If 𝐹 is a continuous function, then 𝑔 ↦ 𝐹 ∘ 𝑔 is a continuous function on function spaces. (Contributed by Mario Carneiro, 23-Mar-2015.)
(πœ‘ β†’ 𝑅 ∈ Top)    &   (πœ‘ β†’ 𝐹 ∈ (𝑆 Cn 𝑇))    β‡’   (πœ‘ β†’ (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)) ∈ ((𝑆 ↑ko 𝑅) Cn (𝑇 ↑ko 𝑅)))
 
Theoremxkococnlem 22932* 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) ↦ (𝑓 ∘ 𝑔))    &   (πœ‘ β†’ 𝑆 ∈ 𝑛-Locally Comp)    &   (πœ‘ β†’ 𝐾 βŠ† βˆͺ 𝑅)    &   (πœ‘ β†’ (𝑅 β†Ύt 𝐾) ∈ Comp)    &   (πœ‘ β†’ 𝑉 ∈ 𝑇)    &   (πœ‘ β†’ 𝐴 ∈ (𝑆 Cn 𝑇))    &   (πœ‘ β†’ 𝐡 ∈ (𝑅 Cn 𝑆))    &   (πœ‘ β†’ ((𝐴 ∘ 𝐡) β€œ 𝐾) βŠ† 𝑉)    β‡’   (πœ‘ β†’ βˆƒπ‘§ ∈ ((𝑇 ↑ko 𝑆) Γ—t (𝑆 ↑ko 𝑅))(⟨𝐴, 𝐡⟩ ∈ 𝑧 ∧ 𝑧 βŠ† (◑𝐹 β€œ {β„Ž ∈ (𝑅 Cn 𝑇) ∣ (β„Ž β€œ 𝐾) βŠ† 𝑉})))
 
Theoremxkococn 22933* 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) ↦ (𝑓 ∘ 𝑔))    β‡’   ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) β†’ 𝐹 ∈ (((𝑇 ↑ko 𝑆) Γ—t (𝑆 ↑ko 𝑅)) Cn (𝑇 ↑ko 𝑅)))
 
12.1.19  Continuous function-builders
 
Theoremcnmptid 22934* The identity function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ π‘₯) ∈ (𝐽 Cn 𝐽))
 
Theoremcnmptc 22935* A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))    &   (πœ‘ β†’ 𝑃 ∈ π‘Œ)    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝑃) ∈ (𝐽 Cn 𝐾))
 
Theoremcnmpt11 22936* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))    &   (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))    &   (πœ‘ β†’ (𝑦 ∈ π‘Œ ↦ 𝐡) ∈ (𝐾 Cn 𝐿))    &   (𝑦 = 𝐴 β†’ 𝐡 = 𝐢)    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐢) ∈ (𝐽 Cn 𝐿))
 
Theoremcnmpt11f 22937* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))    &   (πœ‘ β†’ 𝐹 ∈ (𝐾 Cn 𝐿))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (πΉβ€˜π΄)) ∈ (𝐽 Cn 𝐿))
 
Theoremcnmpt1t 22938* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐿))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ ⟨𝐴, 𝐡⟩) ∈ (𝐽 Cn (𝐾 Γ—t 𝐿)))
 
Theoremcnmpt12f 22939* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐿))    &   (πœ‘ β†’ 𝐹 ∈ ((𝐾 Γ—t 𝐿) Cn 𝑀))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴𝐹𝐡)) ∈ (𝐽 Cn 𝑀))
 
Theoremcnmpt12 22940* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 12-Jun-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐿))    &   (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))    &   (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘))    &   (πœ‘ β†’ (𝑦 ∈ π‘Œ, 𝑧 ∈ 𝑍 ↦ 𝐢) ∈ ((𝐾 Γ—t 𝐿) Cn 𝑀))    &   ((𝑦 = 𝐴 ∧ 𝑧 = 𝐡) β†’ 𝐢 = 𝐷)    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐷) ∈ (𝐽 Cn 𝑀))
 
Theoremcnmpt1st 22941* The projection onto the first coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ π‘₯) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐽))
 
Theoremcnmpt2nd 22942* The projection onto the second coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝑦) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐾))
 
Theoremcnmpt2c 22943* A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))    &   (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘))    &   (πœ‘ β†’ 𝑃 ∈ 𝑍)    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝑃) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))
 
Theoremcnmpt21 22944* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))    &   (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘))    &   (πœ‘ β†’ (𝑧 ∈ 𝑍 ↦ 𝐡) ∈ (𝐿 Cn 𝑀))    &   (𝑧 = 𝐴 β†’ 𝐡 = 𝐢)    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐢) ∈ ((𝐽 Γ—t 𝐾) Cn 𝑀))
 
Theoremcnmpt21f 22945* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))    &   (πœ‘ β†’ 𝐹 ∈ (𝐿 Cn 𝑀))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ (πΉβ€˜π΄)) ∈ ((𝐽 Γ—t 𝐾) Cn 𝑀))
 
Theoremcnmpt2t 22946* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡) ∈ ((𝐽 Γ—t 𝐾) Cn 𝑀))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ ⟨𝐴, 𝐡⟩) ∈ ((𝐽 Γ—t 𝐾) Cn (𝐿 Γ—t 𝑀)))
 
Theoremcnmpt22 22947* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡) ∈ ((𝐽 Γ—t 𝐾) Cn 𝑀))    &   (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘))    &   (πœ‘ β†’ 𝑀 ∈ (TopOnβ€˜π‘Š))    &   (πœ‘ β†’ (𝑧 ∈ 𝑍, 𝑀 ∈ π‘Š ↦ 𝐢) ∈ ((𝐿 Γ—t 𝑀) Cn 𝑁))    &   ((𝑧 = 𝐴 ∧ 𝑀 = 𝐡) β†’ 𝐢 = 𝐷)    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐷) ∈ ((𝐽 Γ—t 𝐾) Cn 𝑁))
 
Theoremcnmpt22f 22948* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡) ∈ ((𝐽 Γ—t 𝐾) Cn 𝑀))    &   (πœ‘ β†’ 𝐹 ∈ ((𝐿 Γ—t 𝑀) Cn 𝑁))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ (𝐴𝐹𝐡)) ∈ ((𝐽 Γ—t 𝐾) Cn 𝑁))
 
Theoremcnmpt1res 22949* The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 5-Jun-2014.)
𝐾 = (𝐽 β†Ύt π‘Œ)    &   (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ π‘Œ βŠ† 𝑋)    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐿))    β‡’   (πœ‘ β†’ (π‘₯ ∈ π‘Œ ↦ 𝐴) ∈ (𝐾 Cn 𝐿))
 
Theoremcnmpt2res 22950* The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
𝐾 = (𝐽 β†Ύt π‘Œ)    &   (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ π‘Œ βŠ† 𝑋)    &   π‘ = (𝑀 β†Ύt π‘Š)    &   (πœ‘ β†’ 𝑀 ∈ (TopOnβ€˜π‘))    &   (πœ‘ β†’ π‘Š βŠ† 𝑍)    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ∈ ((𝐽 Γ—t 𝑀) Cn 𝐿))    β‡’   (πœ‘ β†’ (π‘₯ ∈ π‘Œ, 𝑦 ∈ π‘Š ↦ 𝐴) ∈ ((𝐾 Γ—t 𝑁) Cn 𝐿))
 
Theoremcnmptcom 22951* The argument converse of a continuous function is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))    β‡’   (πœ‘ β†’ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ ((𝐾 Γ—t 𝐽) Cn 𝐿))
 
Theoremcnmptkc 22952* The curried first projection function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ π‘₯)) ∈ (𝐽 Cn (𝐽 ↑ko 𝐾)))
 
Theoremcnmptkp 22953* 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.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))    &   (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))    &   (πœ‘ β†’ 𝐡 ∈ π‘Œ)    &   (𝑦 = 𝐡 β†’ 𝐴 = 𝐢)    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐢) ∈ (𝐽 Cn 𝐿))
 
Theoremcnmptk1 22954* The composition of a curried function with a one-arg function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))    &   (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))    &   (πœ‘ β†’ (𝑧 ∈ 𝑍 ↦ 𝐡) ∈ (𝐿 Cn 𝑀))    &   (𝑧 = 𝐴 β†’ 𝐡 = 𝐢)    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐢)) ∈ (𝐽 Cn (𝑀 ↑ko 𝐾)))
 
Theoremcnmpt1k 22955* 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.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))    &   (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘))    &   (πœ‘ β†’ 𝑀 ∈ (TopOnβ€˜π‘Š))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐿))    &   (πœ‘ β†’ (𝑦 ∈ π‘Œ ↦ (𝑧 ∈ 𝑍 ↦ 𝐡)) ∈ (𝐾 Cn (𝑀 ↑ko 𝐿)))    &   (𝑧 = 𝐴 β†’ 𝐡 = 𝐢)    β‡’   (πœ‘ β†’ (𝑦 ∈ π‘Œ ↦ (π‘₯ ∈ 𝑋 ↦ 𝐢)) ∈ (𝐾 Cn (𝑀 ↑ko 𝐽)))
 
Theoremcnmptkk 22956* The composition of two curried functions is jointly continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))    &   (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘))    &   (πœ‘ β†’ 𝑀 ∈ (TopOnβ€˜π‘Š))    &   (πœ‘ β†’ 𝐿 ∈ 𝑛-Locally Comp)    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑧 ∈ 𝑍 ↦ 𝐡)) ∈ (𝐽 Cn (𝑀 ↑ko 𝐿)))    &   (𝑧 = 𝐴 β†’ 𝐡 = 𝐢)    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐢)) ∈ (𝐽 Cn (𝑀 ↑ko 𝐾)))
 
Theoremxkofvcn 22957* Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 22929.) (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
𝑋 = βˆͺ 𝑅    &   πΉ = (𝑓 ∈ (𝑅 Cn 𝑆), π‘₯ ∈ 𝑋 ↦ (π‘“β€˜π‘₯))    β‡’   ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ 𝐹 ∈ (((𝑆 ↑ko 𝑅) Γ—t 𝑅) Cn 𝑆))
 
Theoremcnmptk1p 22958* 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.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))    &   (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘))    &   (πœ‘ β†’ 𝐾 ∈ 𝑛-Locally Comp)    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐾))    &   (𝑦 = 𝐡 β†’ 𝐴 = 𝐢)    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐢) ∈ (𝐽 Cn 𝐿))
 
Theoremcnmptk2 22959* The uncurrying of a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))    &   (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘))    &   (πœ‘ β†’ 𝐾 ∈ 𝑛-Locally Comp)    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))
 
Theoremxkoinjcn 22960* Continuity of "injection", i.e. currying, as a function on continuous function spaces. (Contributed by Mario Carneiro, 23-Mar-2015.)
𝐹 = (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩))    β‡’   ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝐹 ∈ (𝑅 Cn ((𝑆 Γ—t 𝑅) ↑ko 𝑆)))
 
Theoremcnmpt2k 22961* The currying of a two-argument function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
 
Theoremtxconn 22962 The topological product of two connected spaces is connected. (Contributed by Mario Carneiro, 29-Mar-2015.)
((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) β†’ (𝑅 Γ—t 𝑆) ∈ Conn)
 
Theoremimasnopn 22963 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 = βˆͺ 𝐽    β‡’   (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 Γ—t 𝐾) ∧ 𝐴 ∈ 𝑋)) β†’ (𝑅 β€œ {𝐴}) ∈ 𝐾)
 
Theoremimasncld 22964 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 = βˆͺ 𝐽    β‡’   (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (Clsdβ€˜(𝐽 Γ—t 𝐾)) ∧ 𝐴 ∈ 𝑋)) β†’ (𝑅 β€œ {𝐴}) ∈ (Clsdβ€˜πΎ))
 
Theoremimasncls 22965 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 = βˆͺ 𝐽    &   π‘Œ = βˆͺ 𝐾    β‡’   (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 βŠ† (𝑋 Γ— π‘Œ) ∧ 𝐴 ∈ 𝑋)) β†’ ((clsβ€˜πΎ)β€˜(𝑅 β€œ {𝐴})) βŠ† (((clsβ€˜(𝐽 Γ—t 𝐾))β€˜π‘…) β€œ {𝐴}))
 
12.1.20  Quotient maps and quotient topology
 
Syntaxckq 22966 Extend class notation with the Kolmogorov quotient function.
class KQ
 
Definitiondf-kq 22967* 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 (π‘₯ ∈ βˆͺ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ π‘₯ ∈ 𝑦})))
 
Theoremqtopval 22968* Value of the quotient topology function. (Contributed by Mario Carneiro, 23-Mar-2015.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ 𝑉 ∧ 𝐹 ∈ π‘Š) β†’ (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹 β€œ 𝑋) ∣ ((◑𝐹 β€œ 𝑠) ∩ 𝑋) ∈ 𝐽})
 
Theoremqtopval2 22969* Value of the quotient topology function when 𝐹 is a function on the base set. (Contributed by Mario Carneiro, 23-Mar-2015.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–ontoβ†’π‘Œ ∧ 𝑍 βŠ† 𝑋) β†’ (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 π‘Œ ∣ (◑𝐹 β€œ 𝑠) ∈ 𝐽})
 
Theoremelqtop 22970 Value of the quotient topology function. (Contributed by Mario Carneiro, 23-Mar-2015.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–ontoβ†’π‘Œ ∧ 𝑍 βŠ† 𝑋) β†’ (𝐴 ∈ (𝐽 qTop 𝐹) ↔ (𝐴 βŠ† π‘Œ ∧ (◑𝐹 β€œ 𝐴) ∈ 𝐽)))
 
Theoremqtopres 22971 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 = βˆͺ 𝐽    β‡’   (𝐹 ∈ 𝑉 β†’ (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 β†Ύ 𝑋)))
 
Theoremqtoptop2 22972 The quotient topology is a topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) β†’ (𝐽 qTop 𝐹) ∈ Top)
 
Theoremqtoptop 22973 The quotient topology is a topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) β†’ (𝐽 qTop 𝐹) ∈ Top)
 
Theoremelqtop2 22974 Value of the quotient topology function. (Contributed by Mario Carneiro, 9-Apr-2015.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ (𝐴 ∈ (𝐽 qTop 𝐹) ↔ (𝐴 βŠ† π‘Œ ∧ (◑𝐹 β€œ 𝐴) ∈ 𝐽)))
 
Theoremqtopuni 22975 The base set of the quotient topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ π‘Œ = βˆͺ (𝐽 qTop 𝐹))
 
Theoremelqtop3 22976 Value of the quotient topology function. (Contributed by Mario Carneiro, 9-Apr-2015.)
((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ (𝐴 ∈ (𝐽 qTop 𝐹) ↔ (𝐴 βŠ† π‘Œ ∧ (◑𝐹 β€œ 𝐴) ∈ 𝐽)))
 
Theoremqtoptopon 22977 The base set of the quotient topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ (𝐽 qTop 𝐹) ∈ (TopOnβ€˜π‘Œ))
 
Theoremqtopid 22978 A quotient map is a continuous function into its quotient topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) β†’ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
 
Theoremidqtop 22979 The quotient topology induced by the identity function is the original topology. (Contributed by Mario Carneiro, 30-Aug-2015.)
(𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 qTop ( I β†Ύ 𝑋)) = 𝐽)
 
Theoremqtopcmplem 22980 Lemma for qtopcmp 22981 and qtopconn 22982. (Contributed by Mario Carneiro, 24-Mar-2015.)
𝑋 = βˆͺ 𝐽    &   (𝐽 ∈ 𝐴 β†’ 𝐽 ∈ Top)    &   ((𝐽 ∈ 𝐴 ∧ 𝐹:𝑋–ontoβ†’βˆͺ (𝐽 qTop 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))) β†’ (𝐽 qTop 𝐹) ∈ 𝐴)    β‡’   ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) β†’ (𝐽 qTop 𝐹) ∈ 𝐴)
 
Theoremqtopcmp 22981 A quotient of a compact space is compact. (Contributed by Mario Carneiro, 24-Mar-2015.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Comp ∧ 𝐹 Fn 𝑋) β†’ (𝐽 qTop 𝐹) ∈ Comp)
 
Theoremqtopconn 22982 A quotient of a connected space is connected. (Contributed by Mario Carneiro, 24-Mar-2015.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Conn ∧ 𝐹 Fn 𝑋) β†’ (𝐽 qTop 𝐹) ∈ Conn)
 
Theoremqtopkgen 22983 A quotient of a compactly generated space is compactly generated. (Contributed by Mario Carneiro, 9-Apr-2015.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ ran π‘˜Gen ∧ 𝐹 Fn 𝑋) β†’ (𝐽 qTop 𝐹) ∈ ran π‘˜Gen)
 
Theorembasqtop 22984 An injection maps bases to bases. (Contributed by Mario Carneiro, 27-Aug-2015.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ (𝐽 qTop 𝐹) ∈ TopBases)
 
Theoremtgqtop 22985 An injection maps generated topologies to each other. (Contributed by Mario Carneiro, 27-Aug-2015.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ ((topGenβ€˜π½) qTop 𝐹) = (topGenβ€˜(𝐽 qTop 𝐹)))
 
Theoremqtopcld 22986 The property of being a closed set in the quotient topology. (Contributed by Mario Carneiro, 24-Mar-2015.)
((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ (𝐴 ∈ (Clsdβ€˜(𝐽 qTop 𝐹)) ↔ (𝐴 βŠ† π‘Œ ∧ (◑𝐹 β€œ 𝐴) ∈ (Clsdβ€˜π½))))
 
Theoremqtopcn 22987 Universal property of a quotient map. (Contributed by Mario Carneiro, 23-Mar-2015.)
(((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘)) ∧ (𝐹:𝑋–ontoβ†’π‘Œ ∧ 𝐺:π‘ŒβŸΆπ‘)) β†’ (𝐺 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ↔ (𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐾)))
 
Theoremqtopss 22988 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 22978, 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 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) β†’ 𝐾 βŠ† (𝐽 qTop 𝐹))
 
Theoremqtopeu 22989* 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.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐹:𝑋–ontoβ†’π‘Œ)    &   (πœ‘ β†’ 𝐺 ∈ (𝐽 Cn 𝐾))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ (πΊβ€˜π‘₯) = (πΊβ€˜π‘¦))    β‡’   (πœ‘ β†’ βˆƒ!𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)𝐺 = (𝑓 ∘ 𝐹))
 
Theoremqtoprest 22990 If 𝐴 is a saturated open or closed set (where saturated means that 𝐴 = (◑𝐹 β€œ π‘ˆ) for some π‘ˆ), 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.)
(πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐹:𝑋–ontoβ†’π‘Œ)    &   (πœ‘ β†’ π‘ˆ βŠ† π‘Œ)    &   (πœ‘ β†’ 𝐴 = (◑𝐹 β€œ π‘ˆ))    &   (πœ‘ β†’ (𝐴 ∈ 𝐽 ∨ 𝐴 ∈ (Clsdβ€˜π½)))    β‡’   (πœ‘ β†’ ((𝐽 qTop 𝐹) β†Ύt π‘ˆ) = ((𝐽 β†Ύt 𝐴) qTop (𝐹 β†Ύ 𝐴)))
 
Theoremqtopomap 22991* 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.)
(πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))    &   (πœ‘ β†’ 𝐹 ∈ (𝐽 Cn 𝐾))    &   (πœ‘ β†’ ran 𝐹 = π‘Œ)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐽) β†’ (𝐹 β€œ π‘₯) ∈ 𝐾)    β‡’   (πœ‘ β†’ 𝐾 = (𝐽 qTop 𝐹))
 
Theoremqtopcmap 22992* 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.)
(πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))    &   (πœ‘ β†’ 𝐹 ∈ (𝐽 Cn 𝐾))    &   (πœ‘ β†’ ran 𝐹 = π‘Œ)    &   ((πœ‘ ∧ π‘₯ ∈ (Clsdβ€˜π½)) β†’ (𝐹 β€œ π‘₯) ∈ (Clsdβ€˜πΎ))    β‡’   (πœ‘ β†’ 𝐾 = (𝐽 qTop 𝐹))
 
Theoremimastopn 22993 The topology of an image structure. (Contributed by Mario Carneiro, 27-Aug-2015.)
(πœ‘ β†’ π‘ˆ = (𝐹 β€œs 𝑅))    &   (πœ‘ β†’ 𝑉 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ 𝐹:𝑉–onto→𝐡)    &   (πœ‘ β†’ 𝑅 ∈ π‘Š)    &   π½ = (TopOpenβ€˜π‘…)    &   π‘‚ = (TopOpenβ€˜π‘ˆ)    β‡’   (πœ‘ β†’ 𝑂 = (𝐽 qTop 𝐹))
 
Theoremimastps 22994 The image of a topological space under a function is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
(πœ‘ β†’ π‘ˆ = (𝐹 β€œs 𝑅))    &   (πœ‘ β†’ 𝑉 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ 𝐹:𝑉–onto→𝐡)    &   (πœ‘ β†’ 𝑅 ∈ TopSp)    β‡’   (πœ‘ β†’ π‘ˆ ∈ TopSp)
 
Theoremqustps 22995 A quotient structure is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
(πœ‘ β†’ π‘ˆ = (𝑅 /s 𝐸))    &   (πœ‘ β†’ 𝑉 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ 𝐸 ∈ π‘Š)    &   (πœ‘ β†’ 𝑅 ∈ TopSp)    β‡’   (πœ‘ β†’ π‘ˆ ∈ TopSp)
 
Theoremkqfval 22996* Value of the function appearing in df-kq 22967. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})    β‡’   ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) β†’ (πΉβ€˜π΄) = {𝑦 ∈ 𝐽 ∣ 𝐴 ∈ 𝑦})
 
Theoremkqfeq 22997* Two points in the Kolmogorov quotient are equal iff the original points are topologically indistinguishable. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})    β‡’   ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((πΉβ€˜π΄) = (πΉβ€˜π΅) ↔ βˆ€π‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 ↔ 𝐡 ∈ 𝑦)))
 
Theoremkqffn 22998* The topological indistinguishability map is a function on the base. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})    β‡’   (𝐽 ∈ 𝑉 β†’ 𝐹 Fn 𝑋)
 
Theoremkqval 22999* Value of the quotient topology function. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})    β‡’   (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (KQβ€˜π½) = (𝐽 qTop 𝐹))
 
Theoremkqtopon 23000* The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})    β‡’   (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (KQβ€˜π½) ∈ (TopOnβ€˜ran 𝐹))
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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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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