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Theorem List for Metamath Proof Explorer - 22101-22200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-xko 22101* Define the compact-open topology, which is the natural topology on the set of continuous functions between two topological spaces. (Contributed by Mario Carneiro, 19-Mar-2015.)
ko = (𝑠 ∈ Top, 𝑟 ∈ Top ↦ (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp}, 𝑣𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣}))))
 
Theoremtxval 22102* Value of the binary topological product operation. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 30-Aug-2015.)
𝐵 = ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))       ((𝑅𝑉𝑆𝑊) → (𝑅 ×t 𝑆) = (topGen‘𝐵))
 
Theoremtxuni2 22103* The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 31-Aug-2015.)
𝐵 = ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))    &   𝑋 = 𝑅    &   𝑌 = 𝑆       (𝑋 × 𝑌) = 𝐵
 
Theoremtxbasex 22104* The basis for the product topology is a set. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐵 = ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))       ((𝑅𝑉𝑆𝑊) → 𝐵 ∈ V)
 
Theoremtxbas 22105* The set of Cartesian products of elements from two topological bases is a basis. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐵 = ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))       ((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) → 𝐵 ∈ TopBases)
 
Theoremeltx 22106* A set in a product is open iff each point is surrounded by an open rectangle. (Contributed by Stefan O'Rear, 25-Jan-2015.)
((𝐽𝑉𝐾𝑊) → (𝑆 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑝𝑆𝑥𝐽𝑦𝐾 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆)))
 
Theoremtxtop 22107 The product of two topologies is a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
 
Theoremptval 22108* The value of the product topology function. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}       ((𝐴𝑉𝐹 Fn 𝐴) → (∏t𝐹) = (topGen‘𝐵))
 
Theoremptpjpre1 22109* The preimage of a projection function can be expressed as an indexed cartesian product. (Contributed by Mario Carneiro, 6-Feb-2015.)
𝑋 = X𝑘𝐴 (𝐹𝑘)       (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝐼𝐴𝑈 ∈ (𝐹𝐼))) → ((𝑤𝑋 ↦ (𝑤𝐼)) “ 𝑈) = X𝑘𝐴 if(𝑘 = 𝐼, 𝑈, (𝐹𝑘)))
 
Theoremelpt 22110* Elementhood in the bases of a product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}       (𝑆𝐵 ↔ ∃(( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦)) ∧ 𝑆 = X𝑦𝐴 (𝑦)))
 
Theoremelptr 22111* A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}       ((𝐴𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) ∈ (𝐹𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑊)(𝐺𝑦) = (𝐹𝑦))) → X𝑦𝐴 (𝐺𝑦) ∈ 𝐵)
 
Theoremelptr2 22112* A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}    &   (𝜑𝐴𝑉)    &   (𝜑𝑊 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝑆 ∈ (𝐹𝑘))    &   ((𝜑𝑘 ∈ (𝐴𝑊)) → 𝑆 = (𝐹𝑘))       (𝜑X𝑘𝐴 𝑆𝐵)
 
Theoremptbasid 22113* The base set of the product topology is a basic open set. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}       ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) ∈ 𝐵)
 
Theoremptuni2 22114* The base set for the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}       ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) = 𝐵)
 
Theoremptbasin 22115* The basis for a product topology is closed under intersections. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}       (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝑌) ∈ 𝐵)
 
Theoremptbasin2 22116* The basis for a product topology is closed under intersections. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}       ((𝐴𝑉𝐹:𝐴⟶Top) → (fi‘𝐵) = 𝐵)
 
Theoremptbas 22117* The basis for a product topology is a basis. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}       ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵 ∈ TopBases)
 
Theoremptpjpre2 22118* The basis for a product topology is a basis. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}    &   𝑋 = X𝑛𝐴 (𝐹𝑛)       (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝐼𝐴𝑈 ∈ (𝐹𝐼))) → ((𝑤𝑋 ↦ (𝑤𝐼)) “ 𝑈) ∈ 𝐵)
 
Theoremptbasfi 22119* The basis for the product topology can also be written as the set of finite intersections of "cylinder sets", the preimages of projections into one factor from open sets in the factor. (We have to add 𝑋 itself to the list because if 𝐴 is empty we get (fi‘∅) = ∅ while 𝐵 = {∅}.) (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}    &   𝑋 = X𝑛𝐴 (𝐹𝑛)       ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵 = (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
 
Theorempttop 22120 The product topology is a topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
((𝐴𝑉𝐹:𝐴⟶Top) → (∏t𝐹) ∈ Top)
 
Theoremptopn 22121* A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶Top)    &   (𝜑𝑊 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝑆 ∈ (𝐹𝑘))    &   ((𝜑𝑘 ∈ (𝐴𝑊)) → 𝑆 = (𝐹𝑘))       (𝜑X𝑘𝐴 𝑆 ∈ (∏t𝐹))
 
Theoremptopn2 22122* A sub-basic open set in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶Top)    &   (𝜑𝑂 ∈ (𝐹𝑌))       (𝜑X𝑘𝐴 if(𝑘 = 𝑌, 𝑂, (𝐹𝑘)) ∈ (∏t𝐹))
 
Theoremxkotf 22123* Functionality of function 𝑇. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑋 = 𝑅    &   𝐾 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}    &   𝑇 = (𝑘𝐾, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})       𝑇:(𝐾 × 𝑆)⟶𝒫 (𝑅 Cn 𝑆)
 
Theoremxkobval 22124* Alternative expression for the subbase of the compact-open topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
𝑋 = 𝑅    &   𝐾 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}    &   𝑇 = (𝑘𝐾, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})       ran 𝑇 = {𝑠 ∣ ∃𝑘 ∈ 𝒫 𝑋𝑣𝑆 ((𝑅t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})}
 
Theoremxkoval 22125* Value of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑋 = 𝑅    &   𝐾 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}    &   𝑇 = (𝑘𝐾, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})       ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) = (topGen‘(fi‘ran 𝑇)))
 
Theoremxkotop 22126 The compact-open topology is a topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) ∈ Top)
 
Theoremxkoopn 22127* A basic open set of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑋 = 𝑅    &   (𝜑𝑅 ∈ Top)    &   (𝜑𝑆 ∈ Top)    &   (𝜑𝐴𝑋)    &   (𝜑 → (𝑅t 𝐴) ∈ Comp)    &   (𝜑𝑈𝑆)       (𝜑 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} ∈ (𝑆ko 𝑅))
 
Theoremtxtopi 22128 The product of two topologies is a topology. (Contributed by Jeff Madsen, 15-Jun-2010.)
𝑅 ∈ Top    &   𝑆 ∈ Top       (𝑅 ×t 𝑆) ∈ Top
 
Theoremtxtopon 22129 The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 22-Aug-2015.) (Revised by Mario Carneiro, 2-Sep-2015.)
((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
 
Theoremtxuni 22130 The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝑋 = 𝑅    &   𝑌 = 𝑆       ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = (𝑅 ×t 𝑆))
 
Theoremtxunii 22131 The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 15-Jun-2010.)
𝑅 ∈ Top    &   𝑆 ∈ Top    &   𝑋 = 𝑅    &   𝑌 = 𝑆       (𝑋 × 𝑌) = (𝑅 ×t 𝑆)
 
Theoremptuni 22132* The base set for the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐽 = (∏t𝐹)       ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑥𝐴 (𝐹𝑥) = 𝐽)
 
Theoremptunimpt 22133* Base set of a product topology given by substitution. (Contributed by Stefan O'Rear, 22-Feb-2015.)
𝐽 = (∏t‘(𝑥𝐴𝐾))       ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐾 ∈ Top) → X𝑥𝐴 𝐾 = 𝐽)
 
Theorempttopon 22134* The base set for the product topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
𝐽 = (∏t‘(𝑥𝐴𝐾))       ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐾 ∈ (TopOn‘𝐵)) → 𝐽 ∈ (TopOn‘X𝑥𝐴 𝐵))
 
Theorempttoponconst 22135 The base set for a product topology when all factors are the same. (Contributed by Mario Carneiro, 22-Aug-2015.)
𝐽 = (∏t‘(𝐴 × {𝑅}))       ((𝐴𝑉𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘(𝑋m 𝐴)))
 
Theoremptuniconst 22136 The base set for a product topology when all factors are the same. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐽 = (∏t‘(𝐴 × {𝑅}))    &   𝑋 = 𝑅       ((𝐴𝑉𝑅 ∈ Top) → (𝑋m 𝐴) = 𝐽)
 
Theoremxkouni 22137 The base set of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝐽 = (𝑆ko 𝑅)       ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = 𝐽)
 
Theoremxkotopon 22138 The base set of the compact-open topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
𝐽 = (𝑆ko 𝑅)       ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝐽 ∈ (TopOn‘(𝑅 Cn 𝑆)))
 
Theoremptval2 22139* The value of the product topology function. (Contributed by Mario Carneiro, 7-Feb-2015.)
𝐽 = (∏t𝐹)    &   𝑋 = 𝐽    &   𝐺 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))       ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐽 = (topGen‘(fi‘({𝑋} ∪ ran 𝐺))))
 
Theoremtxopn 22140 The product of two open sets is open in the product topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
(((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → (𝐴 × 𝐵) ∈ (𝑅 ×t 𝑆))
 
Theoremtxcld 22141 The product of two closed sets is closed in the product topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ∈ (Clsd‘(𝑅 ×t 𝑆)))
 
Theoremtxcls 22142 Closure of a rectangle in the product topology. (Contributed by Mario Carneiro, 17-Sep-2015.)
(((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → ((cls‘(𝑅 ×t 𝑆))‘(𝐴 × 𝐵)) = (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)))
 
Theoremtxss12 22143 Subset property of the topological product. (Contributed by Mario Carneiro, 2-Sep-2015.)
(((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (𝐴 ×t 𝐶) ⊆ (𝐵 ×t 𝐷))
 
Theoremtxbasval 22144 It is sufficient to consider products of the bases for the topologies in the topological product. (Contributed by Mario Carneiro, 25-Aug-2014.)
((𝑅𝑉𝑆𝑊) → ((topGen‘𝑅) ×t (topGen‘𝑆)) = (𝑅 ×t 𝑆))
 
Theoremneitx 22145 The Cartesian product of two neighborhoods is a neighborhood in the product topology. (Contributed by Thierry Arnoux, 13-Jan-2018.)
𝑋 = 𝐽    &   𝑌 = 𝐾       (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐴 × 𝐵) ∈ ((nei‘(𝐽 ×t 𝐾))‘(𝐶 × 𝐷)))
 
Theoremtxcnpi 22146* Continuity of a two-argument function at a point. (Contributed by Mario Carneiro, 20-Sep-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩))    &   (𝜑𝑈𝐿)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑌)    &   (𝜑 → (𝐴𝐹𝐵) ∈ 𝑈)       (𝜑 → ∃𝑢𝐽𝑣𝐾 (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈)))
 
Theoremtx1cn 22147 Continuity of the first projection map of a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
 
Theoremtx2cn 22148 Continuity of the second projection map of a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
 
Theoremptpjcn 22149* Continuity of a projection map into a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Feb-2015.)
𝑌 = 𝐽    &   𝐽 = (∏t𝐹)       ((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) → (𝑥𝑌 ↦ (𝑥𝐼)) ∈ (𝐽 Cn (𝐹𝐼)))
 
Theoremptpjopn 22150* The projection map is an open map. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝑌 = 𝐽    &   𝐽 = (∏t𝐹)       (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ((𝑥𝑌 ↦ (𝑥𝐼)) “ 𝑈) ∈ (𝐹𝐼))
 
Theoremptcld 22151* A closed box in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶Top)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (Clsd‘(𝐹𝑘)))       (𝜑X𝑘𝐴 𝐶 ∈ (Clsd‘(∏t𝐹)))
 
Theoremptcldmpt 22152* A closed box in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐽 ∈ Top)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (Clsd‘𝐽))       (𝜑X𝑘𝐴 𝐶 ∈ (Clsd‘(∏t‘(𝑘𝐴𝐽))))
 
Theoremptclsg 22153* The closure of a box in the product topology is the box formed from the closures of the factors. The proof uses the axiom of choice; the last hypothesis is the choice assumption. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝐽 = (∏t‘(𝑘𝐴𝑅))    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝑅 ∈ (TopOn‘𝑋))    &   ((𝜑𝑘𝐴) → 𝑆𝑋)    &   (𝜑 𝑘𝐴 𝑆AC 𝐴)       (𝜑 → ((cls‘𝐽)‘X𝑘𝐴 𝑆) = X𝑘𝐴 ((cls‘𝑅)‘𝑆))
 
Theoremptcls 22154* The closure of a box in the product topology is the box formed from the closures of the factors. This theorem is an AC equivalent. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐽 = (∏t‘(𝑘𝐴𝑅))    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝑅 ∈ (TopOn‘𝑋))    &   ((𝜑𝑘𝐴) → 𝑆𝑋)       (𝜑 → ((cls‘𝐽)‘X𝑘𝐴 𝑆) = X𝑘𝐴 ((cls‘𝑅)‘𝑆))
 
Theoremdfac14lem 22155* Lemma for dfac14 22156. 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 22154 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 22156* Theorem ptcls 22154 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 22157* 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 (𝑆ko 𝑅)))
 
Theoremtxcnp 22158* 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 22159* Lemma for ptcnp 22160. (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 22160* 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 22161* 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 22162* 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 22163* 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 22164 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 22165* 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 22166 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 22167 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 22168 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 22169 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 22170 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 22171 Lemma for txindis 22172. (Contributed by Mario Carneiro, 14-Aug-2015.)
(( I ‘𝐴) × ( I ‘𝐵)) = ( I ‘(𝐴 × 𝐵))
 
Theoremtxindis 22172 The topological product of indiscrete spaces is indiscrete. (Contributed by Mario Carneiro, 14-Aug-2015.)
({∅, 𝐴} ×t {∅, 𝐵}) = {∅, (𝐴 × 𝐵)}
 
Theoremtxdis1cn 22173* 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 22174* 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 22175* 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 22176 The product of a collection of Hausdorff spaces is Hausdorff. (Contributed by Mario Carneiro, 2-Sep-2015.)
((𝐴𝑉𝐹:𝐴⟶Haus) → (∏t𝐹) ∈ Haus)
 
Theoremptrescn 22177* 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 22178* 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 22179* Lemma for txcmp 22181. (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 22180* Lemma for txcmp 22181. (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 22181 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 22182 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 22183 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 22184 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 22185 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 22186* 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 22187* 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 22188 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 22189 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 22190 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 22191 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 22192 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 22193 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 22194* 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 22222.) (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 22195* If 𝐹 is a continuous function, then 𝑔𝑔𝐹 is a continuous function on function spaces. (The reason we prove this and xkoco2cn 22196 independently of the more general xkococn 22198 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 22196* 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 22197* 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 22198* 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 22199* 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 22200* 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))
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