P. 236 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 23501-23600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	txuni2 23501*	The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ 𝐵 = ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))    &   ⊢ 𝑋 = ∪ 𝑅    &   ⊢ 𝑌 = ∪ 𝑆    ⇒   ⊢ (𝑋 × 𝑌) = ∪ 𝐵
Theorem	txbasex 23502*	The basis for the product topology is a set. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐵 = ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝐵 ∈ V)
Theorem	txbas 23503*	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 (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))    ⇒   ⊢ ((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) → 𝐵 ∈ TopBases)
Theorem	eltx 23504*	A set in a product is open iff each point is surrounded by an open rectangle. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ ((𝐽 ∈ 𝑉 ∧ 𝐾 ∈ 𝑊) → (𝑆 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑝 ∈ 𝑆 ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆)))
Theorem	txtop 23505	The product of two topologies is a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
Theorem	ptval 23506*	The value of the product topology function. (Contributed by Mario Carneiro, 3-Feb-2015.)
⊢ 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) → (∏t‘𝐹) = (topGen‘𝐵))
Theorem	ptpjpre1 23507*	The preimage of a projection function can be expressed as an indexed cartesian product. (Contributed by Mario Carneiro, 6-Feb-2015.)
⊢ 𝑋 = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)    ⇒   ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼)) “ 𝑈) = X𝑘 ∈ 𝐴 if(𝑘 = 𝐼, 𝑈, ∪ (𝐹‘𝑘)))
Theorem	elpt 23508*	Elementhood in the bases of a product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
⊢ 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}    ⇒   ⊢ (𝑆 ∈ 𝐵 ↔ ∃ℎ((ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑆 = X𝑦 ∈ 𝐴 (ℎ‘𝑦)))
Theorem	elptr 23509*	A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
⊢ 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) ∈ (𝐹‘𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑊)(𝐺‘𝑦) = ∪ (𝐹‘𝑦))) → X𝑦 ∈ 𝐴 (𝐺‘𝑦) ∈ 𝐵)
Theorem	elptr2 23510*	A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
⊢ 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑊 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑆 ∈ (𝐹‘𝑘))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝑆 = ∪ (𝐹‘𝑘))    ⇒   ⊢ (𝜑 → X𝑘 ∈ 𝐴 𝑆 ∈ 𝐵)
Theorem	ptbasid 23511*	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 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∈ 𝐵)
Theorem	ptuni2 23512*	The base set for the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
⊢ 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐵)
Theorem	ptbasin 23513*	The basis for a product topology is closed under intersections. (Contributed by Mario Carneiro, 3-Feb-2015.)
⊢ 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}    ⇒   ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ∩ 𝑌) ∈ 𝐵)
Theorem	ptbasin2 23514*	The basis for a product topology is closed under intersections. (Contributed by Mario Carneiro, 19-Mar-2015.)
⊢ 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (fi‘𝐵) = 𝐵)
Theorem	ptbas 23515*	The basis for a product topology is a basis. (Contributed by Mario Carneiro, 3-Feb-2015.)
⊢ 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐵 ∈ TopBases)
Theorem	ptpjpre2 23516*	The basis for a product topology is a basis. (Contributed by Mario Carneiro, 3-Feb-2015.)
⊢ 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}    &   ⊢ 𝑋 = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)    ⇒   ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝐼 ∈ 𝐴 ∧ 𝑈 ∈ (𝐹‘𝐼))) → (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝐼)) “ 𝑈) ∈ 𝐵)
Theorem	ptbasfi 23517*	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 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}    &   ⊢ 𝑋 = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐵 = (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
Theorem	pttop 23518	The product topology is a topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (∏t‘𝐹) ∈ Top)
Theorem	ptopn 23519*	A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶Top)    &   ⊢ (𝜑 → 𝑊 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑆 ∈ (𝐹‘𝑘))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝑆 = ∪ (𝐹‘𝑘))    ⇒   ⊢ (𝜑 → X𝑘 ∈ 𝐴 𝑆 ∈ (∏t‘𝐹))
Theorem	ptopn2 23520*	A sub-basic open set in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶Top)    &   ⊢ (𝜑 → 𝑂 ∈ (𝐹‘𝑌))    ⇒   ⊢ (𝜑 → X𝑘 ∈ 𝐴 if(𝑘 = 𝑌, 𝑂, ∪ (𝐹‘𝑘)) ∈ (∏t‘𝐹))
Theorem	xkotf 23521*	Functionality of function 𝑇. (Contributed by Mario Carneiro, 19-Mar-2015.)
⊢ 𝑋 = ∪ 𝑅    &   ⊢ 𝐾 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}    &   ⊢ 𝑇 = (𝑘 ∈ 𝐾, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})    ⇒   ⊢ 𝑇:(𝐾 × 𝑆)⟶𝒫 (𝑅 Cn 𝑆)
Theorem	xkobval 23522*	Alternative expression for the subbase of the compact-open topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
⊢ 𝑋 = ∪ 𝑅    &   ⊢ 𝐾 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}    &   ⊢ 𝑇 = (𝑘 ∈ 𝐾, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})    ⇒   ⊢ ran 𝑇 = {𝑠 ∣ ∃𝑘 ∈ 𝒫 𝑋∃𝑣 ∈ 𝑆 ((𝑅 ↾t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})}
Theorem	xkoval 23523*	Value of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
⊢ 𝑋 = ∪ 𝑅    &   ⊢ 𝐾 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}    &   ⊢ 𝑇 = (𝑘 ∈ 𝐾, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})    ⇒   ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) = (topGen‘(fi‘ran 𝑇)))
Theorem	xkotop 23524	The compact-open topology is a topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) ∈ Top)
Theorem	xkoopn 23525*	A basic open set of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
⊢ 𝑋 = ∪ 𝑅    &   ⊢ (𝜑 → 𝑅 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ∈ Top)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑋)    &   ⊢ (𝜑 → (𝑅 ↾t 𝐴) ∈ Comp)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    ⇒   ⊢ (𝜑 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝐴) ⊆ 𝑈} ∈ (𝑆 ↑ko 𝑅))
Theorem	txtopi 23526	The product of two topologies is a topology. (Contributed by Jeff Madsen, 15-Jun-2010.)
⊢ 𝑅 ∈ Top    &   ⊢ 𝑆 ∈ Top    ⇒   ⊢ (𝑅 ×t 𝑆) ∈ Top
Theorem	txtopon 23527	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)))
Theorem	txuni 23528	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))
Theorem	txunii 23529	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)
Theorem	ptuni 23530*	The base set for the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
⊢ 𝐽 = (∏t‘𝐹)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑥 ∈ 𝐴 ∪ (𝐹‘𝑥) = ∪ 𝐽)
Theorem	ptunimpt 23531*	Base set of a product topology given by substitution. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ 𝐽 = (∏t‘(𝑥 ∈ 𝐴 ↦ 𝐾))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ Top) → X𝑥 ∈ 𝐴 ∪ 𝐾 = ∪ 𝐽)
Theorem	pttopon 23532*	The base set for the product topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
⊢ 𝐽 = (∏t‘(𝑥 ∈ 𝐴 ↦ 𝐾))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐾 ∈ (TopOn‘𝐵)) → 𝐽 ∈ (TopOn‘X𝑥 ∈ 𝐴 𝐵))
Theorem	pttoponconst 23533	The base set for a product topology when all factors are the same. (Contributed by Mario Carneiro, 22-Aug-2015.)
⊢ 𝐽 = (∏t‘(𝐴 × {𝑅}))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘(𝑋 ↑m 𝐴)))
Theorem	ptuniconst 23534	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    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ Top) → (𝑋 ↑m 𝐴) = ∪ 𝐽)
Theorem	xkouni 23535	The base set of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
⊢ 𝐽 = (𝑆 ↑ko 𝑅)    ⇒   ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = ∪ 𝐽)
Theorem	xkotopon 23536	The base set of the compact-open topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
⊢ 𝐽 = (𝑆 ↑ko 𝑅)    ⇒   ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝐽 ∈ (TopOn‘(𝑅 Cn 𝑆)))
Theorem	ptval2 23537*	The value of the product topology function. (Contributed by Mario Carneiro, 7-Feb-2015.)
⊢ 𝐽 = (∏t‘𝐹)    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐺 = (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐽 = (topGen‘(fi‘({𝑋} ∪ ran 𝐺))))
Theorem	txopn 23538	The product of two open sets is open in the product topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴 × 𝐵) ∈ (𝑅 ×t 𝑆))
Theorem	txcld 23539	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)))
Theorem	txcls 23540	Closure of a rectangle in the product topology. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌)) → ((cls‘(𝑅 ×t 𝑆))‘(𝐴 × 𝐵)) = (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)))
Theorem	txss12 23541	Subset property of the topological product. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ (((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)) → (𝐴 ×t 𝐶) ⊆ (𝐵 ×t 𝐷))
Theorem	txbasval 23542	It is sufficient to consider products of the bases for the topologies in the topological product. (Contributed by Mario Carneiro, 25-Aug-2014.)
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((topGen‘𝑅) ×t (topGen‘𝑆)) = (𝑅 ×t 𝑆))
Theorem	neitx 23543	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)))
Theorem	txcnpi 23544*	Continuity of a two-argument function at a point. (Contributed by Mario Carneiro, 20-Sep-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩))    &   ⊢ (𝜑 → 𝑈 ∈ 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑌)    &   ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝑈)    ⇒   ⊢ (𝜑 → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐹 “ 𝑈)))
Theorem	tx1cn 23545	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))
Theorem	tx2cn 23546	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))
Theorem	ptpjcn 23547*	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 = (∏t‘𝐹)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝑥 ∈ 𝑌 ↦ (𝑥‘𝐼)) ∈ (𝐽 Cn (𝐹‘𝐼)))
Theorem	ptpjopn 23548*	The projection map is an open map. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝑌 = ∪ 𝐽    &   ⊢ 𝐽 = (∏t‘𝐹)    ⇒   ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) → ((𝑥 ∈ 𝑌 ↦ (𝑥‘𝐼)) “ 𝑈) ∈ (𝐹‘𝐼))
Theorem	ptcld 23549*	A closed box in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶Top)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (Clsd‘(𝐹‘𝑘)))    ⇒   ⊢ (𝜑 → X𝑘 ∈ 𝐴 𝐶 ∈ (Clsd‘(∏t‘𝐹)))
Theorem	ptcldmpt 23550*	A closed box in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐽 ∈ Top)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (Clsd‘𝐽))    ⇒   ⊢ (𝜑 → X𝑘 ∈ 𝐴 𝐶 ∈ (Clsd‘(∏t‘(𝑘 ∈ 𝐴 ↦ 𝐽))))
Theorem	ptclsg 23551*	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‘(𝑘 ∈ 𝐴 ↦ 𝑅))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑅 ∈ (TopOn‘𝑋))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑆 ⊆ 𝑋)    &   ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝑆 ∈ AC 𝐴)    ⇒   ⊢ (𝜑 → ((cls‘𝐽)‘X𝑘 ∈ 𝐴 𝑆) = X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆))
Theorem	ptcls 23552*	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‘(𝑘 ∈ 𝐴 ↦ 𝑅))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑅 ∈ (TopOn‘𝑋))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑆 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → ((cls‘𝐽)‘X𝑘 ∈ 𝐴 𝑆) = X𝑘 ∈ 𝐴 ((cls‘𝑅)‘𝑆))
Theorem	dfac14lem 23553*	Lemma for dfac14 23554. By equipping 𝑆 ∪ {𝑃} for some 𝑃 ∉ 𝑆 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 23552 to extract an element of the closure of X𝑘 ∈ 𝐼𝑆. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ≠ ∅)    &   ⊢ 𝑃 = 𝒫 ∪ 𝑆    &   ⊢ 𝑅 = {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃 ∈ 𝑦 → 𝑦 = (𝑆 ∪ {𝑃}))}    &   ⊢ 𝐽 = (∏t‘(𝑥 ∈ 𝐼 ↦ 𝑅))    &   ⊢ (𝜑 → ((cls‘𝐽)‘X𝑥 ∈ 𝐼 𝑆) = X𝑥 ∈ 𝐼 ((cls‘𝑅)‘𝑆))    ⇒   ⊢ (𝜑 → X𝑥 ∈ 𝐼 𝑆 ≠ ∅)
Theorem	dfac14 23554*	Theorem ptcls 23552 is an equivalent of the axiom of choice. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (CHOICE ↔ ∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠 ∈ X 𝑘 ∈ dom 𝑓𝒫 ∪ (𝑓‘𝑘)((cls‘(∏t‘𝑓))‘X𝑘 ∈ dom 𝑓(𝑠‘𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓‘𝑘))‘(𝑠‘𝑘))))
Theorem	xkoccn 23555*	The "constant function" function which maps 𝑥 ∈ 𝑌 to the constant function 𝑧 ∈ 𝑋 ↦ 𝑥 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 𝑥 ∈ 𝑌 ↦ (𝑧 ∈ 𝑋 ↦ 𝑧), 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)) → (𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥})) ∈ (𝑆 Cn (𝑆 ↑ko 𝑅)))
Theorem	txcnp 23556*	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.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))    &   ⊢ (𝜑 → 𝐷 ∈ 𝑋)    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 CnP (𝐾 ×t 𝐿))‘𝐷))
Theorem	ptcnplem 23557*	Lemma for ptcnp 23558. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ 𝐾 = (∏t‘𝐹)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐼⟶Top)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝐷))    &   ⊢ Ⅎ𝑘𝜓    &   ⊢ ((𝜑 ∧ 𝜓) → 𝐺 Fn 𝐼)    &   ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑘 ∈ 𝐼) → (𝐺‘𝑘) ∈ (𝐹‘𝑘))    &   ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ∈ Fin)    &   ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑘 ∈ (𝐼 ∖ 𝑊)) → (𝐺‘𝑘) = ∪ (𝐹‘𝑘))    &   ⊢ ((𝜑 ∧ 𝜓) → ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑘 ∈ 𝐼 (𝐺‘𝑘))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝐺‘𝑘)))
Theorem	ptcnp 23558*	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 = (∏t‘𝐹)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐼⟶Top)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝐷))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝐷))
Theorem	upxp 23559*	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))    ⇒   ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) → ∃!ℎ(ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)))
Theorem	txcnmpt 23560*	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 = (𝑥 ∈ 𝑊 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)    ⇒   ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐻 ∈ (𝑈 Cn (𝑅 ×t 𝑆)))
Theorem	uptx 23561*	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 = (1st ↾ 𝑍)    &   ⊢ 𝑄 = (2nd ↾ 𝑍)    ⇒   ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃!ℎ ∈ (𝑈 Cn 𝑇)(𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ)))
Theorem	txcn 23562	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 = (1st ↾ 𝑍)    &   ⊢ 𝑄 = (2nd ↾ 𝑍)    ⇒   ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) → (𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ↔ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆))))
Theorem	ptcn 23563*	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 = (∏t‘𝐹)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐼⟶Top)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn (𝐹‘𝑘)))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ (𝐽 Cn 𝐾))
Theorem	prdstopn 23564	Topology of a structure product. (Contributed by Mario Carneiro, 27-Aug-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 Fn 𝐼)    &   ⊢ 𝑂 = (TopOpen‘𝑌)    ⇒   ⊢ (𝜑 → 𝑂 = (∏t‘(TopOpen ∘ 𝑅)))
Theorem	prdstps 23565	A structure product of topological spaces is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅:𝐼⟶TopSp)    ⇒   ⊢ (𝜑 → 𝑌 ∈ TopSp)
Theorem	pwstps 23566	A structure power of a topological space is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    ⇒   ⊢ ((𝑅 ∈ TopSp ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ TopSp)
Theorem	txrest 23567	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 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = ((𝑅 ↾t 𝐴) ×t (𝑆 ↾t 𝐵)))
Theorem	txdis 23568	The topological product of discrete spaces is discrete. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐴 ×t 𝒫 𝐵) = 𝒫 (𝐴 × 𝐵))
Theorem	txindislem 23569	Lemma for txindis 23570. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (( I ‘𝐴) × ( I ‘𝐵)) = ( I ‘(𝐴 × 𝐵))
Theorem	txindis 23570	The topological product of indiscrete spaces is indiscrete. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ ({∅, 𝐴} ×t {∅, 𝐵}) = {∅, (𝐴 × 𝐵)}
Theorem	txdis1cn 23571*	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.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐾 ∈ Top)    &   ⊢ (𝜑 → 𝐹 Fn (𝑋 × 𝑌))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → 𝐹 ∈ ((𝒫 𝑋 ×t 𝐽) Cn 𝐾))
Theorem	txlly 23572*	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 ∧ 𝑘 ∈ 𝐴) → (𝑗 ×t 𝑘) ∈ 𝐴)    ⇒   ⊢ ((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) → (𝑅 ×t 𝑆) ∈ Locally 𝐴)
Theorem	txnlly 23573*	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 ∧ 𝑘 ∈ 𝐴) → (𝑗 ×t 𝑘) ∈ 𝐴)    ⇒   ⊢ ((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) → (𝑅 ×t 𝑆) ∈ 𝑛-Locally 𝐴)
Theorem	pthaus 23574	The product of a collection of Hausdorff spaces is Hausdorff. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Haus) → (∏t‘𝐹) ∈ Haus)
Theorem	ptrescn 23575*	Restriction is a continuous function on product topologies. (Contributed by Mario Carneiro, 7-Feb-2015.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐽 = (∏t‘𝐹)    &   ⊢ 𝐾 = (∏t‘(𝐹 ↾ 𝐵))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ 𝑋 ↦ (𝑥 ↾ 𝐵)) ∈ (𝐽 Cn 𝐾))
Theorem	txtube 23576*	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    &   ⊢ (𝜑 → 𝑅 ∈ Comp)    &   ⊢ (𝜑 → 𝑆 ∈ Top)    &   ⊢ (𝜑 → 𝑈 ∈ (𝑅 ×t 𝑆))    &   ⊢ (𝜑 → (𝑋 × {𝐴}) ⊆ 𝑈)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑌)    ⇒   ⊢ (𝜑 → ∃𝑢 ∈ 𝑆 (𝐴 ∈ 𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈))
Theorem	txcmplem1 23577*	Lemma for txcmp 23579. (Contributed by Mario Carneiro, 14-Sep-2014.)
⊢ 𝑋 = ∪ 𝑅    &   ⊢ 𝑌 = ∪ 𝑆    &   ⊢ (𝜑 → 𝑅 ∈ Comp)    &   ⊢ (𝜑 → 𝑆 ∈ Comp)    &   ⊢ (𝜑 → 𝑊 ⊆ (𝑅 ×t 𝑆))    &   ⊢ (𝜑 → (𝑋 × 𝑌) = ∪ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑌)    ⇒   ⊢ (𝜑 → ∃𝑢 ∈ 𝑆 (𝐴 ∈ 𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣))
Theorem	txcmplem2 23578*	Lemma for txcmp 23579. (Contributed by Mario Carneiro, 14-Sep-2014.)
⊢ 𝑋 = ∪ 𝑅    &   ⊢ 𝑌 = ∪ 𝑆    &   ⊢ (𝜑 → 𝑅 ∈ Comp)    &   ⊢ (𝜑 → 𝑆 ∈ Comp)    &   ⊢ (𝜑 → 𝑊 ⊆ (𝑅 ×t 𝑆))    &   ⊢ (𝜑 → (𝑋 × 𝑌) = ∪ 𝑊)    ⇒   ⊢ (𝜑 → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = ∪ 𝑣)
Theorem	txcmp 23579	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)
Theorem	txcmpb 23580	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)))
Theorem	hausdiag 23581	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))))
Theorem	hauseqlcld 23582	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.)
⊢ (𝜑 → 𝐾 ∈ Haus)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → dom (𝐹 ∩ 𝐺) ∈ (Clsd‘𝐽))
Theorem	txhaus 23583	The topological product of two Hausdorff spaces is Hausdorff. (Contributed by Mario Carneiro, 23-Mar-2015.)
⊢ ((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) → (𝑅 ×t 𝑆) ∈ Haus)
Theorem	txlm 23584*	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 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐹:𝑍⟶𝑋)    &   ⊢ (𝜑 → 𝐺:𝑍⟶𝑌)    &   ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩)    ⇒   ⊢ (𝜑 → ((𝐹(⇝𝑡‘𝐽)𝑅 ∧ 𝐺(⇝𝑡‘𝐾)𝑆) ↔ 𝐻(⇝𝑡‘(𝐽 ×t 𝐾))⟨𝑅, 𝑆⟩))
Theorem	lmcn2 23585*	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 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐹:𝑍⟶𝑋)    &   ⊢ (𝜑 → 𝐺:𝑍⟶𝑌)    &   ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑅)    &   ⊢ (𝜑 → 𝐺(⇝𝑡‘𝐾)𝑆)    &   ⊢ (𝜑 → 𝑂 ∈ ((𝐽 ×t 𝐾) Cn 𝑁))    &   ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)𝑂(𝐺‘𝑛)))    ⇒   ⊢ (𝜑 → 𝐻(⇝𝑡‘𝑁)(𝑅𝑂𝑆))
Theorem	tx1stc 23586	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)
Theorem	tx2ndc 23587	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)
Theorem	txkgen 23588	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)
Theorem	xkohaus 23589	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)
Theorem	xkoptsub 23590	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) → (𝐽 ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆 ↑ko 𝑅))
Theorem	xkopt 23591	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) → (𝑅 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝑅})))
Theorem	xkopjcn 23592*	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 23620.) (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ 𝑋 = ∪ 𝑅    ⇒   ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓‘𝐴)) ∈ ((𝑆 ↑ko 𝑅) Cn 𝑆))
Theorem	xkoco1cn 23593*	If 𝐹 is a continuous function, then 𝑔 ↦ 𝑔 ∘ 𝐹 is a continuous function on function spaces. (The reason we prove this and xkoco2cn 23594 independently of the more general xkococn 23596 is because that requires some inconvenient extra assumptions on 𝑆.) (Contributed by Mario Carneiro, 20-Mar-2015.)
⊢ (𝜑 → 𝑇 ∈ Top)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑅 Cn 𝑆))    ⇒   ⊢ (𝜑 → (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔 ∘ 𝐹)) ∈ ((𝑇 ↑ko 𝑆) Cn (𝑇 ↑ko 𝑅)))
Theorem	xkoco2cn 23594*	If 𝐹 is a continuous function, then 𝑔 ↦ 𝐹 ∘ 𝑔 is a continuous function on function spaces. (Contributed by Mario Carneiro, 23-Mar-2015.)
⊢ (𝜑 → 𝑅 ∈ Top)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑆 Cn 𝑇))    ⇒   ⊢ (𝜑 → (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹 ∘ 𝑔)) ∈ ((𝑆 ↑ko 𝑅) Cn (𝑇 ↑ko 𝑅)))
Theorem	xkococnlem 23595*	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))    &   ⊢ (𝜑 → 𝑆 ∈ 𝑛-Locally Comp)    &   ⊢ (𝜑 → 𝐾 ⊆ ∪ 𝑅)    &   ⊢ (𝜑 → (𝑅 ↾t 𝐾) ∈ Comp)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑇)    &   ⊢ (𝜑 → 𝐴 ∈ (𝑆 Cn 𝑇))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑅 Cn 𝑆))    &   ⊢ (𝜑 → ((𝐴 ∘ 𝐵) “ 𝐾) ⊆ 𝑉)    ⇒   ⊢ (𝜑 → ∃𝑧 ∈ ((𝑇 ↑ko 𝑆) ×t (𝑆 ↑ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧 ∧ 𝑧 ⊆ (◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉})))
Theorem	xkococn 23596*	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))    ⇒   ⊢ ((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
Theorem	cnmptid 23597*	The identity function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
Theorem	cnmptc 23598*	A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝑃 ∈ 𝑌)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑃) ∈ (𝐽 Cn 𝐾))
Theorem	cnmpt11 23599*	The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ 𝐵) ∈ (𝐾 Cn 𝐿))    &   ⊢ (𝑦 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) ∈ (𝐽 Cn 𝐿))
Theorem	cnmpt11f 23600*	The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐾 Cn 𝐿))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘𝐴)) ∈ (𝐽 Cn 𝐿))