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Type | Label | Description |
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Statement | ||
Theorem | pttopon 22201* | 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 22202 | 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 22203 | 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 22204 | 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 22205 | 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 22206* | 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 22207 | 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 22208 | 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 22209 | 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 22210 | 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 22211 | 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 22212 | 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 22213* | 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 22214 | 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 22215 | 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 22216* | 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 22217* | 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 22218* | 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 22219* | 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 22220* | 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 22221* | 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 22222* | Lemma for dfac14 22223. 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 22221 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 22223* | Theorem ptcls 22221 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 22224* | 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 22225* | 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 22226* | Lemma for ptcnp 22227. (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 22227* | 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 22228* | 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 22229* | 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 22230* | 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 22231 | 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 22232* | 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 22233 | 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 22234 | 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 22235 | 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 22236 | 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 22237 | 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 22238 | Lemma for txindis 22239. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ (( I ‘𝐴) × ( I ‘𝐵)) = ( I ‘(𝐴 × 𝐵)) | ||
Theorem | txindis 22239 | The topological product of indiscrete spaces is indiscrete. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ ({∅, 𝐴} ×t {∅, 𝐵}) = {∅, (𝐴 × 𝐵)} | ||
Theorem | txdis1cn 22240* | 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 22241* | 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 22242* | 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 22243 | 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 22244* | 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 22245* | 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 22246* | Lemma for txcmp 22248. (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 22247* | Lemma for txcmp 22248. (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 22248 | 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 22249 | 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 22250 | 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 22251 | 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 22252 | 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 22253* | 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 22254* | 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 22255 | 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 22256 | 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 22257 | 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 22258 | 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 22259 | 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 22260 | 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 22261* | 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 22289.) (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 22262* | If 𝐹 is a continuous function, then 𝑔 ↦ 𝑔 ∘ 𝐹 is a continuous function on function spaces. (The reason we prove this and xkoco2cn 22263 independently of the more general xkococn 22265 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 22263* | 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 22264* | 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 22265* | 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))) | ||
Theorem | cnmptid 22266* | 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 22267* | 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 22268* | 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 22269* | 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)) | ||
Theorem | cnmpt1t 22270* | 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 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) ∈ (𝐽 Cn (𝐾 ×t 𝐿))) | ||
Theorem | cnmpt12f 22271* | 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 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿)) & ⊢ (𝜑 → 𝐹 ∈ ((𝐾 ×t 𝐿) Cn 𝑀)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝐽 Cn 𝑀)) | ||
Theorem | cnmpt12 22272* | The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 12-Jun-2014.) (Revised by Mario Carneiro, 22-Aug-2015.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) & ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀)) & ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐷) ∈ (𝐽 Cn 𝑀)) | ||
Theorem | cnmpt1st 22273* | The projection onto the first coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽)) | ||
Theorem | cnmpt2nd 22274* | The projection onto the second coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾)) | ||
Theorem | cnmpt2c 22275* | 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 → 𝐿 ∈ (TopOn‘𝑍)) & ⊢ (𝜑 → 𝑃 ∈ 𝑍) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) | ||
Theorem | cnmpt21 22276* | 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 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) & ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) & ⊢ (𝜑 → (𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀)) & ⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) | ||
Theorem | cnmpt21f 22277* | 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 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) & ⊢ (𝜑 → 𝐹 ∈ (𝐿 Cn 𝑀)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐹‘𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) | ||
Theorem | cnmpt2t 22278* | 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 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝐴, 𝐵〉) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀))) | ||
Theorem | cnmpt22 22279* | 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 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) & ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) & ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑊)) & ⊢ (𝜑 → (𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶) ∈ ((𝐿 ×t 𝑀) Cn 𝑁)) & ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐷) ∈ ((𝐽 ×t 𝐾) Cn 𝑁)) | ||
Theorem | cnmpt22f 22280* | 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 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) & ⊢ (𝜑 → 𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴𝐹𝐵)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁)) | ||
Theorem | cnmpt1res 22281* | The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 5-Jun-2014.) |
⊢ 𝐾 = (𝐽 ↾t 𝑌) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝑌 ⊆ 𝑋) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐿)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) | ||
Theorem | cnmpt2res 22282* | The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.) |
⊢ 𝐾 = (𝐽 ↾t 𝑌) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝑌 ⊆ 𝑋) & ⊢ 𝑁 = (𝑀 ↾t 𝑊) & ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑍)) & ⊢ (𝜑 → 𝑊 ⊆ 𝑍) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ∈ ((𝐽 ×t 𝑀) Cn 𝐿)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴) ∈ ((𝐾 ×t 𝑁) Cn 𝐿)) | ||
Theorem | cnmptcom 22283* | The argument converse of a continuous function is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) ⇒ ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿)) | ||
Theorem | cnmptkc 22284* | The curried first projection function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝑥)) ∈ (𝐽 Cn (𝐽 ↑ko 𝐾))) | ||
Theorem | cnmptkp 22285* | 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.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) & ⊢ (𝜑 → 𝐵 ∈ 𝑌) & ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) ∈ (𝐽 Cn 𝐿)) | ||
Theorem | cnmptk1 22286* | The composition of a curried function with a one-arg function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) & ⊢ (𝜑 → (𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀)) & ⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)) ∈ (𝐽 Cn (𝑀 ↑ko 𝐾))) | ||
Theorem | cnmpt1k 22287* | 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.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) & ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑊)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐿)) & ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑧 ∈ 𝑍 ↦ 𝐵)) ∈ (𝐾 Cn (𝑀 ↑ko 𝐿))) & ⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑥 ∈ 𝑋 ↦ 𝐶)) ∈ (𝐾 Cn (𝑀 ↑ko 𝐽))) | ||
Theorem | cnmptkk 22288* | The composition of two curried functions is jointly continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) & ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑊)) & ⊢ (𝜑 → 𝐿 ∈ 𝑛-Locally Comp) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑧 ∈ 𝑍 ↦ 𝐵)) ∈ (𝐽 Cn (𝑀 ↑ko 𝐿))) & ⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)) ∈ (𝐽 Cn (𝑀 ↑ko 𝐾))) | ||
Theorem | xkofvcn 22289* | Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 22261.) (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.) |
⊢ 𝑋 = ∪ 𝑅 & ⊢ 𝐹 = (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥 ∈ 𝑋 ↦ (𝑓‘𝑥)) ⇒ ⊢ ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝐹 ∈ (((𝑆 ↑ko 𝑅) ×t 𝑅) Cn 𝑆)) | ||
Theorem | cnmptk1p 22290* | 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.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) & ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Comp) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) & ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) ∈ (𝐽 Cn 𝐿)) | ||
Theorem | cnmptk2 22291* | The uncurrying of a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) & ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Comp) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) | ||
Theorem | xkoinjcn 22292* | Continuity of "injection", i.e. currying, as a function on continuous function spaces. (Contributed by Mario Carneiro, 23-Mar-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉)) ⇒ ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝐹 ∈ (𝑅 Cn ((𝑆 ×t 𝑅) ↑ko 𝑆))) | ||
Theorem | cnmpt2k 22293* | The currying of a two-argument function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) | ||
Theorem | txconn 22294 | The topological product of two connected spaces is connected. (Contributed by Mario Carneiro, 29-Mar-2015.) |
⊢ ((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) → (𝑅 ×t 𝑆) ∈ Conn) | ||
Theorem | imasnopn 22295 | 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) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) ∈ 𝐾) | ||
Theorem | imasncld 22296 | 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)) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) ∈ (Clsd‘𝐾)) | ||
Theorem | imasncls 22297 | 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) ∧ 𝐴 ∈ 𝑋)) → ((cls‘𝐾)‘(𝑅 “ {𝐴})) ⊆ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴})) | ||
Syntax | ckq 22298 | Extend class notation with the Kolmogorov quotient function. |
class KQ | ||
Definition | df-kq 22299* | 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 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦}))) | ||
Theorem | qtopval 22300* | Value of the quotient topology function. (Contributed by Mario Carneiro, 23-Mar-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 (𝐹 “ 𝑋) ∣ ((◡𝐹 “ 𝑠) ∩ 𝑋) ∈ 𝐽}) |
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