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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ptuniconst 21401 | 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 21402 | 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 21403 | 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 21404* | 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 21405 | 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 21406 | 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 21407 | 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 21408 | 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 21409 | 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 21410 | 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 21411* | 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 21412 | 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 21413 | 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 21414* | 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 21415* | 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 21416* | 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 21417* | 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 21418* | 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 21419* | 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 21420* | Lemma for dfac14 21421. 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 21419 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 21421* | Theorem ptcls 21419 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 21422* | 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 21423* | 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 21424* | Lemma for ptcnp 21425. (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 21425* | 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 21426* | 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 21427* | 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 21428* | 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 21429 | 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 21430* | 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 21431 | 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 21432 | A structure product of topologies 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 21433 | A structure product of topologies is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) ⇒ ⊢ ((𝑅 ∈ TopSp ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ TopSp) | ||
Theorem | txrest 21434 | 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 21435 | 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 21436 | Lemma for txindis 21437. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ (( I ‘𝐴) × ( I ‘𝐵)) = ( I ‘(𝐴 × 𝐵)) | ||
Theorem | txindis 21437 | The topological product of indiscrete spaces is indiscrete. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ ({∅, 𝐴} ×t {∅, 𝐵}) = {∅, (𝐴 × 𝐵)} | ||
Theorem | txdis1cn 21438* | 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 21439* | 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 21440* | 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 21441 | 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 21442* | 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 21443* | 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 21444* | Lemma for txcmp 21446. (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 21445* | Lemma for txcmp 21446. (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 21446 | 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 21447 | 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 21448 | A topology is Hausdorff iff the diagonal set is closed in the topology's product with itself. EDITORIAL: very clumsy proof, can probably be shortened substantially. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×t 𝐽)))) | ||
Theorem | hauseqlcld 21449 | 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 21450 | 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 21451* | 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 21452* | 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 21453 | 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 21454 | 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 21455 | 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 21456 | 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 21457 | 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 21458 | 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 21459* | 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 21487.) (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 21460* | If 𝐹 is a continuous function, then 𝑔 ↦ 𝑔 ∘ 𝐹 is a continuous function on function spaces. (The reason we prove this and xkoco2cn 21461 independently of the more general xkococn 21463 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 21461* | 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 21462* | 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 21463* | 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 21464* | 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 21465* | 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 21466* | 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 21467* | 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 21468* | 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 21469* | 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 21470* | 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 21471* | 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 21472* | 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 21473* | 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 21474* | 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 21475* | 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 21476* | 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 21477* | 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 21478* | 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 21479* | 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 21480* | 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 21481* | 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 21482* | 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 21483* | 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 21484* | 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 21485* | 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 21486* | 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 21487* | Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 21459.) (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 21488* | 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 21489* | 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 21490* | 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 21491* | 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 21492 | 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 21493 | 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 21494 | 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 21495 | 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 21496 | Extend class notation with the Kolmogorov quotient function. |
class KQ | ||
Definition | df-kq 21497* | 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 21498* | 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}) | ||
Theorem | qtopval2 21499* | Value of the quotient topology function when 𝐹 is a function on the base set. (Contributed by Mario Carneiro, 23-Mar-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 𝑌 ∣ (◡𝐹 “ 𝑠) ∈ 𝐽}) | ||
Theorem | elqtop 21500 | Value of the quotient topology function. (Contributed by Mario Carneiro, 23-Mar-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐴 ∈ (𝐽 qTop 𝐹) ↔ (𝐴 ⊆ 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ 𝐽))) |
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