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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Syntax | ckq 23701 | Extend class notation with the Kolmogorov quotient function. |
| class KQ | ||
| Definition | df-kq 23702* | 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 23703* | 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 23704* | 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 23705 | 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))) | ||
| Theorem | qtopres 23706 | The quotient topology is unaffected by restriction to the base set. This property makes it slightly more convenient to use, since we don't have to require that 𝐹 be a function with domain 𝑋. (Contributed by Mario Carneiro, 23-Mar-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐹 ∈ 𝑉 → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 ↾ 𝑋))) | ||
| Theorem | qtoptop2 23707 | The quotient topology is a topology. (Contributed by Mario Carneiro, 23-Mar-2015.) |
| ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝐽 qTop 𝐹) ∈ Top) | ||
| Theorem | qtoptop 23708 | The quotient topology is a topology. (Contributed by Mario Carneiro, 23-Mar-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top) | ||
| Theorem | elqtop2 23709 | Value of the quotient topology function. (Contributed by Mario Carneiro, 9-Apr-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑋–onto→𝑌) → (𝐴 ∈ (𝐽 qTop 𝐹) ↔ (𝐴 ⊆ 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ 𝐽))) | ||
| Theorem | qtopuni 23710 | The base set of the quotient topology. (Contributed by Mario Carneiro, 23-Mar-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→𝑌) → 𝑌 = ∪ (𝐽 qTop 𝐹)) | ||
| Theorem | elqtop3 23711 | Value of the quotient topology function. (Contributed by Mario Carneiro, 9-Apr-2015.) |
| ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐴 ∈ (𝐽 qTop 𝐹) ↔ (𝐴 ⊆ 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ 𝐽))) | ||
| Theorem | qtoptopon 23712 | The base set of the quotient topology. (Contributed by Mario Carneiro, 22-Aug-2015.) |
| ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌)) | ||
| Theorem | qtopid 23713 | A quotient map is a continuous function into its quotient topology. (Contributed by Mario Carneiro, 23-Mar-2015.) |
| ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))) | ||
| Theorem | idqtop 23714 | The quotient topology induced by the identity function is the original topology. (Contributed by Mario Carneiro, 30-Aug-2015.) |
| ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 qTop ( I ↾ 𝑋)) = 𝐽) | ||
| Theorem | qtopcmplem 23715 | Lemma for qtopcmp 23716 and qtopconn 23717. (Contributed by Mario Carneiro, 24-Mar-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Top) & ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹:𝑋–onto→∪ (𝐽 qTop 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))) → (𝐽 qTop 𝐹) ∈ 𝐴) ⇒ ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ 𝐴) | ||
| Theorem | qtopcmp 23716 | A quotient of a compact space is compact. (Contributed by Mario Carneiro, 24-Mar-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Comp ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Comp) | ||
| Theorem | qtopconn 23717 | A quotient of a connected space is connected. (Contributed by Mario Carneiro, 24-Mar-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Conn ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Conn) | ||
| Theorem | qtopkgen 23718 | A quotient of a compactly generated space is compactly generated. (Contributed by Mario Carneiro, 9-Apr-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ ran 𝑘Gen ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ ran 𝑘Gen) | ||
| Theorem | basqtop 23719 | An injection maps bases to bases. (Contributed by Mario Carneiro, 27-Aug-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝐽 qTop 𝐹) ∈ TopBases) | ||
| Theorem | tgqtop 23720 | An injection maps generated topologies to each other. (Contributed by Mario Carneiro, 27-Aug-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → ((topGen‘𝐽) qTop 𝐹) = (topGen‘(𝐽 qTop 𝐹))) | ||
| Theorem | qtopcld 23721 | The property of being a closed set in the quotient topology. (Contributed by Mario Carneiro, 24-Mar-2015.) |
| ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐴 ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ (𝐴 ⊆ 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽)))) | ||
| Theorem | qtopcn 23722 | Universal property of a quotient map. (Contributed by Mario Carneiro, 23-Mar-2015.) |
| ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑍)) ∧ (𝐹:𝑋–onto→𝑌 ∧ 𝐺:𝑌⟶𝑍)) → (𝐺 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ↔ (𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐾))) | ||
| Theorem | qtopss 23723 | A surjective continuous function from 𝐽 to 𝐾 induces a topology 𝐽 qTop 𝐹 on the base set of 𝐾. This topology is in general finer than 𝐾. Together with qtopid 23713, this implies that 𝐽 qTop 𝐹 is the finest topology making 𝐹 continuous, i.e. the final topology with respect to the family {𝐹}. (Contributed by Mario Carneiro, 24-Mar-2015.) |
| ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → 𝐾 ⊆ (𝐽 qTop 𝐹)) | ||
| Theorem | qtopeu 23724* | Universal property of the quotient topology. If 𝐺 is a function from 𝐽 to 𝐾 which is equal on all equivalent elements under 𝐹, then there is a unique continuous map 𝑓:(𝐽 / 𝐹)⟶𝐾 such that 𝐺 = 𝑓 ∘ 𝐹, and we say that 𝐺 "passes to the quotient". (Contributed by Mario Carneiro, 24-Mar-2015.) |
| ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐺‘𝑥) = (𝐺‘𝑦)) ⇒ ⊢ (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)𝐺 = (𝑓 ∘ 𝐹)) | ||
| Theorem | qtoprest 23725 | If 𝐴 is a saturated open or closed set (where saturated means that 𝐴 = (◡𝐹 “ 𝑈) for some 𝑈), then the restriction of the quotient map 𝐹 to 𝐴 is a quotient map. (Contributed by Mario Carneiro, 24-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.) |
| ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ (𝜑 → 𝑈 ⊆ 𝑌) & ⊢ (𝜑 → 𝐴 = (◡𝐹 “ 𝑈)) & ⊢ (𝜑 → (𝐴 ∈ 𝐽 ∨ 𝐴 ∈ (Clsd‘𝐽))) ⇒ ⊢ (𝜑 → ((𝐽 qTop 𝐹) ↾t 𝑈) = ((𝐽 ↾t 𝐴) qTop (𝐹 ↾ 𝐴))) | ||
| Theorem | qtopomap 23726* | If 𝐹 is a surjective continuous open map, then it is a quotient map. (An open map is a function that maps open sets to open sets.) (Contributed by Mario Carneiro, 24-Mar-2015.) |
| ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → ran 𝐹 = 𝑌) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐾) ⇒ ⊢ (𝜑 → 𝐾 = (𝐽 qTop 𝐹)) | ||
| Theorem | qtopcmap 23727* | If 𝐹 is a surjective continuous closed map, then it is a quotient map. (A closed map is a function that maps closed sets to closed sets.) (Contributed by Mario Carneiro, 24-Mar-2015.) |
| ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → ran 𝐹 = 𝑌) & ⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝐹 “ 𝑥) ∈ (Clsd‘𝐾)) ⇒ ⊢ (𝜑 → 𝐾 = (𝐽 qTop 𝐹)) | ||
| Theorem | imastopn 23728 | The topology of an image structure. (Contributed by Mario Carneiro, 27-Aug-2015.) |
| ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ 𝐽 = (TopOpen‘𝑅) & ⊢ 𝑂 = (TopOpen‘𝑈) ⇒ ⊢ (𝜑 → 𝑂 = (𝐽 qTop 𝐹)) | ||
| Theorem | imastps 23729 | The image of a topological space under a function is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.) |
| ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ (𝜑 → 𝑅 ∈ TopSp) ⇒ ⊢ (𝜑 → 𝑈 ∈ TopSp) | ||
| Theorem | qustps 23730 | A quotient structure is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.) |
| ⊢ (𝜑 → 𝑈 = (𝑅 /s 𝐸)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐸 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ TopSp) ⇒ ⊢ (𝜑 → 𝑈 ∈ TopSp) | ||
| Theorem | kqfval 23731* | Value of the function appearing in df-kq 23702. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ⇒ ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) = {𝑦 ∈ 𝐽 ∣ 𝐴 ∈ 𝑦}) | ||
| Theorem | kqfeq 23732* | Two points in the Kolmogorov quotient are equal iff the original points are topologically indistinguishable. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ⇒ ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐹‘𝐴) = (𝐹‘𝐵) ↔ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ↔ 𝐵 ∈ 𝑦))) | ||
| Theorem | kqffn 23733* | The topological indistinguishability map is a function on the base. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ⇒ ⊢ (𝐽 ∈ 𝑉 → 𝐹 Fn 𝑋) | ||
| Theorem | kqval 23734* | Value of the quotient topology function. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ⇒ ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹)) | ||
| Theorem | kqtopon 23735* | The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ⇒ ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹)) | ||
| Theorem | kqid 23736* | The topological indistinguishability map is a continuous function into the Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ⇒ ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽))) | ||
| Theorem | ist0-4 23737* | The topological indistinguishability map is injective iff the space is T0. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ⇒ ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹:𝑋–1-1→V)) | ||
| Theorem | kqfvima 23738* | When the image set is open, the quotient map satisfies a partial converse to fnfvima 7253, which is normally only true for injective functions. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ 𝑈 ↔ (𝐹‘𝐴) ∈ (𝐹 “ 𝑈))) | ||
| Theorem | kqsat 23739* | Any open set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 23725). (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (◡𝐹 “ (𝐹 “ 𝑈)) = 𝑈) | ||
| Theorem | kqdisj 23740* | A version of imain 6651 for the topological indistinguishability map. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝐴 ∖ 𝑈))) = ∅) | ||
| Theorem | kqcldsat 23741* | Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 23725). (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (◡𝐹 “ (𝐹 “ 𝑈)) = 𝑈) | ||
| Theorem | kqopn 23742* | The topological indistinguishability map is an open map. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝐹 “ 𝑈) ∈ (KQ‘𝐽)) | ||
| Theorem | kqcld 23743* | The topological indistinguishability map is a closed map. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹 “ 𝑈) ∈ (Clsd‘(KQ‘𝐽))) | ||
| Theorem | kqt0lem 23744* | Lemma for kqt0 23754. (Contributed by Mario Carneiro, 23-Mar-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ⇒ ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ Kol2) | ||
| Theorem | isr0 23745* | The property "𝐽 is an R0 space". A space is R0 if any two topologically distinguishable points are separated (there is an open set containing each one and disjoint from the other). Or in contraposition, if every open set which contains 𝑥 also contains 𝑦, so there is no separation, then 𝑥 and 𝑦 are members of the same open sets. We have chosen not to give this definition a name, because it turns out that a space is R0 if and only if its Kolmogorov quotient is T1, so that is what we prove here. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ⇒ ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((KQ‘𝐽) ∈ Fre ↔ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 → 𝑤 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝑤 ∈ 𝑜)))) | ||
| Theorem | r0cld 23746* | The analogue of the T1 axiom (singletons are closed) for an R0 space. In an R0 space the set of all points topologically indistinguishable from 𝐴 is closed. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → {𝑧 ∈ 𝑋 ∣ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)} ∈ (Clsd‘𝐽)) | ||
| Theorem | regr1lem 23747* | Lemma for regr1 23758. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐽 ∈ Reg) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑋) & ⊢ (𝜑 → 𝑈 ∈ 𝐽) & ⊢ (𝜑 → ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝐴) ∈ 𝑚 ∧ (𝐹‘𝐵) ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) ⇒ ⊢ (𝜑 → (𝐴 ∈ 𝑈 → 𝐵 ∈ 𝑈)) | ||
| Theorem | regr1lem2 23748* | A Kolmogorov quotient of a regular space is Hausdorff. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Haus) | ||
| Theorem | kqreglem1 23749* | A Kolmogorov quotient of a regular space is regular. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Reg) | ||
| Theorem | kqreglem2 23750* | If the Kolmogorov quotient of a space is regular then so is the original space. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Reg) | ||
| Theorem | kqnrmlem1 23751* | A Kolmogorov quotient of a normal space is normal. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Nrm) | ||
| Theorem | kqnrmlem2 23752* | If the Kolmogorov quotient of a space is normal then so is the original space. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Nrm) | ||
| Theorem | kqtop 23753 | The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top) | ||
| Theorem | kqt0 23754 | The Kolmogorov quotient is T0 even if the original topology is not. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Kol2) | ||
| Theorem | kqf 23755 | The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ KQ:Top⟶Kol2 | ||
| Theorem | r0sep 23756* | The separation property of an R0 space. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜))) | ||
| Theorem | nrmr0reg 23757 | A normal R0 space is also regular. These spaces are usually referred to as normal regular spaces. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → 𝐽 ∈ Reg) | ||
| Theorem | regr1 23758 | A regular space is R1, which means that any two topologically distinct points can be separated by neighborhoods. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ (𝐽 ∈ Reg → (KQ‘𝐽) ∈ Haus) | ||
| Theorem | kqreg 23759 | The Kolmogorov quotient of a regular space is regular. By regr1 23758 it is also Hausdorff, so we can also say that a space is regular iff the Kolmogorov quotient is regular Hausdorff (T3). (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ (𝐽 ∈ Reg ↔ (KQ‘𝐽) ∈ Reg) | ||
| Theorem | kqnrm 23760 | The Kolmogorov quotient of a normal space is normal. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ (𝐽 ∈ Nrm ↔ (KQ‘𝐽) ∈ Nrm) | ||
| Syntax | chmeo 23761 | Extend class notation with the class of all homeomorphisms. |
| class Homeo | ||
| Syntax | chmph 23762 | Extend class notation with the relation "is homeomorphic to.". |
| class ≃ | ||
| Definition | df-hmeo 23763* | Function returning all the homeomorphisms from topology 𝑗 to topology 𝑘. (Contributed by FL, 14-Feb-2007.) |
| ⊢ Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)}) | ||
| Definition | df-hmph 23764 | Definition of the relation 𝑥 is homeomorphic to 𝑦. (Contributed by FL, 14-Feb-2007.) |
| ⊢ ≃ = (◡Homeo “ (V ∖ 1o)) | ||
| Theorem | hmeofn 23765 | The set of homeomorphisms is a function on topologies. (Contributed by Mario Carneiro, 23-Aug-2015.) |
| ⊢ Homeo Fn (Top × Top) | ||
| Theorem | hmeofval 23766* | The set of all the homeomorphisms between two topologies. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.) |
| ⊢ (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)} | ||
| Theorem | ishmeo 23767 | The predicate F is a homeomorphism between topology 𝐽 and topology 𝐾. Criterion of [BourbakiTop1] p. I.2. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.) |
| ⊢ (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn 𝐽))) | ||
| Theorem | hmeocn 23768 | A homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.) |
| ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾)) | ||
| Theorem | hmeocnvcn 23769 | The converse of a homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.) |
| ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽)) | ||
| Theorem | hmeocnv 23770 | The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 22-Aug-2015.) |
| ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾Homeo𝐽)) | ||
| Theorem | hmeof1o2 23771 | A homeomorphism is a 1-1-onto mapping. (Contributed by Mario Carneiro, 22-Aug-2015.) |
| ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽Homeo𝐾)) → 𝐹:𝑋–1-1-onto→𝑌) | ||
| Theorem | hmeof1o 23772 | A homeomorphism is a 1-1-onto mapping. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 30-May-2014.) |
| ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1-onto→𝑌) | ||
| Theorem | hmeoima 23773 | The image of an open set by a homeomorphism is an open set. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 22-Aug-2015.) |
| ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ∈ 𝐽) → (𝐹 “ 𝐴) ∈ 𝐾) | ||
| Theorem | hmeoopn 23774 | Homeomorphisms preserve openness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ 𝐽 ↔ (𝐹 “ 𝐴) ∈ 𝐾)) | ||
| Theorem | hmeocld 23775 | Homeomorphisms preserve closedness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (𝐹 “ 𝐴) ∈ (Clsd‘𝐾))) | ||
| Theorem | hmeocls 23776 | Homeomorphisms preserve closures. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐾)‘(𝐹 “ 𝐴)) = (𝐹 “ ((cls‘𝐽)‘𝐴))) | ||
| Theorem | hmeontr 23777 | Homeomorphisms preserve interiors. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐾)‘(𝐹 “ 𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴))) | ||
| Theorem | hmeoimaf1o 23778* | The function mapping open sets to their images under a homeomorphism is a bijection of topologies. (Contributed by Mario Carneiro, 10-Sep-2015.) |
| ⊢ 𝐺 = (𝑥 ∈ 𝐽 ↦ (𝐹 “ 𝑥)) ⇒ ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐺:𝐽–1-1-onto→𝐾) | ||
| Theorem | hmeores 23779 | The restriction of a homeomorphism is a homeomorphism. (Contributed by Mario Carneiro, 14-Sep-2014.) (Proof shortened by Mario Carneiro, 22-Aug-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌)Homeo(𝐾 ↾t (𝐹 “ 𝑌)))) | ||
| Theorem | hmeoco 23780 | The composite of two homeomorphisms is a homeomorphism. (Contributed by FL, 9-Mar-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.) |
| ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (𝐺 ∘ 𝐹) ∈ (𝐽Homeo𝐿)) | ||
| Theorem | idhmeo 23781 | The identity function is a homeomorphism. (Contributed by FL, 14-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.) |
| ⊢ (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽Homeo𝐽)) | ||
| Theorem | hmeocnvb 23782 | The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) |
| ⊢ (Rel 𝐹 → (◡𝐹 ∈ (𝐽Homeo𝐾) ↔ 𝐹 ∈ (𝐾Homeo𝐽))) | ||
| Theorem | hmeoqtop 23783 | A homeomorphism is a quotient map. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐾 = (𝐽 qTop 𝐹)) | ||
| Theorem | hmph 23784 | Express the predicate 𝐽 is homeomorphic to 𝐾. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.) |
| ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅) | ||
| Theorem | hmphi 23785 | If there is a homeomorphism between spaces, then the spaces are homeomorphic. (Contributed by Mario Carneiro, 23-Aug-2015.) |
| ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐽 ≃ 𝐾) | ||
| Theorem | hmphtop 23786 | Reverse closure for the homeomorphic predicate. (Contributed by Mario Carneiro, 22-Aug-2015.) |
| ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Top ∧ 𝐾 ∈ Top)) | ||
| Theorem | hmphtop1 23787 | The relation "being homeomorphic to" implies the operands are topologies. (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) |
| ⊢ (𝐽 ≃ 𝐾 → 𝐽 ∈ Top) | ||
| Theorem | hmphtop2 23788 | The relation "being homeomorphic to" implies the operands are topologies. (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) |
| ⊢ (𝐽 ≃ 𝐾 → 𝐾 ∈ Top) | ||
| Theorem | hmphref 23789 | "Is homeomorphic to" is reflexive. (Contributed by FL, 25-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.) |
| ⊢ (𝐽 ∈ Top → 𝐽 ≃ 𝐽) | ||
| Theorem | hmphsym 23790 | "Is homeomorphic to" is symmetric. (Contributed by FL, 8-Mar-2007.) (Proof shortened by Mario Carneiro, 30-May-2014.) |
| ⊢ (𝐽 ≃ 𝐾 → 𝐾 ≃ 𝐽) | ||
| Theorem | hmphtr 23791 | "Is homeomorphic to" is transitive. (Contributed by FL, 9-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) |
| ⊢ ((𝐽 ≃ 𝐾 ∧ 𝐾 ≃ 𝐿) → 𝐽 ≃ 𝐿) | ||
| Theorem | hmpher 23792 | "Is homeomorphic to" is an equivalence relation. (Contributed by FL, 22-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) |
| ⊢ ≃ Er Top | ||
| Theorem | hmphen 23793 | Homeomorphisms preserve the cardinality of the topologies. (Contributed by FL, 1-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) |
| ⊢ (𝐽 ≃ 𝐾 → 𝐽 ≈ 𝐾) | ||
| Theorem | hmphsymb 23794 | "Is homeomorphic to" is symmetric. (Contributed by FL, 22-Feb-2007.) |
| ⊢ (𝐽 ≃ 𝐾 ↔ 𝐾 ≃ 𝐽) | ||
| Theorem | haushmphlem 23795* | Lemma for haushmph 23800 and similar theorems. If the topological property 𝐴 is preserved under injective preimages, then property 𝐴 is preserved under homeomorphisms. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Top) & ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑓:∪ 𝐾–1-1→∪ 𝐽 ∧ 𝑓 ∈ (𝐾 Cn 𝐽)) → 𝐾 ∈ 𝐴) ⇒ ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ 𝐴 → 𝐾 ∈ 𝐴)) | ||
| Theorem | cmphmph 23796 | Compactness is a topological property-that is, for any two homeomorphic topologies, either both are compact or neither is. (Contributed by Jeff Hankins, 30-Jun-2009.) (Revised by Mario Carneiro, 23-Aug-2015.) |
| ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Comp → 𝐾 ∈ Comp)) | ||
| Theorem | connhmph 23797 | Connectedness is a topological property. (Contributed by Jeff Hankins, 3-Jul-2009.) |
| ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Conn → 𝐾 ∈ Conn)) | ||
| Theorem | t0hmph 23798 | T0 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Kol2 → 𝐾 ∈ Kol2)) | ||
| Theorem | t1hmph 23799 | T1 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Fre → 𝐾 ∈ Fre)) | ||
| Theorem | haushmph 23800 | Hausdorff-ness is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.) |
| ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Haus → 𝐾 ∈ Haus)) | ||
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