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Type | Label | Description |
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Statement | ||
Theorem | imasncld 22301 | 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 22302 | 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 22303 | Extend class notation with the Kolmogorov quotient function. |
class KQ | ||
Definition | df-kq 22304* | 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 22305* | 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 22306* | 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 22307 | 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 22308 | 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 22309 | 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 22310 | 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 22311 | 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 22312 | 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 22313 | 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 22314 | 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 22315 | 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 22316 | 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 22317 | Lemma for qtopcmp 22318 and qtopconn 22319. (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 22318 | 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 22319 | 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 22320 | 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 22321 | 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 22322 | 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 22323 | 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 22324 | 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 22325 | 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 22315, 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 22326* | 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 22327 | 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 22328* | 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 22329* | 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 22330 | 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 22331 | 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 22332 | 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 22333* | Value of the function appearing in df-kq 22304. (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 22334* | 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 22335* | 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 22336* | 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 22337* | 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 22338* | 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 22339* | 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 22340* | When the image set is open, the quotient map satisfies a partial converse to fnfvima 6997, 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 22341* | Any open set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 22327). (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 22342* | A version of imain 6441 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 22343* | Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 22327). (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 22344* | 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 22345* | 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 22346* | Lemma for kqt0 22356. (Contributed by Mario Carneiro, 23-Mar-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ⇒ ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ Kol2) | ||
Theorem | isr0 22347* | 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 22348* | 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 22349* | Lemma for regr1 22360. (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 22350* | 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 22351* | 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 22352* | 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 22353* | 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 22354* | 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 22355 | The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top) | ||
Theorem | kqt0 22356 | 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 22357 | The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ KQ:Top⟶Kol2 | ||
Theorem | r0sep 22358* | 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 22359 | 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 22360 | 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 22361 | The Kolmogorov quotient of a regular space is regular. By regr1 22360 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 22362 | The Kolmogorov quotient of a normal space is normal. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ (𝐽 ∈ Nrm ↔ (KQ‘𝐽) ∈ Nrm) | ||
Syntax | chmeo 22363 | Extend class notation with the class of all homeomorphisms. |
class Homeo | ||
Syntax | chmph 22364 | Extend class notation with the relation "is homeomorphic to.". |
class ≃ | ||
Definition | df-hmeo 22365* | 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 22366 | Definition of the relation 𝑥 is homeomorphic to 𝑦. (Contributed by FL, 14-Feb-2007.) |
⊢ ≃ = (◡Homeo “ (V ∖ 1o)) | ||
Theorem | hmeofn 22367 | The set of homeomorphisms is a function on topologies. (Contributed by Mario Carneiro, 23-Aug-2015.) |
⊢ Homeo Fn (Top × Top) | ||
Theorem | hmeofval 22368* | 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 22369 | 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 22370 | A homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.) |
⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾)) | ||
Theorem | hmeocnvcn 22371 | The converse of a homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.) |
⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽)) | ||
Theorem | hmeocnv 22372 | 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 22373 | 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 22374 | 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 22375 | 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 22376 | 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 22377 | 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 22378 | 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 22379 | 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 22380* | 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 22381 | 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 22382 | 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 22383 | 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 22384 | 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 22385 | A homeomorphism is a quotient map. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐾 = (𝐽 qTop 𝐹)) | ||
Theorem | hmph 22386 | 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 22387 | 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 22388 | Reverse closure for the homeomorphic predicate. (Contributed by Mario Carneiro, 22-Aug-2015.) |
⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Top ∧ 𝐾 ∈ Top)) | ||
Theorem | hmphtop1 22389 | 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 22390 | 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 22391 | "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 22392 | "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 22393 | "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 22394 | "Is homeomorphic to" is an equivalence relation. (Contributed by FL, 22-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) |
⊢ ≃ Er Top | ||
Theorem | hmphen 22395 | 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 22396 | "Is homeomorphic to" is symmetric. (Contributed by FL, 22-Feb-2007.) |
⊢ (𝐽 ≃ 𝐾 ↔ 𝐾 ≃ 𝐽) | ||
Theorem | haushmphlem 22397* | Lemma for haushmph 22402 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 22398 | 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 22399 | Connectedness is a topological property. (Contributed by Jeff Hankins, 3-Jul-2009.) |
⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Conn → 𝐾 ∈ Conn)) | ||
Theorem | t0hmph 22400 | T0 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Kol2 → 𝐾 ∈ Kol2)) |
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