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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | hmeof1o2 22301 | 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 22302 | 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 22303 | 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 22304 | 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 22305 | 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 22306 | 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 22307 | 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 22308* | 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 22309 | 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 22310 | 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 22311 | 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 22312 | 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 22313 | A homeomorphism is a quotient map. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐾 = (𝐽 qTop 𝐹)) | ||
Theorem | hmph 22314 | 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 22315 | 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 22316 | Reverse closure for the homeomorphic predicate. (Contributed by Mario Carneiro, 22-Aug-2015.) |
⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Top ∧ 𝐾 ∈ Top)) | ||
Theorem | hmphtop1 22317 | 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 22318 | 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 22319 | "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 22320 | "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 22321 | "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 22322 | "Is homeomorphic to" is an equivalence relation. (Contributed by FL, 22-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) |
⊢ ≃ Er Top | ||
Theorem | hmphen 22323 | 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 22324 | "Is homeomorphic to" is symmetric. (Contributed by FL, 22-Feb-2007.) |
⊢ (𝐽 ≃ 𝐾 ↔ 𝐾 ≃ 𝐽) | ||
Theorem | haushmphlem 22325* | Lemma for haushmph 22330 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 22326 | 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 22327 | Connectedness is a topological property. (Contributed by Jeff Hankins, 3-Jul-2009.) |
⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Conn → 𝐾 ∈ Conn)) | ||
Theorem | t0hmph 22328 | T0 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Kol2 → 𝐾 ∈ Kol2)) | ||
Theorem | t1hmph 22329 | T1 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Fre → 𝐾 ∈ Fre)) | ||
Theorem | haushmph 22330 | Hausdorff-ness is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Haus → 𝐾 ∈ Haus)) | ||
Theorem | reghmph 22331 | Regularity is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Reg → 𝐾 ∈ Reg)) | ||
Theorem | nrmhmph 22332 | Normality is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Nrm → 𝐾 ∈ Nrm)) | ||
Theorem | hmph0 22333 | A topology homeomorphic to the empty set is empty. (Contributed by FL, 18-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) |
⊢ (𝐽 ≃ {∅} ↔ 𝐽 = {∅}) | ||
Theorem | hmphdis 22334 | Homeomorphisms preserve topological discretion. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝐽 = 𝒫 𝑋) | ||
Theorem | hmphindis 22335 | Homeomorphisms preserve topological indiscretion. (Contributed by FL, 18-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ≃ {∅, 𝐴} → 𝐽 = {∅, 𝑋}) | ||
Theorem | indishmph 22336 | Equinumerous sets equipped with their indiscrete topologies are homeomorphic (which means in that particular case that a segment is homeomorphic to a circle contrary to what Wikipedia claims). (Contributed by FL, 17-Aug-2008.) (Proof shortened by Mario Carneiro, 10-Sep-2015.) |
⊢ (𝐴 ≈ 𝐵 → {∅, 𝐴} ≃ {∅, 𝐵}) | ||
Theorem | hmphen2 22337 | Homeomorphisms preserve the cardinality of the underlying sets. (Contributed by FL, 17-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ (𝐽 ≃ 𝐾 → 𝑋 ≈ 𝑌) | ||
Theorem | cmphaushmeo 22338 | A continuous bijection from a compact space to a Hausdorff space is a homeomorphism. (Contributed by Mario Carneiro, 17-Feb-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∈ (𝐽Homeo𝐾) ↔ 𝐹:𝑋–1-1-onto→𝑌)) | ||
Theorem | ordthmeolem 22339 | Lemma for ordthmeo 22340. (Contributed by Mario Carneiro, 9-Sep-2015.) |
⊢ 𝑋 = dom 𝑅 & ⊢ 𝑌 = dom 𝑆 ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝐹 ∈ ((ordTop‘𝑅) Cn (ordTop‘𝑆))) | ||
Theorem | ordthmeo 22340 | An order isomorphism is a homeomorphism on the respective order topologies. (Contributed by Mario Carneiro, 9-Sep-2015.) |
⊢ 𝑋 = dom 𝑅 & ⊢ 𝑌 = dom 𝑆 ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝐹 ∈ ((ordTop‘𝑅)Homeo(ordTop‘𝑆))) | ||
Theorem | txhmeo 22341* | Lift a pair of homeomorphisms on the factors to a homeomorphism of product topologies. (Contributed by Mario Carneiro, 2-Sep-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑌 = ∪ 𝐾 & ⊢ (𝜑 → 𝐹 ∈ (𝐽Homeo𝐿)) & ⊢ (𝜑 → 𝐺 ∈ (𝐾Homeo𝑀)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈(𝐹‘𝑥), (𝐺‘𝑦)〉) ∈ ((𝐽 ×t 𝐾)Homeo(𝐿 ×t 𝑀))) | ||
Theorem | txswaphmeolem 22342* | Show inverse for the "swap components" operation on a Cartesian product. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 〈𝑥, 𝑦〉) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉)) = ( I ↾ (𝑋 × 𝑌)) | ||
Theorem | txswaphmeo 22343* | There is a homeomorphism from 𝑋 × 𝑌 to 𝑌 × 𝑋. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉) ∈ ((𝐽 ×t 𝐾)Homeo(𝐾 ×t 𝐽))) | ||
Theorem | pt1hmeo 22344* | The canonical homeomorphism from a topological product on a singleton to the topology of the factor. (Contributed by Mario Carneiro, 3-Feb-2015.) (Proof shortened by AV, 18-Apr-2021.) |
⊢ 𝐾 = (∏t‘{〈𝐴, 𝐽〉}) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ {〈𝐴, 𝑥〉}) ∈ (𝐽Homeo𝐾)) | ||
Theorem | ptuncnv 22345* | Exhibit the converse function of the map 𝐺 which joins two product topologies on disjoint index sets. (Contributed by Mario Carneiro, 8-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐾 & ⊢ 𝑌 = ∪ 𝐿 & ⊢ 𝐽 = (∏t‘𝐹) & ⊢ 𝐾 = (∏t‘(𝐹 ↾ 𝐴)) & ⊢ 𝐿 = (∏t‘(𝐹 ↾ 𝐵)) & ⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝑥 ∪ 𝑦)) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐶⟶Top) & ⊢ (𝜑 → 𝐶 = (𝐴 ∪ 𝐵)) & ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) ⇒ ⊢ (𝜑 → ◡𝐺 = (𝑧 ∈ ∪ 𝐽 ↦ 〈(𝑧 ↾ 𝐴), (𝑧 ↾ 𝐵)〉)) | ||
Theorem | ptunhmeo 22346* | Define a homeomorphism from a binary product of indexed product topologies to an indexed product topology on the union of the index sets. This is the topological analogue of (𝐴↑𝐵) · (𝐴↑𝐶) = 𝐴↑(𝐵 + 𝐶). (Contributed by Mario Carneiro, 8-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐾 & ⊢ 𝑌 = ∪ 𝐿 & ⊢ 𝐽 = (∏t‘𝐹) & ⊢ 𝐾 = (∏t‘(𝐹 ↾ 𝐴)) & ⊢ 𝐿 = (∏t‘(𝐹 ↾ 𝐵)) & ⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝑥 ∪ 𝑦)) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐶⟶Top) & ⊢ (𝜑 → 𝐶 = (𝐴 ∪ 𝐵)) & ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) ⇒ ⊢ (𝜑 → 𝐺 ∈ ((𝐾 ×t 𝐿)Homeo𝐽)) | ||
Theorem | xpstopnlem1 22347* | The function 𝐹 used in xpsval 16833 is a homeomorphism from the binary product topology to the indexed product topology. (Contributed by Mario Carneiro, 2-Sep-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) ⇒ ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ×t 𝐾)Homeo(∏t‘{〈∅, 𝐽〉, 〈1o, 𝐾〉}))) | ||
Theorem | xpstps 22348 | A binary product of topologies is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.) |
⊢ 𝑇 = (𝑅 ×s 𝑆) ⇒ ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑇 ∈ TopSp) | ||
Theorem | xpstopnlem2 22349* | Lemma for xpstopn 22350. (Contributed by Mario Carneiro, 27-Aug-2015.) |
⊢ 𝑇 = (𝑅 ×s 𝑆) & ⊢ 𝐽 = (TopOpen‘𝑅) & ⊢ 𝐾 = (TopOpen‘𝑆) & ⊢ 𝑂 = (TopOpen‘𝑇) & ⊢ 𝑋 = (Base‘𝑅) & ⊢ 𝑌 = (Base‘𝑆) & ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) ⇒ ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑂 = (𝐽 ×t 𝐾)) | ||
Theorem | xpstopn 22350 | The topology on a binary product of topological spaces, as we have defined it (transferring the indexed product topology on functions on {∅, 1o} to (𝑋 × 𝑌) by the canonical bijection), coincides with the usual topological product (generated by a base of rectangles). (Contributed by Mario Carneiro, 27-Aug-2015.) |
⊢ 𝑇 = (𝑅 ×s 𝑆) & ⊢ 𝐽 = (TopOpen‘𝑅) & ⊢ 𝐾 = (TopOpen‘𝑆) & ⊢ 𝑂 = (TopOpen‘𝑇) ⇒ ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑂 = (𝐽 ×t 𝐾)) | ||
Theorem | ptcmpfi 22351 | A topological product of finitely many compact spaces is compact. This weak version of Tychonoff's theorem does not require the axiom of choice. (Contributed by Mario Carneiro, 8-Feb-2015.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘𝐹) ∈ Comp) | ||
Theorem | xkocnv 22352* | The inverse of the "currying" function 𝐹 is the uncurrying function. (Contributed by Mario Carneiro, 13-Apr-2015.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ 𝐹 = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↦ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))) & ⊢ (𝜑 → 𝐽 ∈ 𝑛-Locally Comp) & ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Comp) & ⊢ (𝜑 → 𝐿 ∈ Top) ⇒ ⊢ (𝜑 → ◡𝐹 = (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) | ||
Theorem | xkohmeo 22353* | The Exponential Law for topological spaces. The "currying" function 𝐹 is a homeomorphism on function spaces when 𝐽 and 𝐾 are exponentiable spaces (by xkococn 22198, it is sufficient to assume that 𝐽, 𝐾 are locally compact to ensure exponentiability). (Contributed by Mario Carneiro, 13-Apr-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ 𝐹 = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↦ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))) & ⊢ (𝜑 → 𝐽 ∈ 𝑛-Locally Comp) & ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Comp) & ⊢ (𝜑 → 𝐿 ∈ Top) ⇒ ⊢ (𝜑 → 𝐹 ∈ ((𝐿 ↑ko (𝐽 ×t 𝐾))Homeo((𝐿 ↑ko 𝐾) ↑ko 𝐽))) | ||
Theorem | qtopf1 22354 | If a quotient map is injective, then it is a homeomorphism. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐹:𝑋–1-1→𝑌) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹))) | ||
Theorem | qtophmeo 22355* | If two functions on a base topology 𝐽 make the same identifications in order to create quotient spaces 𝐽 qTop 𝐹 and 𝐽 qTop 𝐺, then not only are 𝐽 qTop 𝐹 and 𝐽 qTop 𝐺 homeomorphic, but there is a unique homeomorphism that makes the diagram commute. (Contributed by Mario Carneiro, 24-Mar-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ (𝜑 → 𝐺:𝑋–onto→𝑌) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐺‘𝑥) = (𝐺‘𝑦))) ⇒ ⊢ (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹)) | ||
Theorem | t0kq 22356* | A topological space is T0 iff the quotient map is a homeomorphism onto the space's Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ⇒ ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹 ∈ (𝐽Homeo(KQ‘𝐽)))) | ||
Theorem | kqhmph 22357 | A topological space is T0 iff it is homeomorphic to its Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ (𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽)) | ||
Theorem | ist1-5lem 22358 | Lemma for ist1-5 22360 and similar theorems. If 𝐴 is a topological property which implies T0, such as T1 or T2, the property can be "decomposed" into T0 and a non-T0 version of property 𝐴 (which is defined as stating that the Kolmogorov quotient of the space has property 𝐴). For example, if 𝐴 is T1, then the theorem states that a space is T1 iff it is T0 and its Kolmogorov quotient is T1 (we call this property R0). (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Kol2) & ⊢ (𝐽 ≃ (KQ‘𝐽) → (𝐽 ∈ 𝐴 → (KQ‘𝐽) ∈ 𝐴)) & ⊢ ((KQ‘𝐽) ≃ 𝐽 → ((KQ‘𝐽) ∈ 𝐴 → 𝐽 ∈ 𝐴)) ⇒ ⊢ (𝐽 ∈ 𝐴 ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴)) | ||
Theorem | t1r0 22359 | A T1 space is R0. That is, the Kolmogorov quotient of a T1 space is also T1 (because they are homeomorphic). (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ (𝐽 ∈ Fre → (KQ‘𝐽) ∈ Fre) | ||
Theorem | ist1-5 22360 | A topological space is T1 iff it is both T0 and R0. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ Fre)) | ||
Theorem | ishaus3 22361 | A topological space is Hausdorff iff it is both T0 and R1 (where R1 means that any two topologically distinct points are separated by neighborhoods). (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ Haus)) | ||
Theorem | nrmreg 22362 | A normal T1 space is regular Hausdorff. In other words, a T4 space is T3 . One can get away with slightly weaker assumptions; see nrmr0reg 22287. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ ((𝐽 ∈ Nrm ∧ 𝐽 ∈ Fre) → 𝐽 ∈ Reg) | ||
Theorem | reghaus 22363 | A regular T0 space is Hausdorff. In other words, a T3 space is T2 . A regular Hausdorff or T0 space is also known as a T3 space. (Contributed by Mario Carneiro, 24-Aug-2015.) |
⊢ (𝐽 ∈ Reg → (𝐽 ∈ Haus ↔ 𝐽 ∈ Kol2)) | ||
Theorem | nrmhaus 22364 | A T1 normal space is Hausdorff. A Hausdorff or T1 normal space is also known as a T4 space. (Contributed by Mario Carneiro, 24-Aug-2015.) |
⊢ (𝐽 ∈ Nrm → (𝐽 ∈ Haus ↔ 𝐽 ∈ Fre)) | ||
Theorem | elmptrab 22365* | Membership in a one-parameter class of sets. (Contributed by Stefan O'Rear, 28-Jul-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) & ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶) & ⊢ (𝑥 ∈ 𝐷 → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐶 ∧ 𝜓)) | ||
Theorem | elmptrab2 22366* | Membership in a one-parameter class of sets, indexed by arbitrary base sets. (Contributed by Stefan O'Rear, 28-Jul-2015.) (Revised by AV, 26-Mar-2021.) |
⊢ 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) & ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶) & ⊢ 𝐵 ∈ V & ⊢ (𝑌 ∈ 𝐶 → 𝑋 ∈ 𝑊) ⇒ ⊢ (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑌 ∈ 𝐶 ∧ 𝜓)) | ||
Theorem | isfbas 22367* | The predicate "𝐹 is a filter base." Note that some authors require filter bases to be closed under pairwise intersections, but that is not necessary under our definition. One advantage of this definition is that tails in a directed set form a filter base under our meaning. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.) |
⊢ (𝐵 ∈ 𝐴 → (𝐹 ∈ (fBas‘𝐵) ↔ (𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)))) | ||
Theorem | fbasne0 22368 | There are no empty filter bases. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.) |
⊢ (𝐹 ∈ (fBas‘𝐵) → 𝐹 ≠ ∅) | ||
Theorem | 0nelfb 22369 | No filter base contains the empty set. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.) |
⊢ (𝐹 ∈ (fBas‘𝐵) → ¬ ∅ ∈ 𝐹) | ||
Theorem | fbsspw 22370 | A filter base on a set is a subset of the power set. (Contributed by Stefan O'Rear, 28-Jul-2015.) |
⊢ (𝐹 ∈ (fBas‘𝐵) → 𝐹 ⊆ 𝒫 𝐵) | ||
Theorem | fbelss 22371 | An element of the filter base is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.) |
⊢ ((𝐹 ∈ (fBas‘𝐵) ∧ 𝑋 ∈ 𝐹) → 𝑋 ⊆ 𝐵) | ||
Theorem | fbdmn0 22372 | The domain of a filter base is nonempty. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
⊢ (𝐹 ∈ (fBas‘𝐵) → 𝐵 ≠ ∅) | ||
Theorem | isfbas2 22373* | The predicate "𝐹 is a filter base." (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
⊢ (𝐵 ∈ 𝐴 → (𝐹 ∈ (fBas‘𝐵) ↔ (𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))))) | ||
Theorem | fbasssin 22374* | A filter base contains subsets of its pairwise intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Jeff Hankins, 1-Dec-2010.) |
⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵)) | ||
Theorem | fbssfi 22375* | A filter base contains subsets of its finite intersections. (Contributed by Mario Carneiro, 26-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ (fi‘𝐹)) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝐴) | ||
Theorem | fbssint 22376* | A filter base contains subsets of its finite intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
⊢ ((𝐹 ∈ (fBas‘𝐵) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴) | ||
Theorem | fbncp 22377 | A filter base does not contain complements of its elements. (Contributed by Mario Carneiro, 26-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹) → ¬ (𝐵 ∖ 𝐴) ∈ 𝐹) | ||
Theorem | fbun 22378* | A necessary and sufficient condition for the union of two filter bases to also be a filter base. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝐹 ∪ 𝐺) ∈ (fBas‘𝑋) ↔ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦))) | ||
Theorem | fbfinnfr 22379 | No filter base containing a finite element is free. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
⊢ ((𝐹 ∈ (fBas‘𝐵) ∧ 𝑆 ∈ 𝐹 ∧ 𝑆 ∈ Fin) → ∩ 𝐹 ≠ ∅) | ||
Theorem | opnfbas 22380* | The collection of open supersets of a nonempty set in a topology is a neighborhoods of the set, one of the motivations for the filter concept. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∈ (fBas‘𝑋)) | ||
Theorem | trfbas2 22381 | Conditions for the trace of a filter base 𝐹 to be a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ ((𝐹 ∈ (fBas‘𝑌) ∧ 𝐴 ⊆ 𝑌) → ((𝐹 ↾t 𝐴) ∈ (fBas‘𝐴) ↔ ¬ ∅ ∈ (𝐹 ↾t 𝐴))) | ||
Theorem | trfbas 22382* | Conditions for the trace of a filter base 𝐹 to be a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ ((𝐹 ∈ (fBas‘𝑌) ∧ 𝐴 ⊆ 𝑌) → ((𝐹 ↾t 𝐴) ∈ (fBas‘𝐴) ↔ ∀𝑣 ∈ 𝐹 (𝑣 ∩ 𝐴) ≠ ∅)) | ||
Syntax | cfil 22383 | Extend class notation with the set of filters on a set. |
class Fil | ||
Definition | df-fil 22384* | The set of filters on a set. Definition 1 (axioms FI, FIIa, FIIb, FIII) of [BourbakiTop1] p. I.36. Filters are used to define the concept of limit in the general case. They are a generalization of the idea of neighborhoods. Suppose you are in ℝ. With neighborhoods you can express the idea of a variable that tends to a specific number but you can't express the idea of a variable that tends to infinity. Filters relax the "axioms" of neighborhoods and then succeed in expressing the idea of something that tends to infinity. Filters were invented by Cartan in 1937 and made famous by Bourbaki in his treatise. A notion similar to the notion of filter is the concept of net invented by Moore and Smith in 1922. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
⊢ Fil = (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)}) | ||
Theorem | isfil 22385* | The predicate "is a filter." (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.) |
⊢ (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹))) | ||
Theorem | filfbas 22386 | A filter is a filter base. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.) |
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋)) | ||
Theorem | 0nelfil 22387 | The empty set doesn't belong to a filter. (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.) |
⊢ (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹) | ||
Theorem | fileln0 22388 | An element of a filter is nonempty. (Contributed by FL, 24-May-2011.) (Revised by Mario Carneiro, 28-Jul-2015.) |
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ≠ ∅) | ||
Theorem | filsspw 22389 | A filter is a subset of the power set of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.) |
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋) | ||
Theorem | filelss 22390 | An element of a filter is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.) |
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ 𝑋) | ||
Theorem | filss 22391 | A filter is closed under taking supersets. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝐵)) → 𝐵 ∈ 𝐹) | ||
Theorem | filin 22392 | A filter is closed under taking intersections. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → (𝐴 ∩ 𝐵) ∈ 𝐹) | ||
Theorem | filtop 22393 | The underlying set belongs to the filter. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹) | ||
Theorem | isfil2 22394* | Derive the standard axioms of a filter. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
⊢ (𝐹 ∈ (Fil‘𝑋) ↔ ((𝐹 ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ 𝐹 ∧ 𝑋 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹)) | ||
Theorem | isfildlem 22395* | Lemma for isfild 22396. (Contributed by Mario Carneiro, 1-Dec-2013.) |
⊢ (𝜑 → (𝑥 ∈ 𝐹 ↔ (𝑥 ⊆ 𝐴 ∧ 𝜓))) & ⊢ (𝜑 → 𝐴 ∈ V) ⇒ ⊢ (𝜑 → (𝐵 ∈ 𝐹 ↔ (𝐵 ⊆ 𝐴 ∧ [𝐵 / 𝑥]𝜓))) | ||
Theorem | isfild 22396* | Sufficient condition for a set of the form {𝑥 ∈ 𝒫 𝐴 ∣ 𝜑} to be a filter. (Contributed by Mario Carneiro, 1-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
⊢ (𝜑 → (𝑥 ∈ 𝐹 ↔ (𝑥 ⊆ 𝐴 ∧ 𝜓))) & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → [𝐴 / 𝑥]𝜓) & ⊢ (𝜑 → ¬ [∅ / 𝑥]𝜓) & ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝑦) → ([𝑧 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜓)) & ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐴) → (([𝑦 / 𝑥]𝜓 ∧ [𝑧 / 𝑥]𝜓) → [(𝑦 ∩ 𝑧) / 𝑥]𝜓)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (Fil‘𝐴)) | ||
Theorem | filfi 22397 | A filter is closed under taking intersections. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
⊢ (𝐹 ∈ (Fil‘𝑋) → (fi‘𝐹) = 𝐹) | ||
Theorem | filinn0 22398 | The intersection of two elements of a filter can't be empty. (Contributed by FL, 16-Sep-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → (𝐴 ∩ 𝐵) ≠ ∅) | ||
Theorem | filintn0 22399 | A filter has the finite intersection property. Remark below Definition 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 20-Sep-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ⊆ 𝐹 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ 𝐴 ≠ ∅) | ||
Theorem | filn0 22400 | The empty set is not a filter. Remark below Definition 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 30-Oct-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅) |
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