Home Metamath Proof ExplorerTheorem List (p. 217 of 429) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27903) Hilbert Space Explorer (27904-29428) Users' Mathboxes (29429-42879)

Theorem List for Metamath Proof Explorer - 21601-21700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremregr1 21601 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)

Theoremkqreg 21602 The Kolmogorov quotient of a regular space is regular. By regr1 21601 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)

Theoremkqnrm 21603 The Kolmogorov quotient of a normal space is normal. (Contributed by Mario Carneiro, 25-Aug-2015.)
(𝐽 ∈ Nrm ↔ (KQ‘𝐽) ∈ Nrm)

12.1.21  Homeomorphisms

Syntaxchmeo 21604 Extend class notation with the class of all homeomorphisms.
class Homeo

Syntaxchmph 21605 Extend class notation with the relation "is homeomorphic to.".
class

Definitiondf-hmeo 21606* 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)})

Definitiondf-hmph 21607 Definition of the relation 𝑥 is homeomorphic to 𝑦. (Contributed by FL, 14-Feb-2007.)
≃ = (Homeo “ (V ∖ 1𝑜))

Theoremhmeofn 21608 The set of homeomorphisms is a function on topologies. (Contributed by Mario Carneiro, 23-Aug-2015.)
Homeo Fn (Top × Top)

Theoremhmeofval 21609* 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)}

Theoremishmeo 21610 The predicate F is a homeomorphism between topology 𝐽 and topology 𝐾. Proposition 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)))

Theoremhmeocn 21611 A homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.)
(𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))

Theoremhmeocnvcn 21612 The converse of a homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.)
(𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))

Theoremhmeocnv 21613 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))

Theoremhmeof1o2 21614 A homeomorphism is a 1-1-onto mapping. (Contributed by Mario Carneiro, 22-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽Homeo𝐾)) → 𝐹:𝑋1-1-onto𝑌)

Theoremhmeof1o 21615 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:𝑋1-1-onto𝑌)

Theoremhmeoima 21616 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) ∧ 𝐴𝐽) → (𝐹𝐴) ∈ 𝐾)

Theoremhmeoopn 21617 Homeomorphisms preserve openness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
𝑋 = 𝐽       ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐴𝐽 ↔ (𝐹𝐴) ∈ 𝐾))

Theoremhmeocld 21618 Homeomorphisms preserve closedness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
𝑋 = 𝐽       ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (𝐹𝐴) ∈ (Clsd‘𝐾)))

Theoremhmeocls 21619 Homeomorphisms preserve closures. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝑋 = 𝐽       ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((cls‘𝐾)‘(𝐹𝐴)) = (𝐹 “ ((cls‘𝐽)‘𝐴)))

Theoremhmeontr 21620 Homeomorphisms preserve interiors. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝑋 = 𝐽       ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((int‘𝐾)‘(𝐹𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴)))

Theoremhmeoimaf1o 21621* The function mapping open sets to their images under a homeomorphism is a bijection of topologies. (Contributed by Mario Carneiro, 10-Sep-2015.)
𝐺 = (𝑥𝐽 ↦ (𝐹𝑥))       (𝐹 ∈ (𝐽Homeo𝐾) → 𝐺:𝐽1-1-onto𝐾)

Theoremhmeores 21622 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) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐽t 𝑌)Homeo(𝐾t (𝐹𝑌))))

Theoremhmeoco 21623 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)) → (𝐺𝐹) ∈ (𝐽Homeo𝐿))

Theoremidhmeo 21624 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))

Theoremhmeocnvb 21625 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)))

Theoremhmeoqtop 21626 A homeomorphism is a quotient map. (Contributed by Mario Carneiro, 25-Aug-2015.)
(𝐹 ∈ (𝐽Homeo𝐾) → 𝐾 = (𝐽 qTop 𝐹))

Theoremhmph 21627 Express the predicate 𝐽 is homeomorphic to 𝐾. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝐽𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)

Theoremhmphi 21628 If there is a homeomorphism between spaces, then the spaces are homeomorphic. (Contributed by Mario Carneiro, 23-Aug-2015.)
(𝐹 ∈ (𝐽Homeo𝐾) → 𝐽𝐾)

Theoremhmphtop 21629 Reverse closure for the homeomorphic predicate. (Contributed by Mario Carneiro, 22-Aug-2015.)
(𝐽𝐾 → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))

Theoremhmphtop1 21630 The relation "being homeomorphic to" implies the operands are topologies. (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
(𝐽𝐾𝐽 ∈ Top)

Theoremhmphtop2 21631 The relation "being homeomorphic to" implies the operands are topologies. (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
(𝐽𝐾𝐾 ∈ Top)

Theoremhmphref 21632 "Is homeomorphic to" is reflexive. (Contributed by FL, 25-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
(𝐽 ∈ Top → 𝐽𝐽)

Theoremhmphsym 21633 "Is homeomorphic to" is symmetric. (Contributed by FL, 8-Mar-2007.) (Proof shortened by Mario Carneiro, 30-May-2014.)
(𝐽𝐾𝐾𝐽)

Theoremhmphtr 21634 "Is homeomorphic to" is transitive. (Contributed by FL, 9-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
((𝐽𝐾𝐾𝐿) → 𝐽𝐿)

Theoremhmpher 21635 "Is homeomorphic to" is an equivalence relation. (Contributed by FL, 22-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
≃ Er Top

Theoremhmphen 21636 Homeomorphisms preserve the cardinality of the topologies. (Contributed by FL, 1-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
(𝐽𝐾𝐽𝐾)

Theoremhmphsymb 21637 "Is homeomorphic to" is symmetric. (Contributed by FL, 22-Feb-2007.)
(𝐽𝐾𝐾𝐽)

Theoremhaushmphlem 21638* Lemma for haushmph 21643 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.)
(𝐽𝐴𝐽 ∈ Top)    &   ((𝐽𝐴𝑓: 𝐾1-1 𝐽𝑓 ∈ (𝐾 Cn 𝐽)) → 𝐾𝐴)       (𝐽𝐾 → (𝐽𝐴𝐾𝐴))

Theoremcmphmph 21639 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.)
(𝐽𝐾 → (𝐽 ∈ Comp → 𝐾 ∈ Comp))

Theoremconnhmph 21640 Connectedness is a topological property. (Contributed by Jeff Hankins, 3-Jul-2009.)
(𝐽𝐾 → (𝐽 ∈ Conn → 𝐾 ∈ Conn))

Theoremt0hmph 21641 T0 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
(𝐽𝐾 → (𝐽 ∈ Kol2 → 𝐾 ∈ Kol2))

Theoremt1hmph 21642 T1 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
(𝐽𝐾 → (𝐽 ∈ Fre → 𝐾 ∈ Fre))

Theoremhaushmph 21643 Hausdorff-ness is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
(𝐽𝐾 → (𝐽 ∈ Haus → 𝐾 ∈ Haus))

Theoremreghmph 21644 Regularity is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
(𝐽𝐾 → (𝐽 ∈ Reg → 𝐾 ∈ Reg))

Theoremnrmhmph 21645 Normality is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
(𝐽𝐾 → (𝐽 ∈ Nrm → 𝐾 ∈ Nrm))

Theoremhmph0 21646 A topology homeomorphic to the empty set is empty. (Contributed by FL, 18-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
(𝐽 ≃ {∅} ↔ 𝐽 = {∅})

Theoremhmphdis 21647 Homeomorphisms preserve topological discretion. (Contributed by Mario Carneiro, 10-Sep-2015.)
𝑋 = 𝐽       (𝐽 ≃ 𝒫 𝐴𝐽 = 𝒫 𝑋)

Theoremhmphindis 21648 Homeomorphisms preserve topological indiscretion. (Contributed by FL, 18-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
𝑋 = 𝐽       (𝐽 ≃ {∅, 𝐴} → 𝐽 = {∅, 𝑋})

Theoremindishmph 21649 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.)
(𝐴𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})

Theoremhmphen2 21650 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       (𝐽𝐾𝑋𝑌)

Theoremcmphaushmeo 21651 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:𝑋1-1-onto𝑌))

Theoremordthmeolem 21652 Lemma for ordthmeo 21653. (Contributed by Mario Carneiro, 9-Sep-2015.)
𝑋 = dom 𝑅    &   𝑌 = dom 𝑆       ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝐹 ∈ ((ordTop‘𝑅) Cn (ordTop‘𝑆)))

Theoremordthmeo 21653 An order isomorphism is a homeomorphism on the respective order topologies. (Contributed by Mario Carneiro, 9-Sep-2015.)
𝑋 = dom 𝑅    &   𝑌 = dom 𝑆       ((𝑅𝑉𝑆𝑊𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝐹 ∈ ((ordTop‘𝑅)Homeo(ordTop‘𝑆)))

Theoremtxhmeo 21654* 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    &   (𝜑𝐹 ∈ (𝐽Homeo𝐿))    &   (𝜑𝐺 ∈ (𝐾Homeo𝑀))       (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩) ∈ ((𝐽 ×t 𝐾)Homeo(𝐿 ×t 𝑀)))

Theoremtxswaphmeolem 21655* Show inverse for the "swap components" operation on a Cartesian product. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)) = ( I ↾ (𝑋 × 𝑌))

Theoremtxswaphmeo 21656* There is a homeomorphism from 𝑋 × 𝑌 to 𝑌 × 𝑋. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ ((𝐽 ×t 𝐾)Homeo(𝐾 ×t 𝐽)))

Theorempt1hmeo 21657* 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⟩})    &   (𝜑𝐴𝑉)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))       (𝜑 → (𝑥𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽Homeo𝐾))

Theoremptuncnv 21658* 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 = (∏t𝐹)    &   𝐾 = (∏t‘(𝐹𝐴))    &   𝐿 = (∏t‘(𝐹𝐵))    &   𝐺 = (𝑥𝑋, 𝑦𝑌 ↦ (𝑥𝑦))    &   (𝜑𝐶𝑉)    &   (𝜑𝐹:𝐶⟶Top)    &   (𝜑𝐶 = (𝐴𝐵))    &   (𝜑 → (𝐴𝐵) = ∅)       (𝜑𝐺 = (𝑧 𝐽 ↦ ⟨(𝑧𝐴), (𝑧𝐵)⟩))

Theoremptunhmeo 21659* 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 (𝐴𝐵) · (𝐴𝐶) = 𝐴↑(𝐵 + 𝐶). (Contributed by Mario Carneiro, 8-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
𝑋 = 𝐾    &   𝑌 = 𝐿    &   𝐽 = (∏t𝐹)    &   𝐾 = (∏t‘(𝐹𝐴))    &   𝐿 = (∏t‘(𝐹𝐵))    &   𝐺 = (𝑥𝑋, 𝑦𝑌 ↦ (𝑥𝑦))    &   (𝜑𝐶𝑉)    &   (𝜑𝐹:𝐶⟶Top)    &   (𝜑𝐶 = (𝐴𝐵))    &   (𝜑 → (𝐴𝐵) = ∅)       (𝜑𝐺 ∈ ((𝐾 ×t 𝐿)Homeo𝐽))

Theoremxpstopnlem1 21660* The function 𝐹 used in xpsval 16279 is a homeomorphism from the binary product topology to the indexed product topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐹 = (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))       (𝜑𝐹 ∈ ((𝐽 ×t 𝐾)Homeo(∏t({𝐽} +𝑐 {𝐾}))))

Theoremxpstps 21661 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)

Theoremxpstopnlem2 21662* Lemma for xpstopn 21663. (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 = (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))       ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑂 = (𝐽 ×t 𝐾))

Theoremxpstopn 21663 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))

Theoremptcmpfi 21664 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) → (∏t𝐹) ∈ Comp)

Theoremxkocnv 21665* The inverse of the "currying" function 𝐹 is the uncurrying function. (Contributed by Mario Carneiro, 13-Apr-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   𝐹 = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↦ (𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑥𝑓𝑦))))    &   (𝜑𝐽 ∈ 𝑛-Locally Comp)    &   (𝜑𝐾 ∈ 𝑛-Locally Comp)    &   (𝜑𝐿 ∈ Top)       (𝜑𝐹 = (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ↦ (𝑥𝑋, 𝑦𝑌 ↦ ((𝑔𝑥)‘𝑦))))

Theoremxkohmeo 21666* 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 21511, 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.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   𝐹 = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↦ (𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑥𝑓𝑦))))    &   (𝜑𝐽 ∈ 𝑛-Locally Comp)    &   (𝜑𝐾 ∈ 𝑛-Locally Comp)    &   (𝜑𝐿 ∈ Top)       (𝜑𝐹 ∈ ((𝐿 ^ko (𝐽 ×t 𝐾))Homeo((𝐿 ^ko 𝐾) ^ko 𝐽)))

Theoremqtopf1 21667 If a quotient map is injective, then it is a homeomorphism. (Contributed by Mario Carneiro, 25-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹:𝑋1-1𝑌)       (𝜑𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)))

Theoremqtophmeo 21668* 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.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹:𝑋onto𝑌)    &   (𝜑𝐺:𝑋onto𝑌)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐺𝑥) = (𝐺𝑦)))       (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓𝐹))

Theoremt0kq 21669* 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 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹 ∈ (𝐽Homeo(KQ‘𝐽))))

Theoremkqhmph 21670 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))

Theoremist1-5lem 21671 Lemma for ist1-5 21673 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.)
(𝐽𝐴𝐽 ∈ Kol2)    &   (𝐽 ≃ (KQ‘𝐽) → (𝐽𝐴 → (KQ‘𝐽) ∈ 𝐴))    &   ((KQ‘𝐽) ≃ 𝐽 → ((KQ‘𝐽) ∈ 𝐴𝐽𝐴))       (𝐽𝐴 ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴))

Theoremt1r0 21672 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)

Theoremist1-5 21673 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))

Theoremishaus3 21674 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))

Theoremnrmreg 21675 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 21600. (Contributed by Mario Carneiro, 25-Aug-2015.)
((𝐽 ∈ Nrm ∧ 𝐽 ∈ Fre) → 𝐽 ∈ Reg)

Theoremreghaus 21676 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))

Theoremnrmhaus 21677 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))

12.2  Filters and filter bases

12.2.1  Filter bases

Theoremelmptrab 21678* Membership in a one-parameter class of sets. (Contributed by Stefan O'Rear, 28-Jul-2015.)
𝐹 = (𝑥𝐷 ↦ {𝑦𝐵𝜑})    &   ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑𝜓))    &   (𝑥 = 𝑋𝐵 = 𝐶)    &   (𝑥𝐷𝐵𝑉)       (𝑌 ∈ (𝐹𝑋) ↔ (𝑋𝐷𝑌𝐶𝜓))

Theoremelmptrab2 21679* 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 ↦ {𝑦𝐵𝜑})    &   ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑𝜓))    &   (𝑥 = 𝑋𝐵 = 𝐶)    &   𝐵 ∈ V    &   (𝑌𝐶𝑋𝑊)       (𝑌 ∈ (𝐹𝑋) ↔ (𝑌𝐶𝜓))

Theoremisfbas 21680* 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.)
(𝐵𝐴 → (𝐹 ∈ (fBas‘𝐵) ↔ (𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥𝐹𝑦𝐹 (𝐹 ∩ 𝒫 (𝑥𝑦)) ≠ ∅))))

Theoremfbasne0 21681 There are no empty filter bases. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
(𝐹 ∈ (fBas‘𝐵) → 𝐹 ≠ ∅)

Theorem0nelfb 21682 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)

Theoremfbsspw 21683 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)

Theoremfbelss 21684 An element of the filter base is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
((𝐹 ∈ (fBas‘𝐵) ∧ 𝑋𝐹) → 𝑋𝐵)

Theoremfbdmn0 21685 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 ≠ ∅)

Theoremisfbas2 21686* The predicate "𝐹 is a filter base." (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)
(𝐵𝐴 → (𝐹 ∈ (fBas‘𝐵) ↔ (𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥𝐹𝑦𝐹𝑧𝐹 𝑧 ⊆ (𝑥𝑦)))))

Theoremfbasssin 21687* 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) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵))

Theoremfbssfi 21688* 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)) → ∃𝑥𝐹 𝑥𝐴)

Theoremfbssint 21689* 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) ∧ 𝐴𝐹𝐴 ∈ Fin) → ∃𝑥𝐹 𝑥 𝐴)

Theoremfbncp 21690 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) ∧ 𝐴𝐹) → ¬ (𝐵𝐴) ∈ 𝐹)

Theoremfbun 21691* 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)) → ((𝐹𝐺) ∈ (fBas‘𝑋) ↔ ∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))

Theoremfbfinnfr 21692 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) ∧ 𝑆𝐹𝑆 ∈ Fin) → 𝐹 ≠ ∅)

Theoremopnfbas 21693* 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 ∧ 𝑆𝑋𝑆 ≠ ∅) → {𝑥𝐽𝑆𝑥} ∈ (fBas‘𝑋))

Theoremtrfbas2 21694 Conditions for the trace of a filter base 𝐹 to be a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)
((𝐹 ∈ (fBas‘𝑌) ∧ 𝐴𝑌) → ((𝐹t 𝐴) ∈ (fBas‘𝐴) ↔ ¬ ∅ ∈ (𝐹t 𝐴)))

Theoremtrfbas 21695* Conditions for the trace of a filter base 𝐹 to be a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)
((𝐹 ∈ (fBas‘𝑌) ∧ 𝐴𝑌) → ((𝐹t 𝐴) ∈ (fBas‘𝐴) ↔ ∀𝑣𝐹 (𝑣𝐴) ≠ ∅))

12.2.2  Filters

Syntaxcfil 21696 Extend class notation with the set of filters on a set.
class Fil

Definitiondf-fil 21697* 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 . 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) ≠ ∅ → 𝑥𝑓)})

Theoremisfil 21698* 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) ≠ ∅ → 𝑥𝐹)))

Theoremfilfbas 21699 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))

Theorem0nelfil 21700 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)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
 Copyright terms: Public domain < Previous  Next >