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Theorem List for Metamath Proof Explorer - 21501-21600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremqtoprest 21501 If 𝐴 is a saturated open or closed set (where saturated means that 𝐴 = (𝐹𝑈) 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.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹:𝑋onto𝑌)    &   (𝜑𝑈𝑌)    &   (𝜑𝐴 = (𝐹𝑈))    &   (𝜑 → (𝐴𝐽𝐴 ∈ (Clsd‘𝐽)))       (𝜑 → ((𝐽 qTop 𝐹) ↾t 𝑈) = ((𝐽t 𝐴) qTop (𝐹𝐴)))

Theoremqtopomap 21502* 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.)
(𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑 → ran 𝐹 = 𝑌)    &   ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ 𝐾)       (𝜑𝐾 = (𝐽 qTop 𝐹))

Theoremqtopcmap 21503* 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.)
(𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑 → ran 𝐹 = 𝑌)    &   ((𝜑𝑥 ∈ (Clsd‘𝐽)) → (𝐹𝑥) ∈ (Clsd‘𝐾))       (𝜑𝐾 = (𝐽 qTop 𝐹))

Theoremimastopn 21504 The topology of an image structure. (Contributed by Mario Carneiro, 27-Aug-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   (𝜑𝑅𝑊)    &   𝐽 = (TopOpen‘𝑅)    &   𝑂 = (TopOpen‘𝑈)       (𝜑𝑂 = (𝐽 qTop 𝐹))

Theoremimastps 21505 The image of a topological space under a function is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   (𝜑𝑅 ∈ TopSp)       (𝜑𝑈 ∈ TopSp)

Theoremqustps 21506 A quotient structure is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
(𝜑𝑈 = (𝑅 /s 𝐸))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐸𝑊)    &   (𝜑𝑅 ∈ TopSp)       (𝜑𝑈 ∈ TopSp)

Theoremkqfval 21507* Value of the function appearing in df-kq 21478. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽𝑉𝐴𝑋) → (𝐹𝐴) = {𝑦𝐽𝐴𝑦})

Theoremkqfeq 21508* Two points in the Kolmogorov quotient are equal iff the original points are topologically indistinguishable. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽𝑉𝐴𝑋𝐵𝑋) → ((𝐹𝐴) = (𝐹𝐵) ↔ ∀𝑦𝐽 (𝐴𝑦𝐵𝑦)))

Theoremkqffn 21509* The topological indistinguishability map is a function on the base. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       (𝐽𝑉𝐹 Fn 𝑋)

Theoremkqval 21510* Value of the quotient topology function. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))

Theoremkqtopon 21511* The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))

Theoremkqid 21512* The topological indistinguishability map is a continuous function into the Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))

Theoremist0-4 21513* The topological indistinguishability map is injective iff the space is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹:𝑋1-1→V))

Theoremkqfvima 21514* When the image set is open, the quotient map satisfies a partial converse to fnfvima 6481, which is normally only true for injective functions. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑋) → (𝐴𝑈 ↔ (𝐹𝐴) ∈ (𝐹𝑈)))

Theoremkqsat 21515* Any open set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 21501). (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹 “ (𝐹𝑈)) = 𝑈)

Theoremkqdisj 21516* A version of imain 5962 for the topological indistinguishability map. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → ((𝐹𝑈) ∩ (𝐹 “ (𝐴𝑈))) = ∅)

Theoremkqcldsat 21517* Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 21501). (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹 “ (𝐹𝑈)) = 𝑈)

Theoremkqopn 21518* The topological indistinguishability map is an open map. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽) → (𝐹𝑈) ∈ (KQ‘𝐽))

Theoremkqcld 21519* The topological indistinguishability map is a closed map. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝐹𝑈) ∈ (Clsd‘(KQ‘𝐽)))

Theoremkqt0lem 21520* Lemma for kqt0 21530. (Contributed by Mario Carneiro, 23-Mar-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ Kol2)

Theoremisr0 21521* 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 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       (𝐽 ∈ (TopOn‘𝑋) → ((KQ‘𝐽) ∈ Fre ↔ ∀𝑧𝑋𝑤𝑋 (∀𝑜𝐽 (𝑧𝑜𝑤𝑜) → ∀𝑜𝐽 (𝑧𝑜𝑤𝑜))))

Theoremr0cld 21522* 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 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → {𝑧𝑋 ∣ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)} ∈ (Clsd‘𝐽))

Theoremregr1lem 21523* Lemma for regr1 21534. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐽 ∈ Reg)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝑈𝐽)    &   (𝜑 → ¬ ∃𝑚 ∈ (KQ‘𝐽)∃𝑛 ∈ (KQ‘𝐽)((𝐹𝐴) ∈ 𝑚 ∧ (𝐹𝐵) ∈ 𝑛 ∧ (𝑚𝑛) = ∅))       (𝜑 → (𝐴𝑈𝐵𝑈))

Theoremregr1lem2 21524* A Kolmogorov quotient of a regular space is Hausdorff. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Haus)

Theoremkqreglem1 21525* A Kolmogorov quotient of a regular space is regular. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Reg)

Theoremkqreglem2 21526* If the Kolmogorov quotient of a space is regular then so is the original space. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Reg)

Theoremkqnrmlem1 21527* A Kolmogorov quotient of a normal space is normal. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Nrm)

Theoremkqnrmlem2 21528* If the Kolmogorov quotient of a space is normal then so is the original space. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})       ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Nrm)

Theoremkqtop 21529 The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
(𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top)

Theoremkqt0 21530 The Kolmogorov quotient is T0 even if the original topology is not. (Contributed by Mario Carneiro, 25-Aug-2015.)
(𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Kol2)

Theoremkqf 21531 The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ:Top⟶Kol2

Theoremr0sep 21532* The separation property of an R0 space. (Contributed by Mario Carneiro, 25-Aug-2015.)
(((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝐴𝑋𝐵𝑋)) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → ∀𝑜𝐽 (𝐴𝑜𝐵𝑜)))

Theoremnrmr0reg 21533 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)

Theoremregr1 21534 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 21535 The Kolmogorov quotient of a regular space is regular. By regr1 21534 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 21536 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 21537 Extend class notation with the class of all homeomorphisms.
class Homeo

Syntaxchmph 21538 Extend class notation with the relation "is homeomorphic to.".
class

Definitiondf-hmeo 21539* 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 21540 Definition of the relation 𝑥 is homeomorphic to 𝑦. (Contributed by FL, 14-Feb-2007.)
≃ = (Homeo “ (V ∖ 1𝑜))

Theoremhmeofn 21541 The set of homeomorphisms is a function on topologies. (Contributed by Mario Carneiro, 23-Aug-2015.)
Homeo Fn (Top × Top)

Theoremhmeofval 21542* 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 21543 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 21544 A homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.)
(𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))

Theoremhmeocnvcn 21545 The converse of a homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.)
(𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))

Theoremhmeocnv 21546 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 21547 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 21548 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 21549 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 21550 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 21551 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 21552 Homeomorphisms preserve closures. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝑋 = 𝐽       ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((cls‘𝐾)‘(𝐹𝐴)) = (𝐹 “ ((cls‘𝐽)‘𝐴)))

Theoremhmeontr 21553 Homeomorphisms preserve interiors. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝑋 = 𝐽       ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((int‘𝐾)‘(𝐹𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴)))

Theoremhmeoimaf1o 21554* 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 21555 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 21556 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 21557 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 21558 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 21559 A homeomorphism is a quotient map. (Contributed by Mario Carneiro, 25-Aug-2015.)
(𝐹 ∈ (𝐽Homeo𝐾) → 𝐾 = (𝐽 qTop 𝐹))

Theoremhmph 21560 Express the predicate 𝐽 is homeomorphic to 𝐾. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝐽𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)

Theoremhmphi 21561 If there is a homeomorphism between spaces, then the spaces are homeomorphic. (Contributed by Mario Carneiro, 23-Aug-2015.)
(𝐹 ∈ (𝐽Homeo𝐾) → 𝐽𝐾)

Theoremhmphtop 21562 Reverse closure for the homeomorphic predicate. (Contributed by Mario Carneiro, 22-Aug-2015.)
(𝐽𝐾 → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))

Theoremhmphtop1 21563 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 21564 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 21565 "Is homeomorphic to" is reflexive. (Contributed by FL, 25-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
(𝐽 ∈ Top → 𝐽𝐽)

Theoremhmphsym 21566 "Is homeomorphic to" is symmetric. (Contributed by FL, 8-Mar-2007.) (Proof shortened by Mario Carneiro, 30-May-2014.)
(𝐽𝐾𝐾𝐽)

Theoremhmphtr 21567 "Is homeomorphic to" is transitive. (Contributed by FL, 9-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
((𝐽𝐾𝐾𝐿) → 𝐽𝐿)

Theoremhmpher 21568 "Is homeomorphic to" is an equivalence relation. (Contributed by FL, 22-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
≃ Er Top

Theoremhmphen 21569 Homeomorphisms preserve the cardinality of the topologies. (Contributed by FL, 1-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
(𝐽𝐾𝐽𝐾)

Theoremhmphsymb 21570 "Is homeomorphic to" is symmetric. (Contributed by FL, 22-Feb-2007.)
(𝐽𝐾𝐾𝐽)

Theoremhaushmphlem 21571* Lemma for haushmph 21576 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 21572 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 21573 Connectedness is a topological property. (Contributed by Jeff Hankins, 3-Jul-2009.)
(𝐽𝐾 → (𝐽 ∈ Conn → 𝐾 ∈ Conn))

Theoremt0hmph 21574 T0 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
(𝐽𝐾 → (𝐽 ∈ Kol2 → 𝐾 ∈ Kol2))

Theoremt1hmph 21575 T1 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
(𝐽𝐾 → (𝐽 ∈ Fre → 𝐾 ∈ Fre))

Theoremhaushmph 21576 Hausdorff-ness is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
(𝐽𝐾 → (𝐽 ∈ Haus → 𝐾 ∈ Haus))

Theoremreghmph 21577 Regularity is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
(𝐽𝐾 → (𝐽 ∈ Reg → 𝐾 ∈ Reg))

Theoremnrmhmph 21578 Normality is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
(𝐽𝐾 → (𝐽 ∈ Nrm → 𝐾 ∈ Nrm))

Theoremhmph0 21579 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 21580 Homeomorphisms preserve topological discretion. (Contributed by Mario Carneiro, 10-Sep-2015.)
𝑋 = 𝐽       (𝐽 ≃ 𝒫 𝐴𝐽 = 𝒫 𝑋)

Theoremhmphindis 21581 Homeomorphisms preserve topological indiscretion. (Contributed by FL, 18-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
𝑋 = 𝐽       (𝐽 ≃ {∅, 𝐴} → 𝐽 = {∅, 𝑋})

Theoremindishmph 21582 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 21583 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 21584 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 21585 Lemma for ordthmeo 21586. (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 21586 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 21587* 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 21588* 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 21589* 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 21590* 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 21591* 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 21592* 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 21593* The function 𝐹 used in xpsval 16213 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 21594 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 21595* Lemma for xpstopn 21596. (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 21596 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 21597 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 21598* 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 21599* 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 21444, 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 21600 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)))

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