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Theorem List for Metamath Proof Explorer - 17701-17800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcnmptk1 17701* The composition of a curried function with a one-arg function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.)
TopOn       TopOn       TopOn

Theoremcnmpt1k 17702* The composition of a one-arg function with a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn       TopOn       TopOn

Theoremcnmptkk 17703* The composition of two curried functions is jointly continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn       TopOn       TopOn       𝑛Locally

Theoremxkofvcn 17704* Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 17676.) (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
𝑛Locally

Theoremcnmptk1p 17705* The evaluation of a curried function by a one-arg function is jointly continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn       TopOn       𝑛Locally

Theoremcnmptk2 17706* The uncurrying of a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn       TopOn       𝑛Locally

Theoremxkoinjcn 17707* Continuity of "injection", i.e. currying, as a function on continuous function spaces. (Contributed by Mario Carneiro, 23-Mar-2015.)
TopOn TopOn

Theoremcnmpt2k 17708* The currying of a two-argument function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.)
TopOn       TopOn

Theoremtxcon 17709 The topological product of two connected spaces is connected. (Contributed by Mario Carneiro, 29-Mar-2015.)

Theoremimasnopn 17710 If a relation graph is opened, then an image set of a singleton is also opened. Corollary of proposition 4 of [BourbakiTop1] p. I.26. (Contributed by Thierry Arnoux, 14-Jan-2018.)

Theoremimasncld 17711 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.)

Theoremimasncls 17712 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.)

11.1.19  Quotient maps and quotient topology

Syntaxckq 17713 Extend class notation with the Kolmogorov quotient function.
KQ

Definitiondf-kq 17714* 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 qTop

Theoremqtopval 17715* Value of the quotient topology function. (Contributed by Mario Carneiro, 23-Mar-2015.)
qTop

Theoremqtopval2 17716* Value of the quotient topology function when is a function on the base set. (Contributed by Mario Carneiro, 23-Mar-2015.)
qTop

Theoremelqtop 17717 Value of the quotient topology function. (Contributed by Mario Carneiro, 23-Mar-2015.)
qTop

Theoremqtopres 17718 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 be a function with domain . (Contributed by Mario Carneiro, 23-Mar-2015.)
qTop qTop

Theoremqtoptop2 17719 The quotient topology is a topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
qTop

Theoremqtoptop 17720 The quotient topology is a topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
qTop

Theoremelqtop2 17721 Value of the quotient topology function. (Contributed by Mario Carneiro, 9-Apr-2015.)
qTop

Theoremqtopuni 17722 The base set of the quotient topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
qTop

Theoremelqtop3 17723 Value of the quotient topology function. (Contributed by Mario Carneiro, 9-Apr-2015.)
TopOn qTop

Theoremqtoptopon 17724 The base set of the quotient topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
TopOn qTop TopOn

Theoremqtopid 17725 A quotient map is a continuous function into its quotient topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
TopOn qTop

Theoremidqtop 17726 The quotient topology induced by the identity function is the original topology. (Contributed by Mario Carneiro, 30-Aug-2015.)
TopOn qTop

Theoremqtopcmplem 17727 Lemma for qtopcmp 17728 and qtopcon 17729. (Contributed by Mario Carneiro, 24-Mar-2015.)
qTop qTop qTop        qTop

Theoremqtopcmp 17728 A quotient of a compact space is compact. (Contributed by Mario Carneiro, 24-Mar-2015.)
qTop

Theoremqtopcon 17729 A quotient of a connected space is connected. (Contributed by Mario Carneiro, 24-Mar-2015.)
qTop

Theoremqtopkgen 17730 A quotient of a compactly generated space is compactly generated. (Contributed by Mario Carneiro, 9-Apr-2015.)
𝑘Gen qTop 𝑘Gen

Theorembasqtop 17731 An injection maps bases to bases. (Contributed by Mario Carneiro, 27-Aug-2015.)
qTop

Theoremtgqtop 17732 An injection maps generated topologies to each other. (Contributed by Mario Carneiro, 27-Aug-2015.)
qTop qTop

Theoremqtopcld 17733 The property of being a closed set in the quotient topology. (Contributed by Mario Carneiro, 24-Mar-2015.)
TopOn qTop

Theoremqtopcn 17734 Universal property of a quotient map. (Contributed by Mario Carneiro, 23-Mar-2015.)
TopOn TopOn qTop

Theoremqtopss 17735 A surjective continuous function from to induces a topology qTop on the base set of . This topology is in general finer than . Together with qtopid 17725, this implies that qTop is the finest topology making continuous, i.e. the final topology with respect to the family . (Contributed by Mario Carneiro, 24-Mar-2015.)
TopOn qTop

Theoremqtopeu 17736* Universal property of the quotient topology. If is a function from to which is equal on all equivalent elements under , then there is a unique continuous map such that , and we say that "passes to the quotient". (Contributed by Mario Carneiro, 24-Mar-2015.)
TopOn                            qTop

Theoremqtoprest 17737 If is a saturated open or closed set (where saturated means that for some ), then the restriction of the quotient map to is a quotient map. (Contributed by Mario Carneiro, 24-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn                                   qTop t t qTop

Theoremqtopomap 17738* If 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.)
TopOn                            qTop

Theoremqtopcmap 17739* If 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.)
TopOn                            qTop

Theoremimastopn 17740 The topology of an image structure. (Contributed by Mario Carneiro, 27-Aug-2015.)
s                                           qTop

Theoremimastps 17741 The image of a topological space under a function is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
s

Theoremdivstps 17742 A quotient structure is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
s

Theoremkqfval 17743* Value of the function appearing in df-kq 17714. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremkqfeq 17744* Two points in the Kolmogorov quotient are equal iff the original points are topologically indistinguishable. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremkqffn 17745* The topological indistinguishability map is a function on the base. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremkqval 17746* Value of the quotient topology function. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ qTop

Theoremkqtopon 17747* The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ TopOn

Theoremkqid 17748* The topological indistinguishability map is a continuous function into the Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremist0-4 17749* The topological indistinguishability map is injective iff the space is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn

Theoremkqfvima 17750* When the image set is open, the quotient map satisfies a partial converse to fnfvima 5967, which is normally only true for injective functions. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn

Theoremkqsat 17751* Any open set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 17737). (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn

Theoremkqdisj 17752* A version of imain 5520 for the topological indistinguishability map. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn

Theoremkqcldsat 17753* Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 17737). (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn

Theoremkqopn 17754* The topological indistinguishability map is an open map. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremkqcld 17755* The topological indistinguishability map is a closed map. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremkqt0lem 17756* Lemma for kqt0 17766. (Contributed by Mario Carneiro, 23-Mar-2015.)
TopOn KQ

Theoremisr0 17757* The property " 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 also contains , so there is no separation, then and 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.)
TopOn KQ

Theoremr0cld 17758* 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 is closed. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremregr1lem 17759* Lemma for regr1 17770. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn                                   KQ KQ

Theoremregr1lem2 17760* A Kolmogorov quotient of a regular space is Hausdorff. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremkqreglem1 17761* A Kolmogorov quotient of a regular space is regular. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremkqreglem2 17762* If the Kolmogorov quotient of a space is regular then so is the original space. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremkqnrmlem1 17763* A Kolmogorov quotient of a normal space is normal. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremkqnrmlem2 17764* If the Kolmogorov quotient of a space is normal then so is the original space. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremkqtop 17765 The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

Theoremkqt0 17766 The Kolmogorov quotient is T0 even if the original topology is not. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

Theoremkqf 17767 The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

Theoremr0sep 17768* The separation property of an R0 space. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremnrmr0reg 17769 A normal R0 space is also regular. These spaces are usually referred to as normal regular spaces. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

Theoremregr1 17770 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.)
KQ

Theoremkqreg 17771 The Kolmogorov quotient of a regular space is regular. By regr1 17770 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.)
KQ

Theoremkqnrm 17772 The Kolmogorov quotient of a normal space is normal. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

11.1.20  Homeomorphisms

Syntaxchmeo 17773 Extend class notation with the class of all homeomorphisms.

Syntaxchmph 17774 Extend class notation with the relation "is homeomorph to.".

Definitiondf-hmeo 17775* Function returning all the homeomorphisms from topology to topology . (Contributed by FL, 14-Feb-2007.)

Definitiondf-hmph 17776 Definition of the relation is homeomorph to . (Contributed by FL, 14-Feb-2007.)

Theoremhmeofn 17777 The set of homeomorphisms is a function on topologies. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremhmeofval 17778* The set of all the homeomorphisms between two topologies. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)

Theoremishmeo 17779 The predicate F is a homeomorphism between topology and topology . Proposition of [BourbakiTop1] p. I.2. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)

Theoremhmeocn 17780 A homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.)

Theoremhmeocnvcn 17781 The converse of a homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.)

Theoremhmeocnv 17782 The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)

Theoremhmeof1o2 17783 A homeomorphism is a 1-1-onto mapping. (Contributed by Mario Carneiro, 22-Aug-2015.)
TopOn TopOn

Theoremhmeof1o 17784 A homeomorphism is a 1-1-onto mapping. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 30-May-2014.)

Theoremhmeoima 17785 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.)

Theoremhmeoopn 17786 Homeomorphisms preserve openness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)

Theoremhmeocld 17787 Homeomorphisms preserve closedness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)

Theoremhmeocls 17788 Homeomorphisms preserve closures. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremhmeontr 17789 Homeomorphisms preserve interiors. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremhmeoimaf1o 17790* The function mapping open sets to their images under a homeomorphism is a bijection of topologies. (Contributed by Mario Carneiro, 10-Sep-2015.)

Theoremhmeores 17791 The restriction of a homeomorphism is a homeomorphism. (Contributed by Mario Carneiro, 14-Sep-2014.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
t t

Theoremhmeoco 17792 The composite of two homeomorphisms is a homeomorphism. (Contributed by FL, 9-Mar-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)

Theoremidhmeo 17793 The identity function is a homeomorphism. (Contributed by FL, 14-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
TopOn

Theoremhmeocnvb 17794 The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremhmeoqtop 17795 A homeomorphism is a quotient map. (Contributed by Mario Carneiro, 25-Aug-2015.)
qTop

Theoremhmph 17796 Express the predicate is homeomorph to . (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)

Theoremhmphi 17797 If there is a homeomorphism between spaces, then the spaces are homeomorphic. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremhmphtop 17798 Reverse closure for the homeomorphic predicate. (Contributed by Mario Carneiro, 22-Aug-2015.)

Theoremhmphtop1 17799 The relation "being homeomorph to" implies the operands are topologies. (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremhmphtop2 17800 The relation "being homeomorph to" implies the operands are topologies. (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

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