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Theorem List for Metamath Proof Explorer - 17601-17700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremptcls 17601* The closure of a box in the product topology is the box formed from the closures of the factors. This theorem is an AC equivalent. (Contributed by Mario Carneiro, 2-Sep-2015.)
TopOn

Theoremdfac14lem 17602* Lemma for dfac14 17603. By equipping for some with the particular point topology, we can show that is in the closure of ; hence the sequence is in the product of the closures, and we can utilize this instance of ptcls 17601 to extract an element of the closure of . (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremdfac14 17603* Theorem ptcls 17601 is an equivalent of the axiom of choice. (Contributed by Mario Carneiro, 3-Sep-2015.)
CHOICE

Theoremxkoccn 17604* The "constant function" function which maps to the constant function is a continuous function from into the space of continuous functions from to . This can also be understood as the currying of the first projection function. (The currying of the second projection function is , which we already know is continuous because it is a constant function.) (Contributed by Mario Carneiro, 19-Mar-2015.)
TopOn TopOn

Theoremtxcnp 17605* If two functions are continuous at , then the ordered pair of them is continuous at into the product topology. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn       TopOn

Theoremptcnplem 17606* Lemma for ptcnp 17607. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn

Theoremptcnp 17607* If every projection of a function is continuous at , then the function itself is continuous at into the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn

Theoremupxp 17608* Universal property of the Cartesian product considered as a categorical product in the category of sets. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremtxcnmpt 17609* A map into the product of two topological spaces is continuous if both of its projections are continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Aug-2015.)

Theoremuptx 17610* Universal property of the binary topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)

Theoremtxcn 17611 A map into the product of two topological spaces is continuous iff both of its projections are continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)

Theoremptcn 17612* If every projection of a function is continuous, then the function itself is continuous into the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
TopOn

Theoremprdstopn 17613 Topology of a structure product. (Contributed by Mario Carneiro, 27-Aug-2015.)
s

Theoremprdstps 17614 A structure product of topologies is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
s

Theorempwstps 17615 A structure product of topologies is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
s

Theoremtxrest 17616 The subspace of a topological product space induced by a subset with a Cartesian product representation is a topological product of the subspaces induced by the subspaces of the terms of the products. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
t t t

Theoremtxdis 17617 The topological product of discrete spaces is discrete. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremtxindislem 17618 Lemma for txindis 17619. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremtxindis 17619 The topological product of indiscrete spaces is indiscrete. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremtxdis1cn 17620* A function is jointly continuous on a discrete left topology iff it is continuous as a function of its right argument, for each fixed left value. (Contributed by Mario Carneiro, 19-Sep-2015.)
TopOn

Theoremtxlly 17621* If the property is preserved under topological products, then so is the property of being locally . (Contributed by Mario Carneiro, 10-Mar-2015.)
Locally Locally Locally

Theoremtxnlly 17622* If the property is preserved under topological products, then so is the property of being n-locally . (Contributed by Mario Carneiro, 13-Apr-2015.)
𝑛Locally 𝑛Locally 𝑛Locally

Theorempthaus 17623 The product of a collection of Hausdorff spaces is Hausdorff. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremptrescn 17624* Restriction is a continuous function on product topologies. (Contributed by Mario Carneiro, 7-Feb-2015.)

Theoremtxtube 17625* The "tube lemma". If is compact and there is an open set containing the line , then there is a "tube" for some neighborhood of which is entirely contained within . (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremtxcmplem1 17626* Lemma for txcmp 17628. (Contributed by Mario Carneiro, 14-Sep-2014.)

Theoremtxcmplem2 17627* Lemma for txcmp 17628. (Contributed by Mario Carneiro, 14-Sep-2014.)

Theoremtxcmp 17628 The topological product of two compact spaces is compact. (Contributed by Mario Carneiro, 14-Sep-2014.) (Proof shortened 21-Mar-2015.)

Theoremtxcmpb 17629 The topological product of two nonempty topologies is compact iff the component topologies are both compact. (Contributed by Mario Carneiro, 14-Sep-2014.)

Theoremhausdiag 17630 A topology is Hausdorff iff the diagonal set is closed in the topology's product with itself. EDITORIAL: very clumsy proof, can probably be shortened substantially. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Theoremhauseqlcld 17631 In a Hausdorff topology, the equalizer of two continuous functions is closed (thus, two continuous functions which agree on a dense set agree everywhere). (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

Theoremtxhaus 17632 The topological product of two Hausdorff spaces is Hausdorff. (Contributed by Mario Carneiro, 23-Mar-2015.)

Theoremtxlm 17633* Two sequences converge iff the sequence of their ordered pairs converges. Proposition 14-2.6 of [Gleason] p. 230. (Contributed by NM, 16-Jul-2007.) (Revised by Mario Carneiro, 5-May-2014.)
TopOn       TopOn

Theoremlmcn2 17634* The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 15-May-2014.)
TopOn       TopOn

Theoremtx1stc 17635 The topological product of two first-countable spaces is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremtx2ndc 17636 The topological product of two second-countable spaces is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremtxkgen 17637 The topological product of a locally compact space and a compactly generated Hausdorff space is compactly generated. (The condition on can also be replaced with either "comactly generated weak Hausdorff (CGWH)" or "compact Hausdorff-ly generated (CHG)", where WH means that all images of compact Hausdorff spaces are closed and CHG means that a set is open iff it is open in all compact Hausdorff spaces.) (Contributed by Mario Carneiro, 23-Mar-2015.)
𝑛Locally 𝑘Gen 𝑘Gen

Theoremxkohaus 17638 If the codomain space is Hausdorff, then the compact-open topology of continuous functions is also Hausdorff. (Contributed by Mario Carneiro, 19-Mar-2015.)

Theoremxkoptsub 17639 The compact-open topology is finer than the product topology restricted to continuous functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
t

Theoremxkopt 17640 The compact-open topology on a discrete set coincides with the product topology where all the factors are the same. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 12-Sep-2015.)

Theoremxkopjcn 17641* Continuity of a projection map from the space of continuous functions. (This theorem can be strengthened, to joint continuity in both and as a function on , but not without stronger assumptions on ; see xkofvcn 17669.) (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

Theoremxkoco1cn 17642* If is a continuous function, then is a continuous function on function spaces. (The reason we prove this and xkoco2cn 17643 independently of the more general xkococn 17645 is because that requires some inconvenient extra assumptions on .) (Contributed by Mario Carneiro, 20-Mar-2015.)

Theoremxkoco2cn 17643* If is a continuous function, then is a continuous function on function spaces. (Contributed by Mario Carneiro, 23-Mar-2015.)

Theoremxkococnlem 17644* Continuity of the composition operation as a function on continuous function spaces. (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
𝑛Locally               t

Theoremxkococn 17645* Continuity of the composition operation as a function on continuous function spaces. (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
𝑛Locally

11.1.17  Continuous function-builders

Theoremcnmptid 17646* The identity function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn

Theoremcnmptc 17647* A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn

Theoremcnmpt11 17648* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn              TopOn

Theoremcnmpt11f 17649* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn

Theoremcnmpt1t 17650* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn

Theoremcnmpt12f 17651* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn

Theoremcnmpt12 17652* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 12-Jun-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn                     TopOn       TopOn

Theoremcnmpt1st 17653* The projection onto the first coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn

Theoremcnmpt2nd 17654* The projection onto the second coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn

Theoremcnmpt2c 17655* A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn       TopOn

Theoremcnmpt21 17656* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn              TopOn

Theoremcnmpt21f 17657* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn

Theoremcnmpt2t 17658* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn

Theoremcnmpt22 17659* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn                     TopOn       TopOn

Theoremcnmpt22f 17660* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn

Theoremcnmpt1res 17661* The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 5-Jun-2014.)
t        TopOn

Theoremcnmpt2res 17662* The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
t        TopOn              t        TopOn

Theoremcnmptcom 17663* The argument converse of a continuous function is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
TopOn       TopOn

Theoremcnmptkc 17664* The curried first projection function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn

Theoremcnmptkp 17665* The evaluation of the inner function in a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn       TopOn

Theoremcnmptk1 17666* The composition of a curried function with a one-arg function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.)
TopOn       TopOn       TopOn

Theoremcnmpt1k 17667* 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 17668* 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 17669* Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 17641.) (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
𝑛Locally

Theoremcnmptk1p 17670* 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 17671* 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 17672* Continuity of "injection", i.e. currying, as a function on continuous function spaces. (Contributed by Mario Carneiro, 23-Mar-2015.)
TopOn TopOn

Theoremcnmpt2k 17673* The currying of a two-argument function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.)
TopOn       TopOn

Theoremtxcon 17674 The topological product of two connected spaces is connected. (Contributed by Mario Carneiro, 29-Mar-2015.)

Theoremimasnopn 17675 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 17676 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 17677 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.)

11.1.18  Quotient maps and quotient topology

Syntaxckq 17678 Extend class notation with the Kolmogorov quotient function.
KQ

Definitiondf-kq 17679* 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 17680* Value of the quotient topology function. (Contributed by Mario Carneiro, 23-Mar-2015.)
qTop

Theoremqtopval2 17681* Value of the quotient topology function when is a function on the base set. (Contributed by Mario Carneiro, 23-Mar-2015.)
qTop

Theoremelqtop 17682 Value of the quotient topology function. (Contributed by Mario Carneiro, 23-Mar-2015.)
qTop

Theoremqtopres 17683 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 17684 The quotient topology is a topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
qTop

Theoremqtoptop 17685 The quotient topology is a topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
qTop

Theoremelqtop2 17686 Value of the quotient topology function. (Contributed by Mario Carneiro, 9-Apr-2015.)
qTop

Theoremqtopuni 17687 The base set of the quotient topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
qTop

Theoremelqtop3 17688 Value of the quotient topology function. (Contributed by Mario Carneiro, 9-Apr-2015.)
TopOn qTop

Theoremqtoptopon 17689 The base set of the quotient topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
TopOn qTop TopOn

Theoremqtopid 17690 A quotient map is a continuous function into its quotient topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
TopOn qTop

Theoremidqtop 17691 The quotient topology induced by the identity function is the original topology. (Contributed by Mario Carneiro, 30-Aug-2015.)
TopOn qTop

Theoremqtopcmplem 17692 Lemma for qtopcmp 17693 and qtopcon 17694. (Contributed by Mario Carneiro, 24-Mar-2015.)
qTop qTop qTop        qTop

Theoremqtopcmp 17693 A quotient of a compact space is compact. (Contributed by Mario Carneiro, 24-Mar-2015.)
qTop

Theoremqtopcon 17694 A quotient of a connected space is connected. (Contributed by Mario Carneiro, 24-Mar-2015.)
qTop

Theoremqtopkgen 17695 A quotient of a compactly generated space is compactly generated. (Contributed by Mario Carneiro, 9-Apr-2015.)
𝑘Gen qTop 𝑘Gen

Theorembasqtop 17696 An injection maps bases to bases. (Contributed by Mario Carneiro, 27-Aug-2015.)
qTop

Theoremtgqtop 17697 An injection maps generated topologies to each other. (Contributed by Mario Carneiro, 27-Aug-2015.)
qTop qTop

Theoremqtopcld 17698 The property of being a closed set in the quotient topology. (Contributed by Mario Carneiro, 24-Mar-2015.)
TopOn qTop

Theoremqtopcn 17699 Universal property of a quotient map. (Contributed by Mario Carneiro, 23-Mar-2015.)
TopOn TopOn qTop

Theoremqtopss 17700 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 17690, 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

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