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Theorem List for Intuitionistic Logic Explorer - 12501-12600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremeltx 12501* A set in a product is open iff each point is surrounded by an open rectangle. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Theoremtxtop 12502 The product of two topologies is a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremtxtopi 12503 The product of two topologies is a topology. (Contributed by Jeff Madsen, 15-Jun-2010.)

Theoremtxtopon 12504 The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 22-Aug-2015.) (Revised by Mario Carneiro, 2-Sep-2015.)
TopOn TopOn TopOn

Theoremtxuni 12505 The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremtxunii 12506 The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 15-Jun-2010.)

Theoremtxopn 12507 The product of two open sets is open in the product topology. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremtxss12 12508 Subset property of the topological product. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtxbasval 12509 It is sufficient to consider products of the bases for the topologies in the topological product. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremneitx 12510 The Cartesian product of two neighborhoods is a neighborhood in the product topology. (Contributed by Thierry Arnoux, 13-Jan-2018.)

Theoremtx1cn 12511 Continuity of the first projection map of a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
TopOn TopOn

Theoremtx2cn 12512 Continuity of the second projection map of a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
TopOn TopOn

Theoremtxcnp 12513* 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

Theoremupxp 12514* 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 12515* 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 12516* Universal property of the binary topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)

Theoremtxcn 12517 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.)

Theoremtxrest 12518 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 12519 The topological product of discrete spaces is discrete. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremtxdis1cn 12520* 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

Theoremtxlm 12521* 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 12522* 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

7.1.9  Continuous function-builders

Theoremcnmptid 12523* The identity function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn

Theoremcnmptc 12524* A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn

Theoremcnmpt11 12525* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn              TopOn

Theoremcnmpt11f 12526* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn

Theoremcnmpt1t 12527* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn

Theoremcnmpt12f 12528* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn

Theoremcnmpt12 12529* 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 12530* 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 12531* 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 12532* A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn       TopOn

Theoremcnmpt21 12533* 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 12534* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn

Theoremcnmpt2t 12535* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn

Theoremcnmpt22 12536* 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 12537* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn

Theoremcnmpt1res 12538* The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 5-Jun-2014.)
t        TopOn

Theoremcnmpt2res 12539* The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
t        TopOn              t        TopOn

Theoremcnmptcom 12540* The argument converse of a continuous function is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
TopOn       TopOn

Theoremimasnopn 12541 If a relation graph is open, then an image set of a singleton is also open. Corollary of Proposition 4 of [BourbakiTop1] p. I.26. (Contributed by Thierry Arnoux, 14-Jan-2018.)

7.1.10  Homeomorphisms

Syntaxchmeo 12542 Extend class notation with the class of all homeomorphisms.

Definitiondf-hmeo 12543* Function returning all the homeomorphisms from topology to topology . (Contributed by FL, 14-Feb-2007.)

Theoremhmeofn 12544 The set of homeomorphisms is a function on topologies. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremhmeofvalg 12545* The set of all the homeomorphisms between two topologies. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)

Theoremishmeo 12546 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 12547 A homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.)

Theoremhmeocnvcn 12548 The converse of a homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.)

Theoremhmeocnv 12549 The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)

Theoremhmeof1o2 12550 A homeomorphism is a 1-1-onto mapping. (Contributed by Mario Carneiro, 22-Aug-2015.)
TopOn TopOn

Theoremhmeof1o 12551 A homeomorphism is a 1-1-onto mapping. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 30-May-2014.)

Theoremhmeoima 12552 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 12553 Homeomorphisms preserve openness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)

Theoremhmeocld 12554 Homeomorphisms preserve closedness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)

Theoremhmeontr 12555 Homeomorphisms preserve interiors. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremhmeoimaf1o 12556* The function mapping open sets to their images under a homeomorphism is a bijection of topologies. (Contributed by Mario Carneiro, 10-Sep-2015.)

Theoremhmeores 12557 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 12558 The composite of two homeomorphisms is a homeomorphism. (Contributed by FL, 9-Mar-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)

Theoremidhmeo 12559 The identity function is a homeomorphism. (Contributed by FL, 14-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
TopOn

Theoremhmeocnvb 12560 The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremtxhmeo 12561* Lift a pair of homeomorphisms on the factors to a homeomorphism of product topologies. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtxswaphmeolem 12562* Show inverse for the "swap components" operation on a Cartesian product. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremtxswaphmeo 12563* There is a homeomorphism from to . (Contributed by Mario Carneiro, 21-Mar-2015.)
TopOn TopOn

7.2  Metric spaces

7.2.1  Pseudometric spaces

Theorempsmetrel 12564 The class of pseudometrics is a relation. (Contributed by Jim Kingdon, 24-Apr-2023.)
PsMet

Theoremispsmet 12565* Express the predicate " is a pseudometric." (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet

Theorempsmetdmdm 12566 Recover the base set from a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet

Theorempsmetf 12567 The distance function of a pseudometric as a function. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet

Theorempsmetcl 12568 Closure of the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet

Theorempsmet0 12569 The distance function of a pseudometric space is zero if its arguments are equal. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet

Theorempsmettri2 12570 Triangle inequality for the distance function of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
PsMet

Theorempsmetsym 12571 The distance function of a pseudometric is symmetrical. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet

Theorempsmettri 12572 Triangle inequality for the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 11-Feb-2018.)
PsMet

Theorempsmetge0 12573 The distance function of a pseudometric space is nonnegative. (Contributed by Thierry Arnoux, 7-Feb-2018.) (Revised by Jim Kingdon, 19-Apr-2023.)
PsMet

Theorempsmetxrge0 12574 The distance function of a pseudometric space is a function into the nonnegative extended real numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.)
PsMet

Theorempsmetres2 12575 Restriction of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
PsMet PsMet

Theorempsmetlecl 12576 Real closure of an extended metric value that is upper bounded by a real. (Contributed by Thierry Arnoux, 11-Mar-2018.)
PsMet

Theoremdistspace 12577 A set together with a (distance) function which is a pseudometric is a distance space (according to E. Deza, M.M. Deza: "Dictionary of Distances", Elsevier, 2006), i.e. a (base) set equipped with a distance , which is a mapping of two elements of the base set to the (extended) reals and which is nonnegative, symmetric and equal to 0 if the two elements are equal. (Contributed by AV, 15-Oct-2021.) (Revised by AV, 5-Jul-2022.)
PsMet

7.2.2  Basic metric space properties

Syntaxcxms 12578 Extend class notation with the class of extended metric spaces.

Syntaxcms 12579 Extend class notation with the class of metric spaces.

Syntaxctms 12580 Extend class notation with the function mapping a metric to the metric space it defines.
toMetSp

Definitiondf-xms 12581 Define the (proper) class of extended metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)

Definitiondf-ms 12582 Define the (proper) class of metric spaces. (Contributed by NM, 27-Aug-2006.)

Definitiondf-tms 12583 Define the function mapping a metric to the metric space which it defines. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp sSet TopSet

Theoremmetrel 12584 The class of metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)

Theoremxmetrel 12585 The class of extended metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)

Theoremismet 12586* Express the predicate " is a metric." (Contributed by NM, 25-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theoremisxmet 12587* Express the predicate " is an extended metric." (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremismeti 12588* Properties that determine a metric. (Contributed by NM, 17-Nov-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theoremisxmetd 12589* Properties that determine an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremisxmet2d 12590* It is safe to only require the triangle inequality when the values are real (so that we can use the standard addition over the reals), but in this case the nonnegativity constraint cannot be deduced and must be provided separately. (Counterexample: satisfies all hypotheses except nonnegativity.) (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmetflem 12591* Lemma for metf 12593 and others. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theoremxmetf 12592 Mapping of the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmetf 12593 Mapping of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.)

Theoremxmetcl 12594 Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.)

Theoremmetcl 12595 Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.)

Theoremismet2 12596 An extended metric is a metric exactly when it takes real values for all values of the arguments. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmetxmet 12597 A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmetdmdm 12598 Recover the base set from an extended metric. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremmetdmdm 12599 Recover the base set from a metric. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxmetunirn 12600 Two ways to express an extended metric on an unspecified base. (Contributed by Mario Carneiro, 13-Oct-2015.)

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