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Theorem List for Intuitionistic Logic Explorer - 12201-12300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlmcn 12201 The image of a convergent sequence under a continuous map is convergent to the image of the original point. (Contributed by Mario Carneiro, 3-May-2014.)

6.1.8  Product topologies

Syntaxctx 12202 Extend class notation with the binary topological product operation.

Definitiondf-tx 12203* Define the binary topological product, which is homeomorphic to the general topological product over a two element set, but is more convenient to use. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremtxvalex 12204 Existence of the binary topological product. If and are known to be topologies, see txtop 12210. (Contributed by Jim Kingdon, 3-Aug-2023.)

Theoremtxval 12205* Value of the binary topological product operation. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 30-Aug-2015.)

Theoremtxuni2 12206* The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremtxbasex 12207* The basis for the product topology is a set. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtxbas 12208* The set of Cartesian products of elements from two topological bases is a basis. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremeltx 12209* 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 12210 The product of two topologies is a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremtxtopi 12211 The product of two topologies is a topology. (Contributed by Jeff Madsen, 15-Jun-2010.)

Theoremtxtopon 12212 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 12213 The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremtxunii 12214 The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 15-Jun-2010.)

Theoremtxopn 12215 The product of two open sets is open in the product topology. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremtxss12 12216 Subset property of the topological product. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtxbasval 12217 It is sufficient to consider products of the bases for the topologies in the topological product. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremneitx 12218 The Cartesian product of two neighborhoods is a neighborhood in the product topology. (Contributed by Thierry Arnoux, 13-Jan-2018.)

Theoremtx1cn 12219 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 12220 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 12221* 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 12222* 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 12223* 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 12224* Universal property of the binary topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)

Theoremtxcn 12225 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 12226 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 12227 The topological product of discrete spaces is discrete. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremtxdis1cn 12228* 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 12229* 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 12230* 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

6.1.9  Continuous function-builders

Theoremcnmptid 12231* The identity function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn

Theoremcnmptc 12232* A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn

Theoremcnmpt11 12233* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn              TopOn

Theoremcnmpt11f 12234* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn

Theoremcnmpt1t 12235* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn

Theoremcnmpt12f 12236* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn

Theoremcnmpt12 12237* 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 12238* 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 12239* 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 12240* A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn       TopOn

Theoremcnmpt21 12241* 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 12242* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn

Theoremcnmpt2t 12243* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn

Theoremcnmpt22 12244* 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 12245* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn

Theoremcnmpt1res 12246* The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 5-Jun-2014.)
t        TopOn

Theoremcnmpt2res 12247* The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
t        TopOn              t        TopOn

Theoremcnmptcom 12248* The argument converse of a continuous function is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
TopOn       TopOn

Theoremimasnopn 12249 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.)

6.2  Metric spaces

6.2.1  Pseudometric spaces

Theorempsmetrel 12250 The class of pseudometrics is a relation. (Contributed by Jim Kingdon, 24-Apr-2023.)
PsMet

Theoremispsmet 12251* Express the predicate " is a pseudometric." (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet

Theorempsmetdmdm 12252 Recover the base set from a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet

Theorempsmetf 12253 The distance function of a pseudometric as a function. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet

Theorempsmetcl 12254 Closure of the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet

Theorempsmet0 12255 The distance function of a pseudometric space is zero if its arguments are equal. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet

Theorempsmettri2 12256 Triangle inequality for the distance function of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
PsMet

Theorempsmetsym 12257 The distance function of a pseudometric is symmetrical. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet

Theorempsmettri 12258 Triangle inequality for the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 11-Feb-2018.)
PsMet

Theorempsmetge0 12259 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 12260 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 12261 Restriction of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
PsMet PsMet

Theorempsmetlecl 12262 Real closure of an extended metric value that is upper bounded by a real. (Contributed by Thierry Arnoux, 11-Mar-2018.)
PsMet

Theoremdistspace 12263 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

6.2.2  Basic metric space properties

Syntaxcxms 12264 Extend class notation with the class of extended metric spaces.

Syntaxcms 12265 Extend class notation with the class of metric spaces.

Syntaxctms 12266 Extend class notation with the function mapping a metric to the metric space it defines.
toMetSp

Definitiondf-xms 12267 Define the (proper) class of extended metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)

Definitiondf-ms 12268 Define the (proper) class of metric spaces. (Contributed by NM, 27-Aug-2006.)

Definitiondf-tms 12269 Define the function mapping a metric to the metric space which it defines. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp sSet TopSet

Theoremmetrel 12270 The class of metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)

Theoremxmetrel 12271 The class of extended metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)

Theoremismet 12272* Express the predicate " is a metric." (Contributed by NM, 25-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theoremisxmet 12273* Express the predicate " is an extended metric." (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremismeti 12274* Properties that determine a metric. (Contributed by NM, 17-Nov-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theoremisxmetd 12275* Properties that determine an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremisxmet2d 12276* 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 12277* Lemma for metf 12279 and others. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theoremxmetf 12278 Mapping of the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmetf 12279 Mapping of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.)

Theoremxmetcl 12280 Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.)

Theoremmetcl 12281 Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.)

Theoremismet2 12282 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 12283 A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmetdmdm 12284 Recover the base set from an extended metric. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremmetdmdm 12285 Recover the base set from a metric. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxmetunirn 12286 Two ways to express an extended metric on an unspecified base. (Contributed by Mario Carneiro, 13-Oct-2015.)

Theoremxmeteq0 12287 The value of an extended metric is zero iff its arguments are equal. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmeteq0 12288 The value of a metric is zero iff its arguments are equal. Property M2 of [Kreyszig] p. 4. (Contributed by NM, 30-Aug-2006.)

Theoremxmettri2 12289 Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmettri2 12290 Triangle inequality for the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 20-Aug-2015.)

Theoremxmet0 12291 The distance function of a metric space is zero if its arguments are equal. Definition 14-1.1(a) of [Gleason] p. 223. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmet0 12292 The distance function of a metric space is zero if its arguments are equal. Definition 14-1.1(a) of [Gleason] p. 223. (Contributed by NM, 30-Aug-2006.)

Theoremxmetge0 12293 The distance function of a metric space is nonnegative. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmetge0 12294 The distance function of a metric space is nonnegative. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theoremxmetlecl 12295 Real closure of an extended metric value that is upper bounded by a real. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmetsym 12296 The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmetpsmet 12297 An extended metric is a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet

Theoremxmettpos 12298 The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.)
tpos

Theoremmetsym 12299 The distance function of a metric space is symmetric. Definition 14-1.1(c) of [Gleason] p. 223. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 20-Aug-2015.)

Theoremxmettri 12300 Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by Mario Carneiro, 20-Aug-2015.)

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