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Theorem List for Metamath Proof Explorer - 18201-18300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremustref 18201 Any element of the base set is "near" itself, i.e. entourages are reflexive. (Contributed by Thierry Arnoux, 16-Nov-2017.)
UnifOn

Theoremust0 18202 The unique uniform structure of the empty set is the empty set. Remark 3 of [BourbakiTop1] p. II.2. (Contributed by Thierry Arnoux, 15-Nov-2017.)
UnifOn

Theoremustn0 18203 The empty set is not an uniform structure. (Contributed by Thierry Arnoux, 3-Dec-2017.)
UnifOn

Theoremustund 18204 If two intersecting sets and are both small in , their union is small in . Proposition 1 of [BourbakiTop1] p. II.12. This proposition actually does not require any axiom of the defintion of unifom structures. (Contributed by Thierry Arnoux, 17-Nov-2017.)

Theoremustelimasn 18205 Any point is near enough to itself. (Contributed by Thierry Arnoux, 18-Nov-2017.)
UnifOn

Theoremustneism 18206 For a point in , is small enough in . This proposition actually does not require any axiom of the defintion of unifom structures. (Contributed by Thierry Arnoux, 18-Nov-2017.)

Theoremelrnust 18207 First direction for ustbas 18210. (Contributed by Thierry Arnoux, 16-Nov-2017.)
UnifOn UnifOn

Theoremustbas2 18208 Second direction for ustbas 18210. (Contributed by Thierry Arnoux, 16-Nov-2017.)
UnifOn

Theoremustuni 18209 The set union of a uniform structure is the cross product of its base. (Contributed by Thierry Arnoux, 5-Dec-2017.)
UnifOn

Theoremustbas 18210 Recover the base of an uniform structure . UnifOn is to UnifOn what is to TopOn. (Contributed by Thierry Arnoux, 16-Nov-2017.)
UnifOn UnifOn

Theoremustimasn 18211 Lemma for ustuqtop 18229 (Contributed by Thierry Arnoux, 5-Dec-2017.)
UnifOn

Theoremtrust 18212 The trace of a uniform structure on a subset is a uniform structure on . Definition 3 of [BourbakiTop1] p. II.9. (Contributed by Thierry Arnoux, 2-Dec-2017.)
UnifOn t UnifOn

11.3.2  The topology induced by an uniform structure

Syntaxcutop 18213 Extend class notation with the function inducing a topology from a uniform structure.
unifTop

Definitiondf-utop 18214* Definition of a topology induced by a uniform structure. Definition 3 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 17-Nov-2017.)
unifTop UnifOn

Theoremutopval 18215* The topology induced by a uniform structure . (Contributed by Thierry Arnoux, 30-Nov-2017.)
UnifOn unifTop

Theoremelutop 18216* Open sets in the topology induced by an uniform structure on (Contributed by Thierry Arnoux, 30-Nov-2017.)
UnifOn unifTop

Theoremutoptop 18217 The topology induced by a uniform structure is a topology. (Contributed by Thierry Arnoux, 30-Nov-2017.)
UnifOn unifTop

Theoremutopbas 18218 The base of the topology induced by a uniform structure . (Contributed by Thierry Arnoux, 5-Dec-2017.)
UnifOn unifTop

Theoremutoptopon 18219 Topology induced by a uniform structure with its base set. (Contributed by Thierry Arnoux, 5-Jan-2018.)
UnifOn unifTop TopOn

Theoremrestutop 18220 Restriction of a topology induced by an uniform structure (Contributed by Thierry Arnoux, 12-Dec-2017.)
UnifOn unifTopt unifTopt

Theoremrestutopopn 18221 The restriction of the topology induced by an uniform structure to an open set. (Contributed by Thierry Arnoux, 16-Dec-2017.)
UnifOn unifTop unifTopt unifTopt

Theoremustuqtoplem 18222* Lemma for ustuqtop 18229 (Contributed by Thierry Arnoux, 11-Jan-2018.)
UnifOn

Theoremustuqtop0 18223* Lemma for ustuqtop 18229 (Contributed by Thierry Arnoux, 11-Jan-2018.)
UnifOn

Theoremustuqtop1 18224* Lemma for ustuqtop 18229, similar to ssnei2 17135 (Contributed by Thierry Arnoux, 11-Jan-2018.)
UnifOn

Theoremustuqtop2 18225* Lemma for ustuqtop 18229 (Contributed by Thierry Arnoux, 11-Jan-2018.)
UnifOn

Theoremustuqtop3 18226* Lemma for ustuqtop 18229, similar to elnei 17130 (Contributed by Thierry Arnoux, 11-Jan-2018.)
UnifOn

Theoremustuqtop4 18227* Lemma for ustuqtop 18229 (Contributed by Thierry Arnoux, 11-Jan-2018.)
UnifOn

Theoremustuqtop5 18228* Lemma for ustuqtop 18229 (Contributed by Thierry Arnoux, 11-Jan-2018.)
UnifOn

Theoremustuqtop 18229* For a given uniform structure on a set , there is a unique topology such that the set is the filter of the neighbourhoods of for that topology. Proposition 1 of [BourbakiTop1] p. II.3. (Contributed by Thierry Arnoux, 11-Jan-2018.)
UnifOn TopOn

Theoremutopsnneiplem 18230* The neighborhoods of a point for the topology induced by an uniform space . (Contributed by Thierry Arnoux, 11-Jan-2018.)
unifTop                     UnifOn

Theoremutopsnneip 18231* The neighborhoods of a point for the topology induced by an uniform space . (Contributed by Thierry Arnoux, 13-Jan-2018.)
unifTop       UnifOn

Theoremutopsnnei 18232 Images of singletons by entourages are neighborhoods of those singletons. (Contributed by Thierry Arnoux, 13-Jan-2018.)
unifTop       UnifOn

Theoremutop2nei 18233 For any symmetrical entourage and any relation , build a neighborhood of . First part of proposition 2 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 14-Jan-2018.)
unifTop       UnifOn

Theoremutop3cls 18234 Relation between a topological closure and a symmetric entourage in an uniform space. Second part of proposition 2 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 17-Jan-2018.)
unifTop       UnifOn

Theoremutopreg 18235 All Hausdorff uniform spaces are regular. Proposition 3 of [BourbakiTop1] p. II.5. (Contributed by Thierry Arnoux, 16-Jan-2018.)
unifTop       UnifOn

11.3.3  Uniform Spaces

Syntaxcuss 18236 Extend class notation with the Uniform Structure extractor function.
UnifSt

Syntaxcusp 18237 Extend class notation with the class of uniform spaces.
UnifSp

Syntaxctus 18238 Extend class notation with the function mapping a uniform structure to a uniform space.
toUnifSp

Definitiondf-uss 18239 Define the uniform structure extractor function. Similarly with df-topn 13606 this differs from df-unif 13507 when a structure has been restricted using df-ress 13431; in this case the component will still have a uniform set over the larger set, and this function fixes this by restricting the uniform set as well. (Contributed by Thierry Arnoux, 1-Dec-2017.)
UnifSt t

Definitiondf-usp 18240 Definition of a uniform space, i.e. a base set with an uniform structure and its induced topology. Derived from definition 3 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 17-Nov-2017.)
UnifSp UnifSt UnifOn unifTopUnifSt

Definitiondf-tus 18241 Define the function mapping a uniform structure to a uniform space. (Contributed by Thierry Arnoux, 17-Nov-2017.)
toUnifSp UnifOn sSet TopSet unifTop

Theoremussval 18242 The uniform structure on uniform space . This proof uses a trick with fvprc 5681 to avoid requiring to be a set. (Contributed by Thierry Arnoux, 3-Dec-2017.)
t UnifSt

Theoremussid 18243 In case the base of the UnifSt element of the uniform space is the base of its element structure, then UnifSt does not restrict it further. (Contributed by Thierry Arnoux, 4-Dec-2017.)
UnifSt

Theoremisusp 18244 The predicate is a uniform space. (Contributed by Thierry Arnoux, 4-Dec-2017.)
UnifSt              UnifSp UnifOn unifTop

Theoremressunif 18245 is unaffected by restriction. (Contributed by Thierry Arnoux, 7-Dec-2017.)
s

Theoremressuss 18246 Value of the uniform structure of a restricted space. (Contributed by Thierry Arnoux, 12-Dec-2017.)
UnifSts UnifStt

Theoremressust 18247 The uniform structure of a restricted space. (Contributed by Thierry Arnoux, 22-Jan-2018.)
UnifSts        UnifSp UnifOn

Theoremressusp 18248 The restriction of a uniform topological space to an open set is a uniform space. (Contributed by Thierry Arnoux, 16-Dec-2017.)
UnifSp s UnifSp

Theoremtusval 18249 The value of the uniform space mapping function. (Contributed by Thierry Arnoux, 5-Dec-2017.)
UnifOn toUnifSp sSet TopSet unifTop

Theoremtuslem 18250 Lemma for tusbas 18251, tusunif 18252, and tustopn 18254. (Contributed by Thierry Arnoux, 5-Dec-2017.)
toUnifSp       UnifOn unifTop

Theoremtusbas 18251 The base set of a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
toUnifSp       UnifOn

Theoremtusunif 18252 The uniform structure of a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
toUnifSp       UnifOn

Theoremtususs 18253 The uniform structure of a constructed uniform space. (Contributed by Thierry Arnoux, 15-Dec-2017.)
toUnifSp       UnifOn UnifSt

Theoremtustopn 18254 The topology induced by a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
toUnifSp       unifTop       UnifOn

Theoremtususp 18255 A constructed uniform space is an uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
toUnifSp       UnifOn UnifSp

Theoremtustps 18256 A constructed uniform space is a topological space. (Contributed by Thierry Arnoux, 25-Jan-2018.)
toUnifSp       UnifOn

Theoremuspreg 18257 If a uniform space is Hausdorff, it is regular. Proposition 3 of [BourbakiTop1] p. II.5. (Contributed by Thierry Arnoux, 4-Jan-2018.)
UnifSp

11.3.4  Uniform continuity

Syntaxcucn 18258 Extend class notation with the uniform continuity operation.
Cnu

Definitiondf-ucn 18259* Define a function on two uniform structures which value is the set of uniformly continuous functions from the first uniform structure to the second. A function is uniformly continuous if, roughly speaking, it is possible to guarantee that and be as close to each other as we please by requiring only that and are sufficiently close to each other; unlike ordinary continuity, the maximum distance between and cannot depend on and themselves. This formulation is the definition 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 16-Nov-2017.)
Cnu UnifOn UnifOn

Theoremucnval 18260* The set of all uniformly continuous function from uniform space to uniform space . (Contributed by Thierry Arnoux, 16-Nov-2017.)
UnifOn UnifOn Cnu

Theoremisucn 18261* The predicate " is a uniformly continuous function from uniform space to uniform space ." (Contributed by Thierry Arnoux, 16-Nov-2017.)
UnifOn UnifOn Cnu

Theoremisucn2 18262* The predicate " is a uniformly continuous function from uniform space to uniform space ." , expressed with filter bases for the entourages. (Contributed by Thierry Arnoux, 26-Jan-2018.)
UnifOn       UnifOn                     Cnu

Theoremucnimalem 18263* Reformulate the function as a mapping with one variable. (Contributed by Thierry Arnoux, 19-Nov-2017.)
UnifOn       UnifOn       Cnu

Theoremucnima 18264* An equivalent statement of the definition of uniformly continuous function. (Contributed by Thierry Arnoux, 19-Nov-2017.)
UnifOn       UnifOn       Cnu

Theoremucnprima 18265* The preimage by a uniformly continuous function of an entourage of is an entourage of . Note of the definition 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 19-Nov-2017.)
UnifOn       UnifOn       Cnu

Theoremiducn 18266 The identity is uniformly continuous from a uniform structure to itself. Example 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 16-Nov-2017.)
UnifOn Cnu

Theoremcstucnd 18267 A constant function is uniformly continuous. Deduction form. Example 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 16-Nov-2017.)
UnifOn       UnifOn              Cnu

Theoremucncn 18268 Uniform continuity implies continuity. Deduction form. Proposition 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 30-Nov-2017.)
UnifSp       UnifSp                     UnifSt CnuUnifSt

11.3.5  Cauchy filters in uniform spaces

Syntaxccfilu 18269 Extend class notation with the set of Cauchy filter bases.
CauFilu

Definitiondf-cfilu 18270* Define the set of Cauchy filter bases on a uniform space. A Cauchy filter base is a filter base on the set such that for every entourage , there is an element of the filter "small enough in " i.e. such that every pair of points in is related by ". Definition 2 of [BourbakiTop1] p. II.13. (Contributed by Thierry Arnoux, 16-Nov-2017.)
CauFilu UnifOn

Theoremiscfilu 18271* The predicate " is a Cauchy filter base on uniform space ." (Contributed by Thierry Arnoux, 18-Nov-2017.)
UnifOn CauFilu

Theoremcfilufbas 18272 A Cauchy filter base is a filter base. (Contributed by Thierry Arnoux, 19-Nov-2017.)
UnifOn CauFilu

Theoremcfiluexsm 18273* For a Cauchy filter base and any entourage , there is an element of the filter small in . (Contributed by Thierry Arnoux, 19-Nov-2017.)
UnifOn CauFilu

Theoremfmucndlem 18274* Lemma for fmucnd 18275. (Contributed by Thierry Arnoux, 19-Nov-2017.)

Theoremfmucnd 18275* The image of a Cauchy filter base by an uniformly continuous function is a Cauchy filter base. Deduction form. Proposition 3 of [BourbakiTop1] p. II.13. (Contributed by Thierry Arnoux, 18-Nov-2017.)
UnifOn       UnifOn       Cnu       CauFilu              CauFilu

Theoremcfilufg 18276 The filter generated by a Cauchy filter base is still a Cauchy filter base. (Contributed by Thierry Arnoux, 24-Jan-2018.)
UnifOn CauFilu CauFilu

Theoremtrcfilu 18277 Condition for the trace of a Cauchy filter base to be a Cauchy filter base for the restricted uniform structure. (Contributed by Thierry Arnoux, 24-Jan-2018.)
UnifOn CauFilu t t CauFilut

Theoremcfiluweak 18278 A Cauchy filter base is also a Cauchy filter base on any coarser uniform structure. (Contributed by Thierry Arnoux, 24-Jan-2018.)
UnifOn CauFilut CauFilu

Theoremneipcfilu 18279 In an uniform space, a neighboring filter is a Cauchy filter base. (Contributed by Thierry Arnoux, 24-Jan-2018.)
UnifSt       UnifSp CauFilu

11.3.6  Complete uniform spaces

Syntaxccusp 18280 Extend class notation with the class of all complete uniform spaces.
CUnifSp

Definitiondf-cusp 18281* Define the class of all complete uniform spaces. Definition 3 of [BourbakiTop1] p. II.15. (Contributed by Thierry Arnoux, 1-Dec-2017.)
CUnifSp UnifSp CauFiluUnifSt

Theoremiscusp 18282* The predicate " is a complete uniform space." (Contributed by Thierry Arnoux, 3-Dec-2017.)
CUnifSp UnifSp CauFiluUnifSt

Theoremcuspusp 18283 A complete uniform space is an uniform space. (Contributed by Thierry Arnoux, 3-Dec-2017.)
CUnifSp UnifSp

Theoremcuspcvg 18284 In a complete uniform space, any Cauchy filter has a limit. (Contributed by Thierry Arnoux, 3-Dec-2017.)
CUnifSp CauFiluUnifSt

Theoremiscusp2 18285* The predicate " is a complete uniform space." (Contributed by Thierry Arnoux, 15-Dec-2017.)
UnifSt              CUnifSp UnifSp CauFilu

Theoremcnextucn 18286* Extension by continuity. Proposition 11 of [BourbakiTop1] p. II.20. Given a topology on , a subset dense in , this states a condition for from to a space Hausdorff and complete to be extensible by continuity (Contributed by Thierry Arnoux, 4-Dec-2017.)
UnifSt                     CUnifSp                                   t CauFilu       CnExt

Theoremucnextcn 18287 Extension by continuity. Theorem 2 of [BourbakiTop1] p. II.20. Given an uniform space on a set , a subset dense in , and a function uniformly continuous from to , that function can be extended by continuity to the whole , and its extension is uniformly continuous. (Contributed by Thierry Arnoux, 25-Jan-2018.)
UnifSt       UnifSts        UnifSt              UnifSp              CUnifSp                     Cnu              CnExt

11.4  Metric spaces

11.4.1  Pseudometric spaces

Theoremispsmet 18288* Express the predicate " is a pseudometric." (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet

Theorempsmetdmdm 18289 Recover the base set from a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet

Theorempsmetf 18290 The distance function of a pseudometric as a function. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet

Theorempsmetcl 18291 Closure of the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet

Theorempsmet0 18292 The distance function of a pseudometric space is zero if its arguments are equal. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet

Theorempsmettri2 18293 Triangle inequality for the distance function of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
PsMet

Theorempsmetsym 18294 The distance function of a pseudometric is symmetrical. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet

Theorempsmettri 18295 Triangle inequality for the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 11-Feb-2018.)
PsMet

Theorempsmetge0 18296 The distance function of a pseudometric space is nonnegative. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet

Theorempsmetxrge0 18297 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 18298 Restriction of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
PsMet PsMet

Theorempsmetlecl 18299 Real closure of an extended metric value that is upper bounded by a real. (Contributed by Thierry Arnoux, 11-Mar-2018.)
PsMet

11.4.2  Basic metric space properties

Syntaxcxme 18300 Extend class notation with the class of all extended metric spaces.

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