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Theorem List for Metamath Proof Explorer - 23501-23600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcnextfun 23501 If the target space is Hausdorff, a continuous extension is a function (Contributed by Thierry Arnoux, 20-Dec-2017.)
CnExt

Theoremcnextfvval 23502* The value of the continuous extension of a given function at a point . (Contributed by Thierry Arnoux, 21-Dec-2017.)
t        CnExt t

Theoremcnextf 23503* Extension by continuity. The extension by continuity is a function. (Contributed by Thierry Arnoux, 25-Dec-2017.)
t        CnExt

Theoremcnextcn 23504* Extension by continuity. Theorem 1 of [BourbakiTop1] p. I.57. Given a topology on , a subset dense in , this states a condition for from to a regular space to be extensible by continuity (Contributed by Thierry Arnoux, 1-Jan-2018.)
t               CnExt

18.3.9  Uniform Stuctures and Spaces

18.3.9.1  Uniform structures

Syntaxcust 23505 Extend class notation with the class function of uniform structures.
UnifOn

Definitiondf-ust 23506* Definition of a uniform structure. Definition 1 of [BourbakiTop1] p. II.1. A uniform structure is used to give a generalization of the idea of Cauchy's sequence. This defintion is analogous to TopOn. Elements of an uniform structure are called entourages. (Contributed by FL, 29-May-2014.) (Revised by Thierry Arnoux, 15-Nov-2017.)
UnifOn

Theoremustfn 23507 The defined uniform structure as a function. (Contributed by Thierry Arnoux, 15-Nov-2017.)
UnifOn

Theoremustval 23508* The class of all uniform structures for a base . (Contributed by Thierry Arnoux, 15-Nov-2017.)
UnifOn

Theoremisust 23509* The predicate " is a uniform structure with base ." (Contributed by Thierry Arnoux, 15-Nov-2017.)
UnifOn

Theoremustssxp 23510 Entourages are subsets of the cross product of the base set. (Contributed by Thierry Arnoux, 19-Nov-2017.)
UnifOn

Theoremustssel 23511 A uniform structure is upward closed. Condition FI of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 19-Nov-2017.)
UnifOn

Theoremustbasel 23512 The full set is always an entourage. Condition FIIb of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 19-Nov-2017.)
UnifOn

Theoremustincl 23513 A uniform structure is closed under finite intersection. Condition FII of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 30-Nov-2017.)
UnifOn

Theoremustdiag 23514 The diagonal set is included in any entourage, i.e. any point is -close to itself. Condition UI of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)
UnifOn

Theoremustinvel 23515 If is an entourage, so is its inverse. Condition UII of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)
UnifOn

Theoremustexhalf 23516* For each entourage there is an entourage that is "not more than half as large". Condition UIII of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)
UnifOn

Theoremustrel 23517 The elements of uniform structures, called entourages, are relations. (Contributed by Thierry Arnoux, 15-Nov-2017.)
UnifOn

Theoremustfilxp 23518 A uniform structure on a non-empty base is a filter. Remark 3 of [BourbakiTop1] p. II.2. (Contributed by Thierry Arnoux, 15-Nov-2017.)
UnifOn

Theoremustne0 23519 A uniform structure cannot be empty. (Contributed by Thierry Arnoux, 16-Nov-2017.)
UnifOn

Theoremustssco 23520 In an uniform structure, any entourage is a subset of its composition with itself. (Contributed by Thierry Arnoux, 5-Jan-2018.)
UnifOn

Theoremustexsym 23521* In an uniform structure, for any entourage , there exists a smaller symmetrical entourage. (Contributed by Thierry Arnoux, 4-Jan-2018.)
UnifOn

Theoremustref 23522 Any element of the base set is "near" itself, i.e. entourages are reflexive. (Contributed by Thierry Arnoux, 16-Nov-2017.)
UnifOn

Theoremust0 23523 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 23524 The empty set is not an uniform structure. (Contributed by Thierry Arnoux, 3-Dec-2017.)
UnifOn

Theoremxpco 23525 Composition of two cross products. (Contributed by Thierry Arnoux, 17-Nov-2017.)

Theoremustund 23526 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.)

Theoremustneism 23527 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 23528 First direction for ustbas 23531. (Contributed by Thierry Arnoux, 16-Nov-2017.)
UnifOn UnifOn

Theoremustbas2 23529 Second direction for ustbas 23531. (Contributed by Thierry Arnoux, 16-Nov-2017.)
UnifOn

Theoremustuni 23530 The set union of a uniform structure is the cross product of its base. (Contributed by Thierry Arnoux, 5-Dec-2017.)
UnifOn

Theoremustbas 23531 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 23532 Lemma for ustuqtop 23550 (Contributed by Thierry Arnoux, 5-Dec-2017.)
UnifOn

Theoremtrust 23533 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

18.3.9.2  The topology induced by an uniform structure

Syntaxcutop 23534 Extend class notation with the function inducing a topology from a uniform structure.
unifTop

Definitiondf-utop 23535* 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 23536* The topology induced by a uniform structure . (Contributed by Thierry Arnoux, 30-Nov-2017.)
UnifOn unifTop

Theoremelutop 23537* Open sets in the topology induced by an uniform structure on (Contributed by Thierry Arnoux, 30-Nov-2017.)
UnifOn unifTop

Theoremutoptop 23538 The topology induced by a uniform structure is a topology. (Contributed by Thierry Arnoux, 30-Nov-2017.)
UnifOn unifTop

Theoremutopbas 23539 The base of the topology induced by a uniform structure . (Contributed by Thierry Arnoux, 5-Dec-2017.)
UnifOn unifTop

Theoremutoptopon 23540 Topology induced by a uniform structure with its base set. (Contributed by Thierry Arnoux, 5-Jan-2018.)
UnifOn unifTop TopOn

Theoremrestutop 23541 Restriction of a topology induced by an uniform structure (Contributed by Thierry Arnoux, 12-Dec-2017.)
UnifOn unifTopt unifTopt

Theoremrestutopopn 23542 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 23543* Lemma for ustuqtop 23550 (Contributed by Thierry Arnoux, 11-Jan-2018.)
UnifOn

Theoremustuqtop0 23544* Lemma for ustuqtop 23550 (Contributed by Thierry Arnoux, 11-Jan-2018.)
UnifOn

Theoremustuqtop1 23545* Lemma for ustuqtop 23550 (Contributed by Thierry Arnoux, 11-Jan-2018.)
UnifOn

Theoremustuqtop2 23546* Lemma for ustuqtop 23550 (Contributed by Thierry Arnoux, 11-Jan-2018.)
UnifOn

Theoremustuqtop3 23547* Lemma for ustuqtop 23550 (Contributed by Thierry Arnoux, 11-Jan-2018.)
UnifOn

Theoremustuqtop4 23548* Lemma for ustuqtop 23550 (Contributed by Thierry Arnoux, 11-Jan-2018.)
UnifOn

Theoremustuqtop5 23549* Lemma for ustuqtop 23550 (Contributed by Thierry Arnoux, 11-Jan-2018.)
UnifOn

Theoremustuqtop 23550* 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 23551* The neighborhoods of a point for the topology induced by an uniform space . (Contributed by Thierry Arnoux, 11-Jan-2018.)
unifTop                     UnifOn

18.3.9.3  Uniform Spaces

Syntaxcuss 23552 Extend class notation with the Uniform Structure extractor function.
UnifSt

Syntaxcusp 23553 Extend class notation with the class of uniform spaces.
UnifSp

Syntaxctus 23554 Extend class notation with the function mapping a uniform structure to a uniform space.
toUnifSp

Definitiondf-uss 23555 Define the uniform structure extractor function. Similarly with df-topn 13427 this differs from df-unif 13328 when a structure has been restricted using df-ress 13252; 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 23556 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 23557 Define the function mapping a uniform structure to a uniform space. (Contributed by Thierry Arnoux, 17-Nov-2017.)
toUnifSp UnifOn sSet TopSet unifTop

Theoremussval 23558 The uniform structure on uniform space . This proof uses a trick with fvprc 5602 to avoid requiring to be a set. (Contributed by Thierry Arnoux, 3-Dec-2017.)
t UnifSt

Theoremussid 23559 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 23560 The predicate is a uniform space. (Contributed by Thierry Arnoux, 4-Dec-2017.)
UnifSt              UnifSp UnifOn unifTop

Theoremressunif 23561 is unaffected by restriction. (Contributed by Thierry Arnoux, 7-Dec-2017.)
s

Theoremressuss 23562 Value of the uniform structure of a restricted space. (Contributed by Thierry Arnoux, 12-Dec-2017.)
UnifSts UnifStt

Theoremressusp 23563 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 23564 The value of the uniform space mapping function. (Contributed by Thierry Arnoux, 5-Dec-2017.)
UnifOn toUnifSp sSet TopSet unifTop

Theoremtuslem 23565 Lemma for tusbas 23566, tusunif 23567, and tustopn 23569. (Contributed by Thierry Arnoux, 5-Dec-2017.)
toUnifSp       UnifOn unifTop

Theoremtusbas 23566 The base set of a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
toUnifSp       UnifOn

Theoremtusunif 23567 The uniform structure of a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
toUnifSp       UnifOn

Theoremtususs 23568 The uniform structure of a constructed uniform space. (Contributed by Thierry Arnoux, 15-Dec-2017.)
toUnifSp       UnifOn UnifSt

Theoremtustopn 23569 The topology induced by a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
toUnifSp       unifTop       UnifOn

Theoremtususp 23570 A constructed uniform space is an uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
toUnifSp       UnifOn UnifSp

18.3.9.4  Uniform continuity

Syntaxcucn 23571 Extend class notation with the uniform continuity operation.
Cnu

Definitiondf-ucn 23572* 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 23573* The set of all uniformly continuous function from uniform space to uniform space . (Contributed by Thierry Arnoux, 16-Nov-2017.)
UnifOn UnifOn Cnu

Theoremisucn 23574* The predicate " is a uniformly continuous function from uniform space to uniform space ." (Contributed by Thierry Arnoux, 16-Nov-2017.)
UnifOn UnifOn Cnu

Theoremucnimalem 23575* Reformulate the function as a mapping with one variable. (Contributed by Thierry Arnoux, 19-Nov-2017.)
UnifOn       UnifOn       Cnu

Theoremucnima 23576* An equivalent statement of the definition of uniformly continuous function. (Contributed by Thierry Arnoux, 19-Nov-2017.)
UnifOn       UnifOn       Cnu

Theoremucnprima 23577* 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 23578 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 23579 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

18.3.9.5  Cauchy filters in uniform spaces

Syntaxccfilu 23580 Extend class notation with the set of Cauchy filters.
CauFilu

Definitiondf-cfilu 23581* Define the set of Cauchy filters on a uniform space. A Cauchy filter is a filter 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 v." . Definition 2 of [BourbakiTop1] p. II.13. (Contributed by Thierry Arnoux, 16-Nov-2017.)
CauFilu UnifOn

Theoremiscfilu 23582* The predicate " is a Cauchy filter base on uniform space ." (Contributed by Thierry Arnoux, 18-Nov-2017.)
UnifOn CauFilu

Theoremcfilufbas 23583 A Cauchy filter base is a filter base. (Contributed by Thierry Arnoux, 19-Nov-2017.)
UnifOn CauFilu

Theoremcfiluexsm 23584* 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 23585* Lemma for fmucnd 23586. (Contributed by Thierry Arnoux, 19-Nov-2017.)

Theoremfmucnd 23586* 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

18.3.9.6  Complete uniform spaces

Syntaxccusp 23587 Extend class notation with the class of all complete uniform spaces.
CUnifSp

Definitiondf-cusp 23588* 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 23589* The predicate " is a complete uniform space." (Contributed by Thierry Arnoux, 3-Dec-2017.)
CUnifSp UnifSp CauFiluUnifSt

Theoremcuspusp 23590 A complete uniform space is an uniform space. (Contributed by Thierry Arnoux, 3-Dec-2017.)
CUnifSp UnifSp

Theoremcuspcvg 23591 In a complete uniform space, any Cauchy filter has a limit. (Contributed by Thierry Arnoux, 3-Dec-2017.)
CUnifSp CauFiluUnifSt

Theoremiscusp2 23592* The predicate " is a complete uniform space." (Contributed by Thierry Arnoux, 15-Dec-2017.)
UnifSt              CUnifSp UnifSp CauFilu

18.3.9.7  The uniform structure generated by a metric

Theoremmetuval 23593* Value of the uniform structure generated by metric . (Contributed by Thierry Arnoux, 1-Dec-2017.)
metUnif

Theoremmetustel 23594* Define a filter base generated by a metric . (Contributed by Thierry Arnoux, 22-Nov-2017.)

Theoremmetustss 23595* Range of the elements of the filter base generated by the metric . (Contributed by Thierry Arnoux, 28-Nov-2017.)

Theoremmetustrel 23596* Elements of the filter base generated by the metric are relations. (Contributed by Thierry Arnoux, 28-Nov-2017.)

Theoremmetustto 23597* Any two elements of the filter base generated by the metric can be compared, like for RR+ (i.e. it's totally ordered). (Contributed by Thierry Arnoux, 22-Nov-2017.)

Theoremmetustid 23598* The identity diagonal is included in all elements of the filter base generated by the metric . (Contributed by Thierry Arnoux, 22-Nov-2017.)

Theoremmetustsym 23599* Elements of the filter base generated by the metric are symmetric. (Contributed by Thierry Arnoux, 28-Nov-2017.)

Theoremmetustexhalf 23600* For any element of the filter base generated by the metric , the half element (corresponding to half the distance) is also in this base. (Contributed by Thierry Arnoux, 28-Nov-2017.)

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