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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.)
 |-  C  =  U. J   &    |-  B  =  U. K   =>    |-  ( ( ( J  e.  Top  /\  K  e.  Haus
 )  /\  ( F : A --> B  /\  A  C_  C ) )  ->  Fun  ( ( JCnExt K ) `  F ) )
 
Theoremcnextfvval 23502* The value of the continuous extension of a given function  F at a point  X. (Contributed by Thierry Arnoux, 21-Dec-2017.)
 |-  C  =  U. J   &    |-  B  =  U. K   &    |-  ( ph  ->  J  e.  Top )   &    |-  ( ph  ->  K  e.  Haus )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  A  C_  C )   &    |-  ( ph  ->  (
 ( cls `  J ) `  A )  =  C )   &    |-  ( ( ph  /\  x  e.  C )  ->  (
 ( K  fLimf  ( ( ( nei `  J ) `  { x }
 )t 
 A ) ) `  F )  =/=  (/) )   =>    |-  ( ( ph  /\  X  e.  C ) 
 ->  ( ( ( JCnExt
 K ) `  F ) `  X )  = 
 U. ( ( K 
 fLimf  ( ( ( nei `  J ) `  { X } )t  A ) ) `  F ) )
 
Theoremcnextf 23503* Extension by continuity. The extension by continuity is a function. (Contributed by Thierry Arnoux, 25-Dec-2017.)
 |-  C  =  U. J   &    |-  B  =  U. K   &    |-  ( ph  ->  J  e.  Top )   &    |-  ( ph  ->  K  e.  Haus )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  A  C_  C )   &    |-  ( ph  ->  (
 ( cls `  J ) `  A )  =  C )   &    |-  ( ( ph  /\  x  e.  C )  ->  (
 ( K  fLimf  ( ( ( nei `  J ) `  { x }
 )t 
 A ) ) `  F )  =/=  (/) )   =>    |-  ( ph  ->  ( ( JCnExt K ) `
  F ) : C --> B )
 
Theoremcnextcn 23504* Extension by continuity. Theorem 1 of [BourbakiTop1] p. I.57. Given a topology  J on  C, a subset  A dense in  C, this states a condition for  F from  A to a regular space  K to be extensible by continuity (Contributed by Thierry Arnoux, 1-Jan-2018.)
 |-  C  =  U. J   &    |-  B  =  U. K   &    |-  ( ph  ->  J  e.  Top )   &    |-  ( ph  ->  K  e.  Haus )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  A  C_  C )   &    |-  ( ph  ->  (
 ( cls `  J ) `  A )  =  C )   &    |-  ( ( ph  /\  x  e.  C )  ->  (
 ( K  fLimf  ( ( ( nei `  J ) `  { x }
 )t 
 A ) ) `  F )  =/=  (/) )   &    |-  ( ph  ->  K  e.  Reg )   =>    |-  ( ph  ->  (
 ( JCnExt K ) `  F )  e.  ( J  Cn  K ) )
 
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.
 class 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  =  ( x  e.  _V  |->  { u  |  ( u 
 C_  ~P ( x  X.  x )  /\  ( x  X.  x )  e.  u  /\  A. v  e.  u  ( A. w  e.  ~P  ( x  X.  x ) ( v  C_  w  ->  w  e.  u )  /\  A. w  e.  u  ( v  i^i  w )  e.  u  /\  (
 (  _I  |`  x ) 
 C_  v  /\  `' v  e.  u  /\  E. w  e.  u  ( w  o.  w ) 
 C_  v ) ) ) } )
 
Theoremustfn 23507 The defined uniform structure as a function. (Contributed by Thierry Arnoux, 15-Nov-2017.)
 |- UnifOn  Fn  _V
 
Theoremustval 23508* The class of all uniform structures for a base  X. (Contributed by Thierry Arnoux, 15-Nov-2017.)
 |-  ( X  e.  _V  ->  (UnifOn `  X )  =  { u  |  ( u  C_ 
 ~P ( X  X.  X )  /\  ( X  X.  X )  e.  u  /\  A. v  e.  u  ( A. w  e.  ~P  ( X  X.  X ) ( v  C_  w  ->  w  e.  u )  /\  A. w  e.  u  ( v  i^i  w )  e.  u  /\  (
 (  _I  |`  X ) 
 C_  v  /\  `' v  e.  u  /\  E. w  e.  u  ( w  o.  w ) 
 C_  v ) ) ) } )
 
Theoremisust 23509* The predicate " U is a uniform structure with base  X." (Contributed by Thierry Arnoux, 15-Nov-2017.)
 |-  ( X  e.  _V  ->  ( U  e.  (UnifOn `  X ) 
 <->  ( U  C_  ~P ( X  X.  X )  /\  ( X  X.  X )  e.  U  /\  A. v  e.  U  ( A. w  e.  ~P  ( X  X.  X ) ( v  C_  w  ->  w  e.  U ) 
 /\  A. w  e.  U  ( v  i^i  w )  e.  U  /\  (
 (  _I  |`  X ) 
 C_  v  /\  `' v  e.  U  /\  E. w  e.  U  ( w  o.  w ) 
 C_  v ) ) ) ) )
 
Theoremustssxp 23510 Entourages are subsets of the cross product of the base set. (Contributed by Thierry Arnoux, 19-Nov-2017.)
 |-  (
 ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( X  X.  X ) )
 
Theoremustssel 23511 A uniform structure is upward closed. Condition FI of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 19-Nov-2017.)
 |-  (
 ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X ) )  ->  ( V  C_  W  ->  W  e.  U ) )
 
Theoremustbasel 23512 The full set is always an entourage. Condition FIIb of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 19-Nov-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  ( X  X.  X )  e.  U )
 
Theoremustincl 23513 A uniform structure is closed under finite intersection. Condition FII of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 30-Nov-2017.)
 |-  (
 ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  e.  U )  ->  ( V  i^i  W )  e.  U )
 
Theoremustdiag 23514 The diagonal set is included in any entourage, i.e. any point is  V -close to itself. Condition UI of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)
 |-  (
 ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  (  _I  |`  X )  C_  V )
 
Theoremustinvel 23515 If  V is an entourage, so is its inverse. Condition UII of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)
 |-  (
 ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  `' V  e.  U )
 
Theoremustexhalf 23516* For each entourage  V there is an entourage  w that is "not more than half as large". Condition UIII of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)
 |-  (
 ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  E. w  e.  U  ( w  o.  w )  C_  V )
 
Theoremustrel 23517 The elements of uniform structures, called entourages, are relations. (Contributed by Thierry Arnoux, 15-Nov-2017.)
 |-  (
 ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  Rel  V )
 
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.)
 |-  (
 ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
 )  ->  U  e.  ( Fil `  ( X  X.  X ) ) )
 
Theoremustne0 23519 A uniform structure cannot be empty. (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  U  =/=  (/) )
 
Theoremustssco 23520 In an uniform structure, any entourage  V is a subset of its composition with itself. (Contributed by Thierry Arnoux, 5-Jan-2018.)
 |-  (
 ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( V  o.  V ) )
 
Theoremustexsym 23521* In an uniform structure, for any entourage  V, there exists a smaller symmetrical entourage. (Contributed by Thierry Arnoux, 4-Jan-2018.)
 |-  (
 ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  E. w  e.  U  ( `' w  =  w  /\  w  C_  V ) )
 
Theoremustref 23522 Any element of the base set is "near" itself, i.e. entourages are reflexive. (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |-  (
 ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  A  e.  X )  ->  A V A )
 
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.)
 |-  -.  (/) 
 e.  U. ran UnifOn
 
Theoremxpco 23525 Composition of two cross products. (Contributed by Thierry Arnoux, 17-Nov-2017.)
 |-  ( B  =/=  (/)  ->  ( ( B  X.  C )  o.  ( A  X.  B ) )  =  ( A  X.  C ) )
 
Theoremustund 23526 If two intersecting sets  A and  B are both small in  V, their union is small in  ( V ^ 2 ). 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.)
 |-  ( ph  ->  ( A  X.  A )  C_  V )   &    |-  ( ph  ->  ( B  X.  B )  C_  V )   &    |-  ( ph  ->  ( A  i^i  B )  =/=  (/) )   =>    |-  ( ph  ->  (
 ( A  u.  B )  X.  ( A  u.  B ) )  C_  ( V  o.  V ) )
 
Theoremustneism 23527 For a point  A in  X,  ( V " { A } ) is small enough in  ( V  o.  `' V ). This proposition actually does not require any axiom of the defintion of unifom structures. (Contributed by Thierry Arnoux, 18-Nov-2017.)
 |-  (
 ( V  C_  ( X  X.  X )  /\  A  e.  X )  ->  ( ( V " { A } )  X.  ( V " { A } ) )  C_  ( V  o.  `' V ) )
 
Theoremelrnust 23528 First direction for ustbas 23531. (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  U  e.  U. ran UnifOn )
 
Theoremustbas2 23529 Second direction for ustbas 23531. (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  X  =  dom  U. U )
 
Theoremustuni 23530 The set union of a uniform structure is the cross product of its base. (Contributed by Thierry Arnoux, 5-Dec-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  U. U  =  ( X  X.  X ) )
 
Theoremustbas 23531 Recover the base of an uniform structure  U.  U. ran UnifOn is to UnifOn what  Top is to TopOn. (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |-  X  =  dom  U. U   =>    |-  ( U  e.  U. ran UnifOn  <->  U  e.  (UnifOn `  X ) )
 
Theoremustimasn 23532 Lemma for ustuqtop 23550 (Contributed by Thierry Arnoux, 5-Dec-2017.)
 |-  (
 ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  P  e.  X )  ->  ( V
 " { P }
 )  C_  X )
 
Theoremtrust 23533 The trace of a uniform structure  U on a subset  A is a uniform structure on  A. Definition 3 of [BourbakiTop1] p. II.9. (Contributed by Thierry Arnoux, 2-Dec-2017.)
 |-  (
 ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  ( Ut  ( A  X.  A ) )  e.  (UnifOn `  A ) )
 
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.
 class 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  =  ( u  e.  U. ran UnifOn  |->  { a  e.  ~P dom  U. u  |  A. x  e.  a  E. v  e.  u  ( v " { x } )  C_  a } )
 
Theoremutopval 23536* The topology induced by a uniform structure  U. (Contributed by Thierry Arnoux, 30-Nov-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  (unifTop `  U )  =  { a  e.  ~P X  |  A. x  e.  a  E. v  e.  U  ( v " { x } )  C_  a } )
 
Theoremelutop 23537* Open sets in the topology induced by an uniform structure  U on  X (Contributed by Thierry Arnoux, 30-Nov-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  ( A  e.  (unifTop `  U )  <->  ( A  C_  X  /\  A. x  e.  A  E. v  e.  U  ( v " { x } )  C_  A ) ) )
 
Theoremutoptop 23538 The topology induced by a uniform structure  U is a topology. (Contributed by Thierry Arnoux, 30-Nov-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  (unifTop `  U )  e.  Top )
 
Theoremutopbas 23539 The base of the topology induced by a uniform structure  U. (Contributed by Thierry Arnoux, 5-Dec-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  X  =  U. (unifTop `  U ) )
 
Theoremutoptopon 23540 Topology induced by a uniform structure  U with its base set. (Contributed by Thierry Arnoux, 5-Jan-2018.)
 |-  ( U  e.  (UnifOn `  X )  ->  (unifTop `  U )  e.  (TopOn `  X )
 )
 
Theoremrestutop 23541 Restriction of a topology induced by an uniform structure (Contributed by Thierry Arnoux, 12-Dec-2017.)
 |-  (
 ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  (
 (unifTop `  U )t  A ) 
 C_  (unifTop `  ( Ut  ( A  X.  A ) ) ) )
 
Theoremrestutopopn 23542 The restriction of the topology induced by an uniform structure to an open set. (Contributed by Thierry Arnoux, 16-Dec-2017.)
 |-  (
 ( U  e.  (UnifOn `  X )  /\  A  e.  (unifTop `  U )
 )  ->  ( (unifTop `  U )t  A )  =  (unifTop `  ( Ut  ( A  X.  A ) ) ) )
 
Theoremustuqtoplem 23543* Lemma for ustuqtop 23550 (Contributed by Thierry Arnoux, 11-Jan-2018.)
 |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )   =>    |-  ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  /\  A  e.  V )  ->  ( A  e.  ( N `  P )  <->  E. w  e.  U  A  =  ( w " { P } )
 ) )
 
Theoremustuqtop0 23544* Lemma for ustuqtop 23550 (Contributed by Thierry Arnoux, 11-Jan-2018.)
 |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )   =>    |-  ( U  e.  (UnifOn `  X )  ->  N : X --> ~P ~P X )
 
Theoremustuqtop1 23545* Lemma for ustuqtop 23550 (Contributed by Thierry Arnoux, 11-Jan-2018.)
 |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )   =>    |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  C_  b  /\  b  C_  X ) 
 /\  a  e.  ( N `  p ) ) 
 ->  b  e.  ( N `  p ) )
 
Theoremustuqtop2 23546* Lemma for ustuqtop 23550 (Contributed by Thierry Arnoux, 11-Jan-2018.)
 |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )   =>    |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( fi `  ( N `  p ) ) 
 C_  ( N `  p ) )
 
Theoremustuqtop3 23547* Lemma for ustuqtop 23550 (Contributed by Thierry Arnoux, 11-Jan-2018.)
 |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )   =>    |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p ) ) 
 ->  p  e.  a
 )
 
Theoremustuqtop4 23548* Lemma for ustuqtop 23550 (Contributed by Thierry Arnoux, 11-Jan-2018.)
 |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )   =>    |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p ) ) 
 ->  E. b  e.  ( N `  p ) A. q  e.  b  a  e.  ( N `  q
 ) )
 
Theoremustuqtop5 23549* Lemma for ustuqtop 23550 (Contributed by Thierry Arnoux, 11-Jan-2018.)
 |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )   =>    |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  X  e.  ( N `
  p ) )
 
Theoremustuqtop 23550* For a given uniform structure  U on a set  X, there is a unique topology  j such that the set  ran  ( v  e.  U  |->  ( v
" { p }
) ) is the filter of the neighbourhoods of  p for that topology. Proposition 1 of [BourbakiTop1] p. II.3. (Contributed by Thierry Arnoux, 11-Jan-2018.)
 |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )   =>    |-  ( U  e.  (UnifOn `  X )  ->  E! j  e.  (TopOn `  X ) A. p  e.  X  ( N `  p )  =  ( ( nei `  j ) `  { p } ) )
 
Theoremutopsnneiplem 23551* The neighborhoods of a point  P for the topology induced by an uniform space  U. (Contributed by Thierry Arnoux, 11-Jan-2018.)
 |-  J  =  (unifTop `  U )   &    |-  K  =  { a  e.  ~P X  |  A. p  e.  a  a  e.  ( N `  p ) }   &    |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )   =>    |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  ( ( nei `  J ) `  { P }
 )  =  ran  (
 v  e.  U  |->  ( v " { P } ) ) )
 
18.3.9.3  Uniform Spaces
 
Syntaxcuss 23552 Extend class notation with the Uniform Structure extractor function.
 class UnifSt
 
Syntaxcusp 23553 Extend class notation with the class of uniform spaces.
 class UnifSp
 
Syntaxctus 23554 Extend class notation with the function mapping a uniform structure to a uniform space.
 class 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  UnifSet 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  =  ( f  e.  _V  |->  ( ( UnifSet `  f )t  (
 ( Base `  f )  X.  ( Base `  f )
 ) ) )
 
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  =  {
 f  |  ( (UnifSt `  f )  e.  (UnifOn `  ( Base `  f )
 )  /\  ( TopOpen `  f )  =  (unifTop `  (UnifSt `  f )
 ) ) }
 
Definitiondf-tus 23557 Define the function mapping a uniform structure to a uniform space. (Contributed by Thierry Arnoux, 17-Nov-2017.)
 |- toUnifSp  =  ( u  e.  U. ran UnifOn  |->  ( { <. ( Base `  ndx ) ,  dom  U. u >. ,  <. ( UnifSet `  ndx ) ,  u >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  u ) >. ) )
 
Theoremussval 23558 The uniform structure on uniform space  W. This proof uses a trick with fvprc 5602 to avoid requiring  W to be a set. (Contributed by Thierry Arnoux, 3-Dec-2017.)
 |-  B  =  ( Base `  W )   &    |-  U  =  ( UnifSet `  W )   =>    |-  ( Ut  ( B  X.  B ) )  =  (UnifSt `  W )
 
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.)
 |-  B  =  ( Base `  W )   &    |-  U  =  ( UnifSet `  W )   =>    |-  (
 ( B  X.  B )  =  U. U  ->  U  =  (UnifSt `  W ) )
 
Theoremisusp 23560 The predicate  W is a uniform space. (Contributed by Thierry Arnoux, 4-Dec-2017.)
 |-  B  =  ( Base `  W )   &    |-  U  =  (UnifSt `  W )   &    |-  J  =  ( TopOpen `  W )   =>    |-  ( W  e. UnifSp  <->  ( U  e.  (UnifOn `  B )  /\  J  =  (unifTop `  U ) ) )
 
Theoremressunif 23561  UnifSet is unaffected by restriction. (Contributed by Thierry Arnoux, 7-Dec-2017.)
 |-  H  =  ( Gs  A )   &    |-  U  =  (
 UnifSet `  G )   =>    |-  ( A  e.  V  ->  U  =  (
 UnifSet `  H ) )
 
Theoremressuss 23562 Value of the uniform structure of a restricted space. (Contributed by Thierry Arnoux, 12-Dec-2017.)
 |-  ( A  e.  V  ->  (UnifSt `  ( Ws  A ) )  =  ( (UnifSt `  W )t  ( A  X.  A ) ) )
 
Theoremressusp 23563 The restriction of a uniform topological space to an open set is a uniform space. (Contributed by Thierry Arnoux, 16-Dec-2017.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   =>    |-  (
 ( W  e. UnifSp  /\  W  e.  TopSp  /\  A  e.  J )  ->  ( Ws  A )  e. UnifSp )
 
Theoremtusval 23564 The value of the uniform space mapping function. (Contributed by Thierry Arnoux, 5-Dec-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  (toUnifSp `  U )  =  ( { <. ( Base ` 
 ndx ) ,  dom  U. U >. ,  <. ( UnifSet `  ndx ) ,  U >. } sSet  <. (TopSet `  ndx ) ,  (unifTop `  U ) >. ) )
 
Theoremtuslem 23565 Lemma for tusbas 23566, tusunif 23567, and tustopn 23569. (Contributed by Thierry Arnoux, 5-Dec-2017.)
 |-  K  =  (toUnifSp `  U )   =>    |-  ( U  e.  (UnifOn `  X )  ->  ( X  =  ( Base `  K )  /\  U  =  ( UnifSet `  K )  /\  (unifTop `  U )  =  ( TopOpen `  K ) ) )
 
Theoremtusbas 23566 The base set of a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
 |-  K  =  (toUnifSp `  U )   =>    |-  ( U  e.  (UnifOn `  X )  ->  X  =  (
 Base `  K ) )
 
Theoremtusunif 23567 The uniform structure of a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
 |-  K  =  (toUnifSp `  U )   =>    |-  ( U  e.  (UnifOn `  X )  ->  U  =  (
 UnifSet `  K ) )
 
Theoremtususs 23568 The uniform structure of a constructed uniform space. (Contributed by Thierry Arnoux, 15-Dec-2017.)
 |-  K  =  (toUnifSp `  U )   =>    |-  ( U  e.  (UnifOn `  X )  ->  U  =  (UnifSt `  K ) )
 
Theoremtustopn 23569 The topology induced by a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
 |-  K  =  (toUnifSp `  U )   &    |-  J  =  (unifTop `  U )   =>    |-  ( U  e.  (UnifOn `  X )  ->  J  =  (
 TopOpen `  K ) )
 
Theoremtususp 23570 A constructed uniform space is an uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
 |-  K  =  (toUnifSp `  U )   =>    |-  ( U  e.  (UnifOn `  X )  ->  K  e. UnifSp )
 
18.3.9.4  Uniform continuity
 
Syntaxcucn 23571 Extend class notation with the uniform continuity operation.
 class 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  f is uniformly continuous if, roughly speaking, it is possible to guarantee that  ( f `  x
) and  ( f `  y ) be as close to each other as we please by requiring only that  x and  y are sufficiently close to each other; unlike ordinary continuity, the maximum distance between  ( f `  x
) and  ( f `  y ) cannot depend on  x and  y themselves. This formulation is the definition 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |- Cnu  =  ( u  e.  U. ran UnifOn ,  v  e.  U. ran UnifOn  |->  { f  e.  ( dom  U. v  ^m  dom  U. u )  |  A. s  e.  v  E. r  e.  u  A. x  e. 
 dom  U. u A. y  e.  dom  U. u ( x r y  ->  (
 f `  x )
 s ( f `  y ) ) }
 )
 
Theoremucnval 23573* The set of all uniformly continuous function from uniform space  U to uniform space  V. (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |-  (
 ( U  e.  (UnifOn `  X )  /\  V  e.  (UnifOn `  Y )
 )  ->  ( U Cnu V )  =  { f  e.  ( Y  ^m  X )  |  A. s  e.  V  E. r  e.  U  A. x  e.  X  A. y  e.  X  ( x r y  ->  ( f `  x ) s ( f `  y ) ) } )
 
Theoremisucn 23574* The predicate " F is a uniformly continuous function from uniform space  U to uniform space  V." (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |-  (
 ( U  e.  (UnifOn `  X )  /\  V  e.  (UnifOn `  Y )
 )  ->  ( F  e.  ( U Cnu V )  <->  ( F : X
 --> Y  /\  A. s  e.  V  E. r  e.  U  A. x  e.  X  A. y  e.  X  ( x r y  ->  ( F `  x ) s ( F `  y ) ) ) ) )
 
Theoremucnimalem 23575* Reformulate the  G function as a mapping with one variable. (Contributed by Thierry Arnoux, 19-Nov-2017.)
 |-  ( ph  ->  U  e.  (UnifOn `  X ) )   &    |-  ( ph  ->  V  e.  (UnifOn `  Y ) )   &    |-  ( ph  ->  F  e.  ( U Cnu
 V ) )   &    |-  ( ph  ->  W  e.  V )   &    |-  G  =  ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y ) >. )   =>    |-  G  =  ( p  e.  ( X  X.  X )  |->  <. ( F `  ( 1st `  p )
 ) ,  ( F `
  ( 2nd `  p ) ) >. )
 
Theoremucnima 23576* An equivalent statement of the definition of uniformly continuous function. (Contributed by Thierry Arnoux, 19-Nov-2017.)
 |-  ( ph  ->  U  e.  (UnifOn `  X ) )   &    |-  ( ph  ->  V  e.  (UnifOn `  Y ) )   &    |-  ( ph  ->  F  e.  ( U Cnu
 V ) )   &    |-  ( ph  ->  W  e.  V )   &    |-  G  =  ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y ) >. )   =>    |-  ( ph  ->  E. r  e.  U  ( G "
 r )  C_  W )
 
Theoremucnprima 23577* The preimage by a uniformly continuous function  F of an entourage  W of  Y is an entourage of  X. Note of the definition 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 19-Nov-2017.)
 |-  ( ph  ->  U  e.  (UnifOn `  X ) )   &    |-  ( ph  ->  V  e.  (UnifOn `  Y ) )   &    |-  ( ph  ->  F  e.  ( U Cnu
 V ) )   &    |-  ( ph  ->  W  e.  V )   &    |-  G  =  ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y ) >. )   =>    |-  ( ph  ->  ( `' G " W )  e.  U )
 
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.)
 |-  ( U  e.  (UnifOn `  X )  ->  (  _I  |`  X )  e.  ( U Cnu U ) )
 
Theoremcstucnd 23579 A constant function is uniformly continuous. Deduction form. Example 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |-  ( ph  ->  U  e.  (UnifOn `  X ) )   &    |-  ( ph  ->  V  e.  (UnifOn `  Y ) )   &    |-  ( ph  ->  A  e.  Y )   =>    |-  ( ph  ->  ( X  X.  { A }
 )  e.  ( U Cnu V ) )
 
18.3.9.5  Cauchy filters in uniform spaces
 
Syntaxccfilu 23580 Extend class notation with the set of Cauchy filters.
 class 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  v, there is an element  a of the filter "small enough in  v", i.e. such that every pair  { x ,  y } of points in  a is related by v." . Definition 2 of [BourbakiTop1] p. II.13. (Contributed by Thierry Arnoux, 16-Nov-2017.)
 |- CauFilu  =  ( u  e.  U. ran UnifOn  |->  { f  e.  ( fBas `  dom  U. u )  |  A. v  e.  u  E. a  e.  f  ( a  X.  a )  C_  v }
 )
 
Theoremiscfilu 23582* The predicate " F is a Cauchy filter base on uniform space  U." (Contributed by Thierry Arnoux, 18-Nov-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  ( F  e.  (CauFilu `  U )  <->  ( F  e.  ( fBas `  X )  /\  A. v  e.  U  E. a  e.  F  ( a  X.  a
 )  C_  v )
 ) )
 
Theoremcfilufbas 23583 A Cauchy filter base is a filter base. (Contributed by Thierry Arnoux, 19-Nov-2017.)
 |-  (
 ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `
  U ) ) 
 ->  F  e.  ( fBas `  X ) )
 
Theoremcfiluexsm 23584* For a Cauchy filter base and any entourage  V, there is an element of the filter small in  V. (Contributed by Thierry Arnoux, 19-Nov-2017.)
 |-  (
 ( U  e.  (UnifOn `  X )  /\  F  e.  (CauFilu `
  U )  /\  V  e.  U )  ->  E. a  e.  F  ( a  X.  a
 )  C_  V )
 
Theoremfmucndlem 23585* Lemma for fmucnd 23586. (Contributed by Thierry Arnoux, 19-Nov-2017.)
 |-  (
 ( F  Fn  X  /\  A  C_  X )  ->  ( ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y ) >. )
 " ( A  X.  A ) )  =  ( ( F " A )  X.  ( F " A ) ) )
 
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.)
 |-  ( ph  ->  U  e.  (UnifOn `  X ) )   &    |-  ( ph  ->  V  e.  (UnifOn `  Y ) )   &    |-  ( ph  ->  F  e.  ( U Cnu
 V ) )   &    |-  ( ph  ->  C  e.  (CauFilu `  U ) )   &    |-  D  =  ran  ( a  e.  C  |->  ( F "
 a ) )   =>    |-  ( ph  ->  D  e.  (CauFilu `
  V ) )
 
18.3.9.6  Complete uniform spaces
 
Syntaxccusp 23587 Extend class notation with the class of all complete uniform spaces.
 class 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  =  { w  e. UnifSp  |  A. c  e.  ( Fil `  ( Base `  w ) ) ( c  e.  (CauFilu `  (UnifSt `  w ) ) 
 ->  ( ( TopOpen `  w )  fLim  c )  =/=  (/) ) }
 
Theoremiscusp 23589* The predicate " W is a complete uniform space." (Contributed by Thierry Arnoux, 3-Dec-2017.)
 |-  ( W  e. CUnifSp  <->  ( W  e. UnifSp  /\ 
 A. c  e.  ( Fil `  ( Base `  W ) ) ( c  e.  (CauFilu `
  (UnifSt `  W ) )  ->  ( (
 TopOpen `  W )  fLim  c )  =/=  (/) ) ) )
 
Theoremcuspusp 23590 A complete uniform space is an uniform space. (Contributed by Thierry Arnoux, 3-Dec-2017.)
 |-  ( W  e. CUnifSp  ->  W  e. UnifSp )
 
Theoremcuspcvg 23591 In a complete uniform space, any Cauchy filter  C has a limit. (Contributed by Thierry Arnoux, 3-Dec-2017.)
 |-  B  =  ( Base `  W )   &    |-  J  =  ( TopOpen `  W )   =>    |-  (
 ( W  e. CUnifSp  /\  C  e.  (CauFilu `
  (UnifSt `  W ) )  /\  C  e.  ( Fil `  B )
 )  ->  ( J  fLim  C )  =/=  (/) )
 
Theoremiscusp2 23592* The predicate " W is a complete uniform space." (Contributed by Thierry Arnoux, 15-Dec-2017.)
 |-  B  =  ( Base `  W )   &    |-  U  =  (UnifSt `  W )   &    |-  J  =  ( TopOpen `  W )   =>    |-  ( W  e. CUnifSp  <->  ( W  e. UnifSp  /\ 
 A. c  e.  ( Fil `  B ) ( c  e.  (CauFilu `  U )  ->  ( J  fLim  c )  =/=  (/) ) ) )
 
18.3.9.7  The uniform structure generated by a metric
 
Theoremmetuval 23593* Value of the uniform structure generated by metric  D. (Contributed by Thierry Arnoux, 1-Dec-2017.)
 |-  ( D  e.  ( * Met `  X )  ->  (metUnif `  D )  =  ( ( X  X.  X ) filGen ran  (
 a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) ) )
 
Theoremmetustel 23594* Define a filter base  F generated by a metric  D. (Contributed by Thierry Arnoux, 22-Nov-2017.)
 |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
 ) ) )   =>    |-  ( D  e.  ( * Met `  X )  ->  ( B  e.  F 
 <-> 
 E. a  e.  RR+  B  =  ( `' D " ( 0 [,) a
 ) ) ) )
 
Theoremmetustss 23595* Range of the elements of the filter base generated by the metric  D. (Contributed by Thierry Arnoux, 28-Nov-2017.)
 |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
 ) ) )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  F )  ->  A  C_  ( X  X.  X ) )
 
Theoremmetustrel 23596* Elements of the filter base generated by the metric  D are relations. (Contributed by Thierry Arnoux, 28-Nov-2017.)
 |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
 ) ) )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  F )  ->  Rel  A )
 
Theoremmetustto 23597* Any two elements of the filter base generated by the metric  D can be compared, like for RR+ (i.e. it's totally ordered). (Contributed by Thierry Arnoux, 22-Nov-2017.)
 |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
 ) ) )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  F  /\  B  e.  F )  ->  ( A 
 C_  B  \/  B  C_  A ) )
 
Theoremmetustid 23598* The identity diagonal is included in all elements of the filter base generated by the metric  D. (Contributed by Thierry Arnoux, 22-Nov-2017.)
 |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
 ) ) )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  F )  ->  (  _I  |`  X )  C_  A )
 
Theoremmetustsym 23599* Elements of the filter base generated by the metric  D are symmetric. (Contributed by Thierry Arnoux, 28-Nov-2017.)
 |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
 ) ) )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  F )  ->  `' A  =  A )
 
Theoremmetustexhalf 23600* For any element  A of the filter base generated by the metric  D, the half element (corresponding to half the distance) is also in this base. (Contributed by Thierry Arnoux, 28-Nov-2017.)
 |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
 ) ) )   =>    |-  ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X ) )  /\  A  e.  F )  ->  E. v  e.  F  ( v  o.  v )  C_  A )
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