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Theorem List for Metamath Proof Explorer - 17701-17800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremflfval 17701 Given a function from a filtered set to a topological space, define the set of limit points of the function. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  ( ( J  fLimf  L ) `  F )  =  ( J  fLim  ( ( X  FilMap  F ) `
  L ) ) )
 
Theoremflfnei 17702* The property of being a limit point of a function in terms of neighborhoods. (Contributed by Jeff Hankins, 9-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  ( A  e.  (
 ( J  fLimf  L ) `
  F )  <->  ( A  e.  X  /\  A. n  e.  ( ( nei `  J ) `  { A }
 ) E. s  e.  L  ( F "
 s )  C_  n ) ) )
 
Theoremflfneii 17703* A neighborhood of a limit point of a function contains the image of a filter element. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  A  e.  ( ( J  fLimf  L ) `  F )  /\  N  e.  ( ( nei `  J ) `  { A }
 ) )  ->  E. s  e.  L  ( F "
 s )  C_  N )
 
Theoremisflf 17704* The property of being a limit point of a function. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  ( A  e.  (
 ( J  fLimf  L ) `
  F )  <->  ( A  e.  X  /\  A. o  e.  J  ( A  e.  o  ->  E. s  e.  L  ( F " s ) 
 C_  o ) ) ) )
 
Theoremflfelbas 17705 A limit point of a function is in the topological space. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
 |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X ) 
 /\  A  e.  (
 ( J  fLimf  L ) `
  F ) ) 
 ->  A  e.  X )
 
Theoremflffbas 17706* Limit points of a function can be defined using filter bases. (Contributed by Jeff Hankins, 9-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
 |-  L  =  ( Y
 filGen B )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  Y )  /\  F : Y --> X )  ->  ( A  e.  (
 ( J  fLimf  L ) `
  F )  <->  ( A  e.  X  /\  A. o  e.  J  ( A  e.  o  ->  E. s  e.  B  ( F " s ) 
 C_  o ) ) ) )
 
Theoremflftg 17707* Limit points of a function can be defined using topological bases. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  J  =  ( topGen `  B )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  ( A  e.  (
 ( J  fLimf  L ) `
  F )  <->  ( A  e.  X  /\  A. o  e.  B  ( A  e.  o  ->  E. s  e.  L  ( F " s ) 
 C_  o ) ) ) )
 
Theoremhausflf 17708* If a function has its values in a Hausdorff space, then it has at most one limit value. (Contributed by FL, 14-Nov-2010.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Haus  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  E* x  x  e.  ( ( J  fLimf  L ) `  F ) )
 
Theoremhausflf2 17709 If a convergent function has its values in a Hausdorff space, then it has a unique limit. (Contributed by FL, 14-Nov-2010.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  ( ( J  fLimf  L ) `  F )  =/=  (/) )  ->  (
 ( J  fLimf  L ) `
  F )  ~~  1o )
 
Theoremcnpflfi 17710 Forward direction of cnpflf 17712. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( ( A  e.  ( J  fLim  L ) 
 /\  F  e.  (
 ( J  CnP  K ) `  A ) ) 
 ->  ( F `  A )  e.  ( ( K  fLimf  L ) `  F ) )
 
Theoremcnpflf2 17711  F is continous at point  A iff a limit of  F when  x tends to  A is  ( F `  A ). Proposition 9 of [BourbakiTop1] p. TG I.50. (Contributed by FL, 29-May-2011.) (Revised by Mario Carneiro, 9-Apr-2015.)
 |-  L  =  ( ( nei `  J ) `  { A } )   =>    |-  (
 ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X ) 
 ->  ( F  e.  (
 ( J  CnP  K ) `  A )  <->  ( F : X
 --> Y  /\  ( F `
  A )  e.  ( ( K  fLimf  L ) `  F ) ) ) )
 
Theoremcnpflf 17712* Continuity of a function at a point in terms of filter limits. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  A )  <-> 
 ( F : X --> Y  /\  A. f  e.  ( Fil `  X ) ( A  e.  ( J  fLim  f ) 
 ->  ( F `  A )  e.  ( ( K  fLimf  f ) `  F ) ) ) ) )
 
Theoremcnflf 17713* A function is continuous iff it respects filter limits. (Contributed by Jeff Hankins, 6-Sep-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  ->  ( F  e.  ( J  Cn  K )  <->  ( F : X
 --> Y  /\  A. f  e.  ( Fil `  X ) A. x  e.  ( J  fLim  f ) ( F `  x )  e.  ( ( K 
 fLimf  f ) `  F ) ) ) )
 
Theoremcnflf2 17714* A function is continuous iff it respects filter limits. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  ->  ( F  e.  ( J  Cn  K )  <->  ( F : X
 --> Y  /\  A. f  e.  ( Fil `  X ) ( F "
 ( J  fLim  f
 ) )  C_  (
 ( K  fLimf  f ) `
  F ) ) ) )
 
Theoremflfcnp 17715 A continuous function preserves filter limits. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X ) 
 /\  ( A  e.  ( ( J  fLimf  L ) `  F ) 
 /\  G  e.  (
 ( J  CnP  K ) `  A ) ) )  ->  ( G `  A )  e.  (
 ( K  fLimf  L ) `
  ( G  o.  F ) ) )
 
Theoremlmflf 17716 The topological limit relation on functions can be written in terms of the filter limit along the filter generated by the upper integer sets. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  L  =  ( Z filGen ( ZZ>= " Z ) )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z --> X )  ->  ( F ( ~~> t `  J ) P  <->  P  e.  (
 ( J  fLimf  L ) `
  F ) ) )
 
Theoremtxflf 17717* Two sequences converge in a filter iff the sequence of their ordered pairs converges. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  ( Fil `  Z ) )   &    |-  ( ph  ->  F : Z --> X )   &    |-  ( ph  ->  G : Z
 --> Y )   &    |-  H  =  ( n  e.  Z  |->  <.
 ( F `  n ) ,  ( G `  n ) >. )   =>    |-  ( ph  ->  (
 <. R ,  S >.  e.  ( ( ( J 
 tX  K )  fLimf  L ) `  H )  <-> 
 ( R  e.  (
 ( J  fLimf  L ) `
  F )  /\  S  e.  ( ( K  fLimf  L ) `  G ) ) ) )
 
Theoremflfcnp2 17718* 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, 19-Sep-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  ( Fil `  Z ) )   &    |-  ( ( ph  /\  x  e.  Z ) 
 ->  A  e.  X )   &    |-  ( ( ph  /\  x  e.  Z )  ->  B  e.  Y )   &    |-  ( ph  ->  R  e.  ( ( J 
 fLimf  L ) `  ( x  e.  Z  |->  A ) ) )   &    |-  ( ph  ->  S  e.  ( ( K 
 fLimf  L ) `  ( x  e.  Z  |->  B ) ) )   &    |-  ( ph  ->  O  e.  ( ( ( J  tX  K )  CnP  N ) `  <. R ,  S >. ) )   =>    |-  ( ph  ->  ( R O S )  e.  ( ( N 
 fLimf  L ) `  ( x  e.  Z  |->  ( A O B ) ) ) )
 
Theoremfclsval 17719* The set of all cluster points of a filter. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  ->  ( J  fClus  F )  =  if ( X  =  Y ,  |^|_ t  e.  F  ( ( cls `  J ) `  t ) ,  (/) ) )
 
Theoremisfcls 17720* A cluster point of a filter. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( A  e.  ( J  fClus  F )  <->  ( J  e.  Top  /\  F  e.  ( Fil `  X )  /\  A. s  e.  F  A  e.  ( ( cls `  J ) `  s ) ) )
 
Theoremfclsfil 17721 Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( A  e.  ( J  fClus  F )  ->  F  e.  ( Fil `  X )
 )
 
Theoremfclstop 17722 Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( A  e.  ( J  fClus  F )  ->  J  e.  Top )
 
Theoremfclstopon 17723 Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( A  e.  ( J  fClus  F )  ->  ( J  e.  (TopOn `  X )  <->  F  e.  ( Fil `  X ) ) )
 
Theoremisfcls2 17724* A cluster point of a filter. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X ) )  ->  ( A  e.  ( J  fClus  F )  <->  A. s  e.  F  A  e.  ( ( cls `  J ) `  s ) ) )
 
Theoremfclsopn 17725* Write the cluster point condition in terms of open sets. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X ) )  ->  ( A  e.  ( J  fClus  F )  <->  ( A  e.  X  /\  A. o  e.  J  ( A  e.  o  ->  A. s  e.  F  ( o  i^i  s )  =/=  (/) ) ) ) )
 
Theoremfclsopni 17726 An open neighborhood of a cluster point of a filter intersects any element of that filter. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( ( A  e.  ( J  fClus  F ) 
 /\  ( U  e.  J  /\  A  e.  U  /\  S  e.  F ) )  ->  ( U  i^i  S )  =/=  (/) )
 
Theoremfclselbas 17727 A cluster point is in the base set. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( A  e.  ( J  fClus  F )  ->  A  e.  X )
 
Theoremfclsneii 17728 A neighborhood of a cluster point of a filter intersects any element of that filter. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( ( A  e.  ( J  fClus  F ) 
 /\  N  e.  (
 ( nei `  J ) `  { A } )  /\  S  e.  F ) 
 ->  ( N  i^i  S )  =/=  (/) )
 
Theoremfclssscls 17729 The set of cluster points is a subset of the closure of any filter element. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( S  e.  F  ->  ( J  fClus  F ) 
 C_  ( ( cls `  J ) `  S ) )
 
Theoremfclsnei 17730* Cluster points in terms of neighborhoods. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X ) )  ->  ( A  e.  ( J  fClus  F )  <->  ( A  e.  X  /\  A. n  e.  ( ( nei `  J ) `  { A }
 ) A. s  e.  F  ( n  i^i  s )  =/=  (/) ) ) )
 
Theoremsupnfcls 17731* The filter of supersets of  X  \  U does not cluster at any point of the open set  U. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  U )  ->  -.  A  e.  ( J  fClus  { x  e. 
 ~P X  |  ( X  \  U ) 
 C_  x } )
 )
 
Theoremfclsbas 17732* Cluster points in terms of filter bases. (Contributed by Jeff Hankins, 13-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  F  =  ( X
 filGen B )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  X ) )  ->  ( A  e.  ( J  fClus  F )  <->  ( A  e.  X  /\  A. o  e.  J  ( A  e.  o  ->  A. s  e.  B  ( o  i^i  s )  =/=  (/) ) ) ) )
 
Theoremfclsss1 17733 A finer topology has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X )  /\  J  C_  K )  ->  ( K  fClus  F )  C_  ( J  fClus  F ) )
 
Theoremfclsss2 17734 A finer filter has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X )  /\  F  C_  G )  ->  ( J  fClus  G )  C_  ( J  fClus  F ) )
 
Theoremfclsrest 17735 The set of cluster points in a restricted topological space. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X )  /\  Y  e.  F )  ->  (
 ( Jt  Y )  fClus  ( Ft  Y ) )  =  ( ( J  fClus  F )  i^i  Y ) )
 
Theoremfclscf 17736* Characterization of fineness of topologies in terms of cluster points. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  ->  ( J 
 C_  K  <->  A. f  e.  ( Fil `  X ) ( K  fClus  f )  C_  ( J  fClus  f ) ) )
 
Theoremflimfcls 17737 A limit point is a cluster point. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( J  fLim  F ) 
 C_  ( J  fClus  F )
 
Theoremfclsfnflim 17738* A filter clusters at a point iff a finer filter converges to it. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  ( A  e.  ( J  fClus  F )  <->  E. g  e.  ( Fil `  X ) ( F  C_  g  /\  A  e.  ( J  fLim  g ) ) ) )
 
Theoremflimfnfcls 17739* A filter converges to a point iff every finer filter clusters there. Along with fclsfnflim 17738, this theorem illustrates the duality between convergence and clustering. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( F  e.  ( Fil `  X )  ->  ( A  e.  ( J  fLim  F )  <->  A. g  e.  ( Fil `  X ) ( F  C_  g  ->  A  e.  ( J  fClus  g ) ) ) )
 
Theoremfclscmpi 17740 Forward direction of fclscmp 17741. Every filter clusters in a compact space. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  ->  ( J  fClus  F )  =/=  (/) )
 
Theoremfclscmp 17741* A space is compact iff every filter clusters. (Contributed by Jeff Hankins, 20-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( J  e.  (TopOn `  X )  ->  ( J  e.  Comp  <->  A. f  e.  ( Fil `  X ) ( J  fClus  f )  =/=  (/) ) )
 
Theoremuffclsflim 17742 The cluster points of an ultrafilter are its limit points. (Contributed by Jeff Hankins, 11-Dec-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
 |-  ( F  e.  ( UFil `  X )  ->  ( J  fClus  F )  =  ( J  fLim  F ) )
 
Theoremufilcmp 17743* A space is compact iff every ultrafilter converges. (Contributed by Jeff Hankins, 11-Dec-2009.) (Proof shortened by Mario Carneiro, 12-Apr-2015.) (Revised by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( X  e. UFL  /\  J  e.  (TopOn `  X ) )  ->  ( J  e.  Comp  <->  A. f  e.  ( UFil `  X ) ( J  fLim  f )  =/= 
 (/) ) )
 
Theoremfcfval 17744 The set of cluster points of a function. (Contributed by Jeff Hankins, 24-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  ( ( J  fClusf  L ) `  F )  =  ( J  fClus  ( ( X  FilMap  F ) `
  L ) ) )
 
Theoremisfcf 17745* The property of being a cluster point of a function. (Contributed by Jeff Hankins, 24-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  ( A  e.  (
 ( J  fClusf  L ) `
  F )  <->  ( A  e.  X  /\  A. o  e.  J  ( A  e.  o  ->  A. s  e.  L  ( o  i^i  ( F
 " s ) )  =/=  (/) ) ) ) )
 
Theoremfcfnei 17746* The property of being a cluster point of a function in terms of neighborhoods. (Contributed by Jeff Hankins, 26-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  ( A  e.  (
 ( J  fClusf  L ) `
  F )  <->  ( A  e.  X  /\  A. n  e.  ( ( nei `  J ) `  { A }
 ) A. s  e.  L  ( n  i^i  ( F
 " s ) )  =/=  (/) ) ) )
 
Theoremfcfelbas 17747 A cluster point of a function is in the base set of the topology. (Contributed by Jeff Hankins, 26-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X ) 
 /\  A  e.  (
 ( J  fClusf  L ) `
  F ) ) 
 ->  A  e.  X )
 
Theoremfcfneii 17748 A neighborhood of a cluster point of a function contains a function value from every tail. (Contributed by Jeff Hankins, 27-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X ) 
 /\  ( A  e.  ( ( J  fClusf  L ) `  F ) 
 /\  N  e.  (
 ( nei `  J ) `  { A } )  /\  S  e.  L ) )  ->  ( N  i^i  ( F " S ) )  =/=  (/) )
 
Theoremflfssfcf 17749 A limit point of a function is a cluster point of the function. (Contributed by Jeff Hankins, 28-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  ( ( J  fLimf  L ) `  F ) 
 C_  ( ( J 
 fClusf  L ) `  F ) )
 
Theoremuffcfflf 17750 If the domain filter is an ultrafilter, the cluster points of the function are the limit points. (Contributed by Jeff Hankins, 12-Dec-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( UFil `  Y )  /\  F : Y --> X )  ->  ( ( J  fClusf  L ) `  F )  =  ( ( J 
 fLimf  L ) `  F ) )
 
Theoremcnpfcfi 17751 Lemma for cnpfcf 17752. If a function is continuous at a point, it respects clustering there. (Contributed by Jeff Hankins, 20-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( ( K  e.  Top  /\  A  e.  ( J 
 fClus  L )  /\  F  e.  ( ( J  CnP  K ) `  A ) )  ->  ( F `  A )  e.  (
 ( K  fClusf  L ) `
  F ) )
 
Theoremcnpfcf 17752* A function  F is continuous at point  A iff  F respects cluster points there. (Contributed by Jeff Hankins, 14-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  A )  <-> 
 ( F : X --> Y  /\  A. f  e.  ( Fil `  X ) ( A  e.  ( J  fClus  f ) 
 ->  ( F `  A )  e.  ( ( K  fClusf  f ) `  F ) ) ) ) )
 
Theoremcnfcf 17753* Continuity of a function in terms of cluster points of a function. (Contributed by Jeff Hankins, 28-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  ->  ( F  e.  ( J  Cn  K )  <->  ( F : X
 --> Y  /\  A. f  e.  ( Fil `  X ) A. x  e.  ( J  fClus  f ) ( F `  x )  e.  ( ( K 
 fClusf  f ) `  F ) ) ) )
 
Theoremalexsublem 17754* Lemma for alexsub 17755. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( ph  ->  X  e. UFL )   &    |-  ( ph  ->  X  =  U. B )   &    |-  ( ph  ->  J  =  ( topGen `  ( fi `  B ) ) )   &    |-  ( ( ph  /\  ( x  C_  B  /\  X  =  U. x ) ) 
 ->  E. y  e.  ( ~P x  i^i  Fin ) X  =  U. y )   &    |-  ( ph  ->  F  e.  ( UFil `  X )
 )   &    |-  ( ph  ->  ( J  fLim  F )  =  (/) )   =>    |- 
 -.  ph
 
Theoremalexsub 17755* The Alexander Subbase Theorem: If 
B is a subbase for the topology  J, and any cover taken from  B has a finite subcover, then the generated topology is compact. This proof uses the ultrafilter lemma; see alexsubALT 17761 for a proof using Zorn's lemma. (Contributed by Jeff Hankins, 24-Jan-2010.) (Revised by Mario Carneiro, 26-Aug-2015.)
 |-  ( ph  ->  X  e. UFL )   &    |-  ( ph  ->  X  =  U. B )   &    |-  ( ph  ->  J  =  ( topGen `  ( fi `  B ) ) )   &    |-  ( ( ph  /\  ( x  C_  B  /\  X  =  U. x ) ) 
 ->  E. y  e.  ( ~P x  i^i  Fin ) X  =  U. y )   =>    |-  ( ph  ->  J  e.  Comp
 )
 
Theoremalexsubb 17756* Biconditional form of the Alexander Subbase Theorem alexsub 17755. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  ( ( X  e. UFL  /\  X  =  U. B )  ->  ( ( topGen `  ( fi `  B ) )  e.  Comp  <->  A. x  e.  ~P  B ( X  =  U. x  ->  E. y  e.  ( ~P x  i^i  Fin ) X  =  U. y ) ) )
 
TheoremalexsubALTlem1 17757* Lemma for alexsubALT 17761. A compact space has a subbase such that every cover taken from it has a finite subcover. (Contributed by Jeff Hankins, 27-Jan-2010.)
 |-  X  =  U. J   =>    |-  ( J  e.  Comp  ->  E. x ( J  =  ( topGen `
  ( fi `  x ) )  /\  A. c  e.  ~P  x ( X  =  U. c  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d ) ) )
 
TheoremalexsubALTlem2 17758* Lemma for alexsubALT 17761. Every subset of a base which has no finite subcover is a subset of a maximal such collection. (Contributed by Jeff Hankins, 27-Jan-2010.)
 |-  X  =  U. J   =>    |-  (
 ( ( J  =  ( topGen `  ( fi `  x ) )  /\  A. c  e.  ~P  x ( X  =  U. c  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d ) 
 /\  a  e.  ~P ( fi `  x ) )  /\  A. b  e.  ( ~P a  i^i 
 Fin )  -.  X  =  U. b )  ->  E. u  e.  ( { z  e.  ~P ( fi `  x )  |  ( a  C_  z  /\  A. b  e.  ( ~P z  i^i 
 Fin )  -.  X  =  U. b ) }  u.  { (/) } ) A. v  e.  ( {
 z  e.  ~P ( fi `  x )  |  ( a  C_  z  /\  A. b  e.  ( ~P z  i^i  Fin )  -.  X  =  U. b
 ) }  u.  { (/)
 } )  -.  u  C.  v )
 
TheoremalexsubALTlem3 17759* Lemma for alexsubALT 17761. If a point is covered by a collection taken from the base with no finite subcover, a set from the subbase can be added that covers the point so that the resulting collection has no finite subcover. (Contributed by Jeff Hankins, 28-Jan-2010.) (Revised by Mario Carneiro, 14-Dec-2013.)
 |-  X  =  U. J   =>    |-  (
 ( ( ( ( J  =  ( topGen `  ( fi `  x ) )  /\  A. c  e.  ~P  x ( X  =  U. c  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d ) 
 /\  a  e.  ~P ( fi `  x ) )  /\  ( u  e.  ~P ( fi
 `  x )  /\  ( a  C_  u  /\  A. b  e.  ( ~P u  i^i  Fin )  -.  X  =  U. b
 ) ) )  /\  w  e.  u )  /\  ( ( t  e.  ( ~P x  i^i  Fin )  /\  w  = 
 |^| t )  /\  ( y  e.  w  /\  -.  y  e.  U. ( x  i^i  u ) ) ) )  ->  E. s  e.  t  A. n  e.  ( ~P ( u  u.  {
 s } )  i^i 
 Fin )  -.  X  =  U. n )
 
TheoremalexsubALTlem4 17760* Lemma for alexsubALT 17761. If any cover taken from a subbase has a finite subcover, any cover taken from the corresponding base has a finite subcover. (Contributed by Jeff Hankins, 28-Jan-2010.) (Revised by Mario Carneiro, 14-Dec-2013.)
 |-  X  =  U. J   =>    |-  ( J  =  ( topGen `  ( fi `  x ) )  ->  ( A. c  e.  ~P  x ( X  =  U. c  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d ) 
 ->  A. a  e.  ~P  ( fi `  x ) ( X  =  U. a  ->  E. b  e.  ( ~P a  i^i  Fin ) X  =  U. b ) ) )
 
TheoremalexsubALT 17761* The Alexander Subbase Theorem: a space is compact iff it has a subbase such that any cover taken from the subbase has a finite subcover. (Contributed by Jeff Hankins, 24-Jan-2010.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  X  =  U. J   =>    |-  ( J  e.  Comp  <->  E. x ( J  =  ( topGen `  ( fi `  x ) ) 
 /\  A. c  e.  ~P  x ( X  =  U. c  ->  E. d  e.  ( ~P c  i^i 
 Fin ) X  =  U. d ) ) )
 
Theoremptcmplem1 17762* Lemma for ptcmp 17768. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  S  =  ( k  e.  A ,  u  e.  ( F `  k
 )  |->  ( `' ( w  e.  X  |->  ( w `
  k ) )
 " u ) )   &    |-  X  =  X_ n  e.  A  U. ( F `
  n )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> Comp )   &    |-  ( ph  ->  X  e.  (UFL  i^i  dom  card
 ) )   =>    |-  ( ph  ->  ( X  =  U. ( ran 
 S  u.  { X } )  /\  ( Xt_ `  F )  =  (
 topGen `  ( fi `  ( ran  S  u.  { X } ) ) ) ) )
 
Theoremptcmplem2 17763* Lemma for ptcmp 17768. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  S  =  ( k  e.  A ,  u  e.  ( F `  k
 )  |->  ( `' ( w  e.  X  |->  ( w `
  k ) )
 " u ) )   &    |-  X  =  X_ n  e.  A  U. ( F `
  n )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> Comp )   &    |-  ( ph  ->  X  e.  (UFL  i^i  dom  card
 ) )   &    |-  ( ph  ->  U 
 C_  ran  S )   &    |-  ( ph  ->  X  =  U. U )   &    |-  ( ph  ->  -. 
 E. z  e.  ( ~P U  i^i  Fin ) X  =  U. z )   =>    |-  ( ph  ->  U_ k  e. 
 { n  e.  A  |  -.  U. ( F `
  n )  ~~  1o } U. ( F `
  k )  e. 
 dom  card )
 
Theoremptcmplem3 17764* Lemma for ptcmp 17768. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  S  =  ( k  e.  A ,  u  e.  ( F `  k
 )  |->  ( `' ( w  e.  X  |->  ( w `
  k ) )
 " u ) )   &    |-  X  =  X_ n  e.  A  U. ( F `
  n )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> Comp )   &    |-  ( ph  ->  X  e.  (UFL  i^i  dom  card
 ) )   &    |-  ( ph  ->  U 
 C_  ran  S )   &    |-  ( ph  ->  X  =  U. U )   &    |-  ( ph  ->  -. 
 E. z  e.  ( ~P U  i^i  Fin ) X  =  U. z )   &    |-  K  =  { u  e.  ( F `  k
 )  |  ( `' ( w  e.  X  |->  ( w `  k ) ) " u )  e.  U }   =>    |-  ( ph  ->  E. f ( f  Fn  A  /\  A. k  e.  A  ( f `  k )  e.  ( U. ( F `  k
 )  \  U. K ) ) )
 
Theoremptcmplem4 17765* Lemma for ptcmp 17768. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  S  =  ( k  e.  A ,  u  e.  ( F `  k
 )  |->  ( `' ( w  e.  X  |->  ( w `
  k ) )
 " u ) )   &    |-  X  =  X_ n  e.  A  U. ( F `
  n )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> Comp )   &    |-  ( ph  ->  X  e.  (UFL  i^i  dom  card
 ) )   &    |-  ( ph  ->  U 
 C_  ran  S )   &    |-  ( ph  ->  X  =  U. U )   &    |-  ( ph  ->  -. 
 E. z  e.  ( ~P U  i^i  Fin ) X  =  U. z )   &    |-  K  =  { u  e.  ( F `  k
 )  |  ( `' ( w  e.  X  |->  ( w `  k ) ) " u )  e.  U }   =>    |-  -.  ph
 
Theoremptcmplem5 17766* Lemma for ptcmp 17768. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  S  =  ( k  e.  A ,  u  e.  ( F `  k
 )  |->  ( `' ( w  e.  X  |->  ( w `
  k ) )
 " u ) )   &    |-  X  =  X_ n  e.  A  U. ( F `
  n )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> Comp )   &    |-  ( ph  ->  X  e.  (UFL  i^i  dom  card
 ) )   =>    |-  ( ph  ->  ( Xt_ `  F )  e. 
 Comp )
 
Theoremptcmpg 17767 Tychonoff's theorem: The product of compact spaces is compact. The choice principles needed are encoded in the last hypothesis: the base set of the product must be well-orderable and satisfy the ultrafilter lemma. Both these assumptions are satisfied if  ~P ~P X is well-orderable, so if we assume the Axiom of Choice we can eliminate them (see ptcmp 17768). (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  J  =  ( Xt_ `  F )   &    |-  X  =  U. J   =>    |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card ) )  ->  J  e.  Comp )
 
Theoremptcmp 17768 Tychonoff's theorem: The product of compact spaces is compact. The proof uses the Axiom of Choice. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( A  e.  V  /\  F : A --> Comp )  ->  ( Xt_ `  F )  e.  Comp )
 
11.2.5  Topological groups
 
Syntaxctmd 17769 Extend class notation with the class of all topological monoids.
 class TopMnd
 
Syntaxctgp 17770 Extend class notation with the class of all topological groups.
 class  TopGrp
 
Definitiondf-tmd 17771* Define the class of all topological monoids. A topological monoid is a monoid whose operation is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |- TopMnd  =  { f  e.  ( Mnd  i^i  TopSp )  |  [. ( TopOpen `  f )  /  j ]. ( + f `  f )  e.  ( ( j 
 tX  j )  Cn  j ) }
 
Definitiondf-tgp 17772* Define the class of all topological groups. A topological group is a group whose operation and inverse function are continuous. (Contributed by FL, 18-Apr-2010.)
 |-  TopGrp  =  { f  e.  ( Grp  i^i TopMnd )  | 
 [. ( TopOpen `  f
 )  /  j ]. ( inv g `  f
 )  e.  ( j  Cn  j ) }
 
Theoremistmd 17773 The predicate "is a topological monoid". (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  F  =  ( + f `  G )   &    |-  J  =  ( TopOpen `  G )   =>    |-  ( G  e. TopMnd  <->  ( G  e.  Mnd  /\  G  e.  TopSp  /\  F  e.  ( ( J  tX  J )  Cn  J ) ) )
 
Theoremtmdmnd 17774 A topological monoid is a monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  ( G  e. TopMnd  ->  G  e.  Mnd )
 
Theoremtmdtps 17775 A topological monoid is a topological space. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  ( G  e. TopMnd  ->  G  e.  TopSp )
 
Theoremistgp 17776 The predicate "is a topological group". Definition of [BourbakiTop1] p. III.1 (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  I  =  ( inv g `  G )   =>    |-  ( G  e.  TopGrp  <->  ( G  e.  Grp  /\  G  e. TopMnd  /\  I  e.  ( J  Cn  J ) ) )
 
Theoremtgpgrp 17777 A topological group is a group. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  ( G  e.  TopGrp  ->  G  e.  Grp )
 
Theoremtgptmd 17778 A topological group is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  ( G  e.  TopGrp  ->  G  e. TopMnd )
 
Theoremtgptps 17779 A topological group is a topological space. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  ( G  e.  TopGrp  ->  G  e.  TopSp )
 
Theoremtmdtopon 17780 The topology of a topological monoid. (Contributed by Mario Carneiro, 27-Jun-2014.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  X  =  (
 Base `  G )   =>    |-  ( G  e. TopMnd  ->  J  e.  (TopOn `  X ) )
 
Theoremtgptopon 17781 The topology of a topological group. (Contributed by Mario Carneiro, 27-Jun-2014.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  X  =  (
 Base `  G )   =>    |-  ( G  e.  TopGrp  ->  J  e.  (TopOn `  X ) )
 
Theoremtmdcn 17782 In a topological monoid, the operation  F representing the functionalization of the operator slot  +g is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  F  =  ( + f `  G )   =>    |-  ( G  e. TopMnd  ->  F  e.  ( ( J 
 tX  J )  Cn  J ) )
 
Theoremtgpcn 17783 In a topological group, the operation  F representing the functionalization of the operator slot  +g is continuous. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  F  =  ( + f `  G )   =>    |-  ( G  e.  TopGrp  ->  F  e.  ( ( J  tX  J )  Cn  J ) )
 
Theoremtgpinv 17784 In a topological group, the inverse function is continuous. (Contributed by FL, 21-Jun-2010.) (Revised by FL, 27-Jun-2014.)
 |-  J  =  ( TopOpen `  G )   &    |-  I  =  ( inv g `  G )   =>    |-  ( G  e.  TopGrp  ->  I  e.  ( J  Cn  J ) )
 
Theoremgrpinvhmeo 17785 The inverse function in a topological group is a homeomorphism from the group to itself. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  I  =  ( inv g `  G )   =>    |-  ( G  e.  TopGrp  ->  I  e.  ( J  Homeo  J ) )
 
Theoremcnmpt1plusg 17786* Continuity of the group sum; analogue of cnmpt12f 17376 which cannot be used directly because 
+g is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e. TopMnd )   &    |-  ( ph  ->  K  e.  (TopOn `  X ) )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( K  Cn  J ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( K  Cn  J ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( A  .+  B ) )  e.  ( K  Cn  J ) )
 
Theoremcnmpt2plusg 17787* Continuity of the group sum; analogue of cnmpt22f 17385 which cannot be used directly because  +g is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e. TopMnd )   &    |-  ( ph  ->  K  e.  (TopOn `  X ) )   &    |-  ( ph  ->  L  e.  (TopOn `  Y ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( K  tX  L )  Cn  J ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  B )  e.  ( ( K 
 tX  L )  Cn  J ) )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( A 
 .+  B ) )  e.  ( ( K 
 tX  L )  Cn  J ) )
 
Theoremtmdcn2 17788* Write out the definition of continuity of  +g explicitly. (Contributed by Mario Carneiro, 20-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  J  =  (
 TopOpen `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  (
 ( ( G  e. TopMnd  /\  U  e.  J ) 
 /\  ( X  e.  B  /\  Y  e.  B  /\  ( X  .+  Y )  e.  U )
 )  ->  E. u  e.  J  E. v  e.  J  ( X  e.  u  /\  Y  e.  v  /\  A. x  e.  u  A. y  e.  v  ( x  .+  y )  e.  U ) )
 
Theoremtgpsubcn 17789 In a topological group, the "subtraction" (or "division") is continuous. Axiom GT' of [BourbakiTop1] p. III.1 (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 19-Mar-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( G  e.  TopGrp  -> 
 .-  e.  ( ( J  tX  J )  Cn  J ) )
 
Theoremistgp2 17790 A group with a topology is a topological group iff the subtraction operation is continuous. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( G  e.  TopGrp  <->  ( G  e.  Grp  /\  G  e.  TopSp  /\  .-  e.  ( ( J  tX  J )  Cn  J ) ) )
 
Theoremtmdmulg 17791* In a topological monoid, the n-times group multiple function is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  .x.  =  (.g `  G )   &    |-  B  =  (
 Base `  G )   =>    |-  ( ( G  e. TopMnd  /\  N  e.  NN0 )  ->  ( x  e.  B  |->  ( N  .x.  x ) )  e.  ( J  Cn  J ) )
 
Theoremtgpmulg 17792* In a topological group, the n-times group multiple function is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  .x.  =  (.g `  G )   &    |-  B  =  (
 Base `  G )   =>    |-  ( ( G  e.  TopGrp  /\  N  e.  ZZ )  ->  ( x  e.  B  |->  ( N 
 .x.  x ) )  e.  ( J  Cn  J ) )
 
Theoremtgpmulg2 17793 In a topological monoid, the group multiple function is jointly continuous (although this is not saying much as one of the factors is discrete). Use zdis 18338 to write the left topology as a subset of the complexes. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( G  e.  TopGrp  ->  .x.  e.  ( ( ~P ZZ  tX  J )  Cn  J ) )
 
Theoremtmdgsum 17794* In a topological monoid, the group sum operation is a continuous function from the function space to the base topology. This theorem is not true when  A is infinite, because in this case for any basic open set of the domain one of the factors will be the whole space, so by varying the value of the functions to sum at this index, one can achieve any desired sum. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  B  =  (
 Base `  G )   =>    |-  ( ( G  e. CMnd  /\  G  e. TopMnd  /\  A  e.  Fin )  ->  ( x  e.  ( B  ^m  A )  |->  ( G  gsumg  x ) )  e.  ( ( J  ^ k o  ~P A )  Cn  J ) )
 
Theoremtmdgsum2 17795* For any neighborhood  U of  n X, there is a neighborhood  u of  X such that any sum of  n elements in  u sums to an element of  U. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  B  =  (
 Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e. TopMnd )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  U  e.  J )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  ( ( # `  A )  .x.  X )  e.  U )   =>    |-  ( ph  ->  E. u  e.  J  ( X  e.  u  /\  A. f  e.  ( u  ^m  A ) ( G  gsumg  f )  e.  U ) )
 
Theoremoppgtmd 17796 The opposite of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  O  =  (oppg `  G )   =>    |-  ( G  e. TopMnd  ->  O  e. TopMnd )
 
Theoremoppgtgp 17797 The opposite of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  O  =  (oppg `  G )   =>    |-  ( G  e.  TopGrp  ->  O  e.  TopGrp )
 
Theoremdistgp 17798 Any group equipped with the discrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  B  =  ( Base `  G )   &    |-  J  =  (
 TopOpen `  G )   =>    |-  ( ( G  e.  Grp  /\  J  =  ~P B )  ->  G  e.  TopGrp )
 
Theoremindistgp 17799 Any group equipped with the indiscrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  B  =  ( Base `  G )   &    |-  J  =  (
 TopOpen `  G )   =>    |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B }
 )  ->  G  e.  TopGrp )
 
Theoremsymgtgp 17800 The symmetric group is a topological group. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  G  =  ( SymGrp `  A )   =>    |-  ( A  e.  V  ->  G  e.  TopGrp )
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