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Theorem List for Metamath Proof Explorer - 17701-17800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtxflf 17701* 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 17702* 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 17703* 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 17704* 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 17705 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 17706 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 17707 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 17708* 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 17709* 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 17710 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 17711 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 17712 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 17713 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 17714* 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 17715* 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 17716* 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 17717 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 17718 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 17719 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 17720* 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 17721 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 17722* 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 17723* A filter converges to a point iff every finer filter clusters there. Along with fclsfnflim 17722, 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 17724 Forward direction of fclscmp 17725. 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 17725* 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 17726 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 17727* 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 17728 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 17729* 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 17730* 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 17731 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 17732 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 17733 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 17734 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 17735 Lemma for cnpfcf 17736. 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 17736* 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 17737* 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 17738* Lemma for alexsub 17739. (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 17739* 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 17745 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 17740* Biconditional form of the Alexander Subbase Theorem alexsub 17739. (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 17741* Lemma for alexsubALT 17745. 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 17742* Lemma for alexsubALT 17745. 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 17743* Lemma for alexsubALT 17745. 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 17744* Lemma for alexsubALT 17745. 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 17745* 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 17746* Lemma for ptcmp 17752. (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 17747* Lemma for ptcmp 17752. (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 17748* Lemma for ptcmp 17752. (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 17749* Lemma for ptcmp 17752. (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 17750* Lemma for ptcmp 17752. (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 17751 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 17752). (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 17752 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 17753 Extend class notation with the class of all topological monoids.
 class TopMnd
 
Syntaxctgp 17754 Extend class notation with the class of all topological groups.
 class  TopGrp
 
Definitiondf-tmd 17755* 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 17756* 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 17757 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 17758 A topological monoid is a monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  ( G  e. TopMnd  ->  G  e.  Mnd )
 
Theoremtmdtps 17759 A topological monoid is a topological space. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  ( G  e. TopMnd  ->  G  e.  TopSp )
 
Theoremistgp 17760 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 17761 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 17762 A topological group is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  ( G  e.  TopGrp  ->  G  e. TopMnd )
 
Theoremtgptps 17763 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 17764 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 17765 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 17766 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 17767 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 17768 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 17769 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 17770* Continuity of the group sum; analogue of cnmpt12f 17360 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 17771* Continuity of the group sum; analogue of cnmpt22f 17369 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 17772* 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 17773 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 17774 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 17775* 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 17776* 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 17777 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 18322 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 17778* 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 17779* 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 17780 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 17781 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 17782 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 17783 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 17784 The symmetric group is a topological group. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  G  =  ( SymGrp `  A )   =>    |-  ( A  e.  V  ->  G  e.  TopGrp )
 
Theoremtmdlactcn 17785* The left group action of element  A in a topological monoid 
G is a continuous function. (Contributed by FL, 18-Mar-2008.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  ( A 
 .+  x ) )   &    |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  J  =  ( TopOpen `  G )   =>    |-  (
 ( G  e. TopMnd  /\  A  e.  X )  ->  F  e.  ( J  Cn  J ) )
 
Theoremtgplacthmeo 17786* The left group action of element  A in a topological group 
G is a homeomorphism from the group to itself. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  ( A 
 .+  x ) )   &    |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  J  =  ( TopOpen `  G )   =>    |-  (
 ( G  e.  TopGrp  /\  A  e.  X ) 
 ->  F  e.  ( J 
 Homeo  J ) )
 
Theoremsubmtmd 17787 A submonoid of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  H  =  ( Gs  S )   =>    |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G ) )  ->  H  e. TopMnd )
 
Theoremsubgtgp 17788 A subgroup of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  H  =  ( Gs  S )   =>    |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G ) )  ->  H  e.  TopGrp )
 
Theoremsubgntr 17789 A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 17791, the interior of a subgroup is either a subgroup, or empty. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  J  =  ( TopOpen `  G )   =>    |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )  /\  A  e.  (
 ( int `  J ) `  S ) )  ->  S  e.  J )
 
Theoremopnsubg 17790 An open subgroup of a topological group is also closed. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  J  =  ( TopOpen `  G )   =>    |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )  /\  S  e.  J )  ->  S  e.  ( Clsd `  J ) )
 
Theoremclssubg 17791 The closure of a subgroup in a topological group is a subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  J  =  ( TopOpen `  G )   =>    |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G ) )  ->  ( ( cls `  J ) `  S )  e.  (SubGrp `  G ) )
 
Theoremclsnsg 17792 The closure of a normal subgroup is a normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  J  =  ( TopOpen `  G )   =>    |-  ( ( G  e.  TopGrp  /\  S  e.  (NrmSGrp `  G ) )  ->  ( ( cls `  J ) `  S )  e.  (NrmSGrp `  G ) )
 
Theoremcldsubg 17793 A subgroup of finite index is closed iff it is open. (Contributed by Mario Carneiro, 20-Sep-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  R  =  ( G ~QG 
 S )   &    |-  X  =  (
 Base `  G )   =>    |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )  /\  ( X /. R )  e.  Fin )  ->  ( S  e.  ( Clsd `  J )  <->  S  e.  J ) )
 
Theoremtgpconcompeqg 17794* The connected component containing 
A is the left coset of the identity component containing  A. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  J  =  ( TopOpen `  G )   &    |-  S  =  U. { x  e.  ~P X  |  (  .0.  e.  x  /\  ( Jt  x )  e.  Con ) }   &    |-  .~  =  ( G ~QG  S )   =>    |-  ( ( G  e.  TopGrp  /\  A  e.  X ) 
 ->  [ A ]  .~  =  U. { x  e. 
 ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) } )
 
Theoremtgpconcomp 17795* The identity component, the connected component containing the identity element, is a closed (concompcld 17160) normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  J  =  ( TopOpen `  G )   &    |-  S  =  U. { x  e.  ~P X  |  (  .0.  e.  x  /\  ( Jt  x )  e.  Con ) }   =>    |-  ( G  e.  TopGrp  ->  S  e.  (NrmSGrp `  G )
 )
 
Theoremtgpconcompss 17796* The identity component is a subset of any open subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  J  =  ( TopOpen `  G )   &    |-  S  =  U. { x  e.  ~P X  |  (  .0.  e.  x  /\  ( Jt  x )  e.  Con ) }   =>    |-  (
 ( G  e.  TopGrp  /\  T  e.  (SubGrp `  G )  /\  T  e.  J )  ->  S  C_  T )
 
Theoremghmcnp 17797 A group homomorphism on topological groups is continuous everywhere if it is continuous at any point. (Contributed by Mario Carneiro, 21-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  J  =  (
 TopOpen `  G )   &    |-  K  =  ( TopOpen `  H )   =>    |-  (
 ( G  e. TopMnd  /\  H  e. TopMnd  /\  F  e.  ( G  GrpHom  H ) ) 
 ->  ( F  e.  (
 ( J  CnP  K ) `  A )  <->  ( A  e.  X  /\  F  e.  ( J  Cn  K ) ) ) )
 
Theoremsnclseqg 17798 The coset of the closure of the identity is the closure of a point. (Contributed by Mario Carneiro, 22-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  J  =  (
 TopOpen `  G )   &    |-  .0.  =  ( 0g `  G )   &    |- 
 .~  =  ( G ~QG  S )   &    |-  S  =  ( ( cls `  J ) `  {  .0.  } )   =>    |-  (
 ( G  e.  TopGrp  /\  A  e.  X ) 
 ->  [ A ]  .~  =  ( ( cls `  J ) `  { A }
 ) )
 
Theoremtgphaus 17799 A topological group is Hausdorff iff the identity subgroup is closed. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |- 
 .0.  =  ( 0g `  G )   &    |-  J  =  (
 TopOpen `  G )   =>    |-  ( G  e.  TopGrp  ->  ( J  e.  Haus  <->  {  .0.  }  e.  ( Clsd `  J ) ) )
 
Theoremtgpt1 17800 Hausdorff and T1 are equivalent for topological groups. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |-  J  =  ( TopOpen `  G )   =>    |-  ( G  e.  TopGrp  ->  ( J  e.  Haus  <->  J  e.  Fre ) )
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