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Theorem List for Metamath Proof Explorer - 17501-17600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremneiflim 17501 A point is a limit point of its neighborhood filter. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  A  e.  ( J 
 fLim  ( ( nei `  J ) `  { A }
 ) ) )
 
Theoremflimopn 17502* The condition for being a limit point of a filter still holds if one only considers open neighborhoods. (Contributed by Jeff Hankins, 4-Sep-2009.) (Proof shortened by Mario Carneiro, 9-Apr-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X ) )  ->  ( A  e.  ( J  fLim  F )  <->  ( A  e.  X  /\  A. x  e.  J  ( A  e.  x  ->  x  e.  F ) ) ) )
 
Theoremfbflim 17503* A condition for a filter to converge to a point involving one of its bases. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  F  =  ( X
 filGen B )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  X ) )  ->  ( A  e.  ( J  fLim  F )  <->  ( A  e.  X  /\  A. x  e.  J  ( A  e.  x  ->  E. y  e.  B  y  C_  x ) ) ) )
 
Theoremfbflim2 17504* A condition for a filter base  B to converge to a point 
A. Use neighborhoods instead of open neighborhoods. Compare fbflim 17503. (Contributed by FL, 4-Jul-2011.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  F  =  ( X
 filGen B )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  X ) )  ->  ( A  e.  ( J  fLim  F )  <->  ( A  e.  X  /\  A. n  e.  ( ( nei `  J ) `  { A }
 ) E. x  e.  B  x  C_  n ) ) )
 
Theoremflimclsi 17505 The convergent points of a filter are a subset of the closure of any of the filter sets. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( S  e.  F  ->  ( J  fLim  F ) 
 C_  ( ( cls `  J ) `  S ) )
 
Theoremhausflimlem 17506 If  A and  B are both limits of the same filter, then all neighborhoods of  A and  B intersect. (Contributed by Mario Carneiro, 21-Sep-2015.)
 |-  ( ( ( A  e.  ( J  fLim  F )  /\  B  e.  ( J  fLim  F ) )  /\  ( U  e.  J  /\  V  e.  J )  /\  ( A  e.  U  /\  B  e.  V )
 )  ->  ( U  i^i  V )  =/=  (/) )
 
Theoremhausflimi 17507* One direction of hausflim 17508. A filter in a Hausdorf space has at most one limit. (Contributed by FL, 14-Nov-2010.) (Revised by Mario Carneiro, 21-Sep-2015.)
 |-  ( J  e.  Haus  ->  E* x  x  e.  ( J  fLim  F ) )
 
Theoremhausflim 17508* A condition for a topology to be Hausdorff in terms of filters. A topology is Hausdorff iff every filter has at most one limit point. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( J  e.  Haus  <->  ( J  e.  Top  /\  A. f  e.  ( Fil `  X ) E* x  x  e.  ( J  fLim  f ) ) )
 
Theoremflimcf 17509* Fineness is properly characterized by the property that every limit point of a filter in the finer topology is a limit point in the coarser topology. (Contributed by Jeff Hankins, 28-Sep-2009.) (Revised by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  ->  ( J 
 C_  K  <->  A. f  e.  ( Fil `  X ) ( K  fLim  f )  C_  ( J  fLim  f
 ) ) )
 
Theoremflimrest 17510 The set of limit 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 )  fLim  ( Ft  Y ) )  =  ( ( J  fLim  F )  i^i  Y ) )
 
Theoremflimclslem 17511 Lemma for flimcls 17512. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  F  =  ( X
 filGen ( fi `  (
 ( ( nei `  J ) `  { A }
 )  u.  { S } ) ) )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  A  e.  ( ( cls `  J ) `  S ) ) 
 ->  ( F  e.  ( Fil `  X )  /\  S  e.  F  /\  A  e.  ( J  fLim  F ) ) )
 
Theoremflimcls 17512* Closure in terms of filter convergence. (Contributed by Jeff Hankins, 28-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  ( A  e.  (
 ( cls `  J ) `  S )  <->  E. f  e.  ( Fil `  X ) ( S  e.  f  /\  A  e.  ( J  fLim  f ) ) ) )
 
Theoremflimsncls 17513 If  A is a limit point of the filter  F, then all the points which specialize  A (in the specialization preorder) are also limit points. Thus the set of limit points is a union of closed sets (although this is only nontrivial for non-T1 spaces). (Contributed by Mario Carneiro, 20-Sep-2015.)
 |-  ( A  e.  ( J  fLim  F )  ->  ( ( cls `  J ) `  { A }
 )  C_  ( J  fLim  F ) )
 
Theoremhauspwpwf1 17514* Lemma for hauspwpwdom 17515. Points in the closure of a set in a Hausdorff space are characterized by the open neighborhoods they extend into the generating set. (Contributed by Mario Carneiro, 28-Jul-2015.)
 |-  X  =  U. J   &    |-  F  =  ( x  e.  (
 ( cls `  J ) `  A )  |->  { a  |  E. j  e.  J  ( x  e.  j  /\  a  =  (
 j  i^i  A )
 ) } )   =>    |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  F :
 ( ( cls `  J ) `  A ) -1-1-> ~P ~P A )
 
Theoremhauspwpwdom 17515 If  X is a Hausdorff space, then the cardinality of the closure of a set  A is bounded by the double powerset of  A. In particular, a Hausdorff space with a dense subset  A has cardinality at most  ~P ~P A, and a separable Hausdorff space has cardinality at most  ~P ~P NN. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 28-Jul-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Haus  /\  A  C_  X )  ->  ( ( cls `  J ) `  A )  ~<_  ~P
 ~P A )
 
Theoremflffval 17516* Given a topology and a filtered set, return the convergence function on the functions from the filtered set to the base set of the topological space. (Contributed by Jeff Hankins, 14-Oct-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y ) )  ->  ( J  fLimf  L )  =  ( f  e.  ( X  ^m  Y )  |->  ( J  fLim  ( ( X  FilMap  f ) `
  L ) ) ) )
 
Theoremflfval 17517 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 17518* 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 17519* 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 17520* 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 17521 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 17522* 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 17523* 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 17524* 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 17525 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 17526 Forward direction of cnpflf 17528. (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 17527  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 17528* 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 17529* 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 17530* 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 17531 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 17532 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 17533* 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 17534* 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 17535* 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 17536* 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 17537 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 17538 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 17539 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 17540* 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 17541* 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 17542 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 17543 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 17544 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 17545 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 17546* 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 17547* 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 17548* 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 17549 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 17550 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 17551 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 17552* 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 17553 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 17554* 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 17555* A filter converges to a point iff every finer filter clusters there. Along with fclsfnflim 17554, 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 17556 Forward direction of fclscmp 17557. 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 17557* 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 17558 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 17559* 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 17560 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 17561* 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 17562* 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 17563 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 17564 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 17565 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 17566 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 17567 Lemma for cnpfcf 17568. 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 17568* 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 17569* 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 17570* Lemma for alexsub 17571. (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 17571* 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 17577 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 17572* Biconditional form of the Alexander Subbase Theorem alexsub 17571. (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 17573* Lemma for alexsubALT 17577. 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 17574* Lemma for alexsubALT 17577. 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 17575* Lemma for alexsubALT 17577. 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 17576* Lemma for alexsubALT 17577. 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 17577* 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 17578* Lemma for ptcmp 17584. (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 17579* Lemma for ptcmp 17584. (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 17580* Lemma for ptcmp 17584. (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 17581* Lemma for ptcmp 17584. (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 17582* Lemma for ptcmp 17584. (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 17583 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 17584). (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 17584 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 17585 Extend class notation with the class of all topological monoids.
 class TopMnd
 
Syntaxctgp 17586 Extend class notation with the class of all topological groups.
 class  TopGrp
 
Definitiondf-tmd 17587* 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 17588* 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 17589 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 17590 A topological monoid is a monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  ( G  e. TopMnd  ->  G  e.  Mnd )
 
Theoremtmdtps 17591 A topological monoid is a topological space. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  ( G  e. TopMnd  ->  G  e.  TopSp )
 
Theoremistgp 17592 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 17593 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 17594 A topological group is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  ( G  e.  TopGrp  ->  G  e. TopMnd )
 
Theoremtgptps 17595 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 17596 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 17597 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 17598 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 17599 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 17600 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 ) )
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