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Theorem List for Metamath Proof Explorer - 24901-25000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelioo1t3 24901 If an open interval has an element, then  A  <  B. (Contributed by FL, 14-Aug-2007.)
 |-  ( C  e.  ( A (,) B )  ->  A  <  B )
 
Theoremoisbmi 24902 An open interval with its upper bound equal to  -oo is empty. (Contributed by FL, 12-Sep-2007.)
 |-  ( A (,)  -oo )  =  (/)
 
Theoremoisbmj 24903 An open interval with its lower bound equal to  +oo is empty. (Contributed by FL, 12-Sep-2007.)
 |-  (  +oo (,) A )  =  (/)
 
Theoremtruni1 24904 Closure of translation in a half-infinite interval. (Contributed by FL, 11-Sep-2007.)
 |-  (
 ( A  e.  RR*  /\  D  e.  RR  /\  0  <  D )  ->  ( C  e.  ( A (,)  +oo )  ->  ( C  +  D )  e.  ( A (,)  +oo ) ) )
 
Theoremtruni2 24905 Closure of translation in a half-infinite interval. (Contributed by FL, 26-Jan-2009.)
 |-  (
 ( D  e.  RR  /\  0  <  D ) 
 ->  ( C  e.  ( A (,)  +oo )  ->  ( C  +  D )  e.  ( A (,)  +oo ) ) )
 
Theoremtruni3 24906 Closure of translation in a half-infinite interval. (Contributed by FL, 26-Jan-2009.)
 |-  (
 ( D  e.  RR  /\  0  <  D ) 
 ->  ( C  e.  (  -oo (,) A )  ->  ( C  -  D )  e.  (  -oo (,)
 A ) ) )
 
Theoremcbci 24907 The center belongs to a centered interval. (Contributed by FL, 5-Jan-2009.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR+ )  ->  A  e.  ( ( A  -  B ) (,) ( A  +  B ) ) )
 
Theoremoibbi1 24908 An open interval is included in a bound below interval. (Contributed by FL, 26-Jan-2009.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  ( A (,) B )  C_  ( A (,)  +oo )
 
Theoremoibbi2 24909 An open interval is included in a bound above interval. (Contributed by FL, 26-Jan-2009.)
 |-  ( A (,) B )  C_  (  -oo (,) B )
 
Theoremnelioo5 24910 Membership in an open interval of extended reals. (Contributed by FL, 7-Dec-2010.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  ( -.  C  e.  ( A (,) B )  <->  ( C  <_  A  \/  B  <_  C ) ) )
 
18.12.27  Topology
 
Theoremtopnem 24911 A topology is not empty. (Contributed by FL, 1-Jun-2008.)
 |-  ( J  e.  Top  ->  J  =/= 
 (/) )
 
Theoremclsint 24912 The closure of an intersection is included in the intersection of the closures. (Contributed by FL, 23-Feb-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  X )  ->  ( ( cls `  J ) `  ( S  i^i  T ) )  C_  (
 ( ( cls `  J ) `  S )  i^i  ( ( cls `  J ) `  S ) ) )
 
Theoremislp3 24913* The predicate " P is a limit point of  S " in terms of open sets. see islp2 16871, elcls 16804, islp 16866. (Contributed by FL, 31-Jul-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  X )  ->  ( P  e.  (
 ( limPt `  J ) `  S )  <->  A. x  e.  J  ( P  e.  x  ->  ( x  i^i  ( S  \  { P }
 ) )  =/=  (/) ) ) )
 
Theoreminttop2 24914* The intersection of a family of topologies is a topology. (Contributed by FL, 19-Sep-2011.)
 |-  (
 ( I  =/=  (/)  /\  A. x  e.  I  J  e.  Top )  ->  |^|_ x  e.  I  J  e.  Top )
 
Theoreminttop3 24915 The intersection of a family of topologies is a topology. (Contributed by FL, 19-Sep-2011.)
 |-  (
 ( J  =/=  (/)  /\  J  C_ 
 Top )  ->  |^| J  e.  Top )
 
Theoreminttop4 24916 The intersection of two topologies is a topology. (Contributed by FL, 19-Sep-2011.)
 |-  (
 ( J  e.  Top  /\  K  e.  Top )  ->  ( J  i^i  K )  e.  Top )
 
Theoremunint2t 24917 The intersection of two topologies over the same underlying set  U. J is a topology over  U. J. compare uniin 3848. (Contributed by FL, 27-Nov-2011.)
 |-  (
 ( J  e.  Top  /\  K  e.  Top  /\  U. J  =  U. K )  ->  U. ( J  i^i  K )  =  U. J )
 
Theoremintfmu2 24918* The intersection of a family of topologies over the same underying set  U. J is a topology over  U. J. (Contributed by FL, 27-Nov-2011.)
 |-  (
 ( T  C_  Top  /\  J  e.  T  /\  A. x  e.  T  U. x  =  U. J ) 
 ->  U. |^| T  =  U. J )
 
Theoremapnei 24919* Any point has a neighborhood. (Contributed by FL, 15-Oct-2012.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  A  e.  X ) 
 ->  E. v  v  e.  ( ( nei `  J ) `  { A }
 ) )
 
Theoremnpmp 24920 A neighborhood of a point can't be empty. (Contributed by FL, 15-Oct-2012.)
 |-  (
 ( J  e.  Top  /\  A  e.  X  /\  N  e.  ( ( nei `  J ) `  { A } ) ) 
 ->  N  =/=  (/) )
 
Theorembasexre 24921 A basis for the standard topology over the extended reals. (Contributed by FL, 14-Sep-2013.) (Proof shortened by Mario Carneiro, 30-Jan-2014.)
 |-  ( ran  (,)  u.  { RR* } )  e.  TopBases
 
Theoremstovr 24922 The standard topology over  RR*. (Contributed by FL, 15-Sep-2013.)
 |-  ( topGen `
  ( ran  (,)  u. 
 { RR* } ) )  e.  Top
 
Theoremcldifemp 24923 The closure of a class  S is empty iff  S is empty. (Contributed by FL, 15-Sep-2013.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  S  C_  X )  ->  ( ( ( cls `  J ) `  S )  =  (/)  <->  S  =  (/) ) )
 
18.12.28  Continuous functions
 
Theoremcnrsfin 24924 A mapping remains continuous when the topology associated to its domain is replaced by a finer one. (Contributed by FL, 22-May-2008.)
 |-  (
 ( J  e.  Top  /\  K  e.  Top  /\  L  e.  Top )  ->  ( ( F  e.  ( J  Cn  L ) 
 /\  U. J  =  U. K  /\  J  C_  K )  ->  F  e.  ( K  Cn  L ) ) )
 
Theoremcnrscoa 24925 A mapping remains continuous when the topology associated to its range is replaced by a coarser one. (Contributed by FL, 1-Jun-2008.)
 |-  (
 ( J  e.  Top  /\  K  e.  Top  /\  L  e.  Top )  ->  ( ( F  e.  ( J  Cn  L ) 
 /\  U. L  =  U. K  /\  K  C_  L )  ->  F  e.  ( J  Cn  K ) ) )
 
Theoremmapdiscn 24926 Any mapping whose domain is associated to the discrete topology is continuous. (Contributed by FL, 19-Sep-2011.) (Proof shortened by Mario Carneiro, 7-Apr-2015.)
 |-  B  =  U. J   =>    |-  ( ( J  e.  Top  /\  F : A --> B  /\  A  e.  D )  ->  F  e.  ( ~P A  Cn  J ) )
 
Theoremmapudiscn 24927 Any mapping whose range is associated to the undiscrete topology is continuous. (Contributed by FL, 1-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  A  =  U. J   &    |-  B  e.  _V   =>    |-  (
 ( J  e.  Top  /\  F : A --> B ) 
 ->  F  e.  ( J  Cn  { (/) ,  B } ) )
 
Theoremsallnei 24928* Two ways to state the set of all the neighborhoods. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  U. ran  ( nei `  J )  =  { v  |  ( v  C_  X  /\  E. g  e.  J  g  C_  v ) }
 )
 
Theoremnsn 24929* The neighborhoods of the singletons are neighborhoods. (Contributed by FL, 2-Aug-2009.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  U_ x  e.  X  ( ( nei `  J ) `  { x }
 )  C_  U. ran  ( nei `  J ) )
 
Theoremosneisi 24930* The non empty open sets are neighborhoods of the singletons. (Contributed by FL, 16-Jul-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  A  =/=  (/) )  ->  ( A  e.  J  ->  A  e.  U_ x  e.  X  ( ( nei `  J ) `  { x } ) ) )
 
Theoremelsubops 24931 The elements of a subbase are open sets. (Contributed by FL, 16-Apr-2012.) (Revised by Mario Carneiro, 14-Dec-2013.)
 |-  K  =  ( topGen `  ( fi `  S ) )   =>    |-  ( S  e.  A  ->  S  C_  K )
 
18.12.29  Homeomorphisms
 
Theoremdmhmph 24932  ~= is a relation whose domain is included in  Top. (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 30-May-2014.)
 |-  dom  ~=  C_  Top
 
Theoremrnhmph 24933  ~= is a relation whose range is included in  Top. (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 30-May-2014.)
 |-  ran  ~=  C_  Top
 
Theoremhmeogrplem 24934* Lemma for hmeogrp 24936. (Contributed by FL, 30-May-2014.)
 |-  G  =  ( x  e.  ( J  Homeo  J ) ,  y  e.  ( J 
 Homeo  J )  |->  ( x  o.  y ) )   =>    |-  ( ( A  e.  ( J  Homeo  J ) 
 /\  B  e.  ( J  Homeo  J ) ) 
 ->  ( A G B )  =  ( A  o.  B ) )
 
Theoremhmeogrpi 24935* Lemma for hmeogrp 24936. (Contributed by FL, 31-May-2014.)
 |-  G  =  ( x  e.  ( J  Homeo  J ) ,  y  e.  ( J 
 Homeo  J )  |->  ( x  o.  y ) )   &    |-  J  e.  Top   =>    |-  G  e.  GrpOp
 
Theoremhmeogrp 24936* Homeomorphisms on a topology  J is a group for composition. This means from Felix Klein's point of view that a set equipped with a topology is a geometry, namely the so-called rubber sheet geometry. (Contributed by FL, 3-Feb-2008.) (Proof shortened by Mario Carneiro, 31-May-2014.)
 |-  G  =  ( x  e.  ( J  Homeo  J ) ,  y  e.  ( J 
 Homeo  J )  |->  ( x  o.  y ) )   =>    |-  ( J  e.  Top  ->  G  e.  GrpOp )
 
18.12.30  Initial and final topologies
 
Theoremintopcoaconlem3b 24937* The underlying set of the initial topology is the domain of the mappings  F. (Contributed by FL, 24-Apr-2012.) (Revised by Mario Carneiro, 25-Nov-2013.)
 |-  J  =  ( topGen `  ( fi ` 
 { x  |  E. i  e.  I  E. o  e.  K  x  =  ( `' F "
 o ) } )
 )   =>    |-  ( ( ( I  e.  A  /\  X  e.  B )  /\  I  =/= 
 (/)  /\  A. i  e.  I  ( K  e.  Top  /\  F : X --> U. K ) )  ->  U. J  =  X )
 
Theoremintopcoaconlem3 24938* The underlying set of the initial topology is the domain of the mappings  F. (Contributed by FL, 21-Apr-2012.) (Revised by Mario Carneiro, 25-Nov-2013.)
 |-  J  =  ( topGen `  ( fi ` 
 { x  |  E. i  e.  I  E. o  e.  K  x  =  ( `' F "
 o ) } )
 )   &    |-  I  e.  A   &    |-  X  e.  B   &    |-  A. i  e.  I  ( K  e.  Top  /\  F : X --> U. K )   &    |-  I  =/=  (/)   =>    |- 
 U. J  =  X
 
Theoremintopcoaconb 24939* The initial topology is the coarsest one making the functions  F continuous . (Contributed by FL, 14-May-2012.) (Revised by Mario Carneiro, 26-Nov-2013.)
 |-  J  =  ( topGen `  ( fi ` 
 { x  |  E. i  e.  I  E. o  e.  K  x  =  ( `' F "
 o ) } )
 )   &    |-  I  e.  A   &    |-  X  e.  B   &    |-  A. i  e.  I  ( K  e.  Top  /\  F : X --> U. K )   &    |-  I  =/=  (/)   =>    |-  J  =  |^| { t  e.  Top  |  A. i  e.  I  F  e.  ( t  Cn  K ) }
 
Theoremintopcoaconc 24940* The initial topology makes the functions  F continuous. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 26-Nov-2013.)
 |-  J  =  ( topGen `  ( fi ` 
 { x  |  E. i  e.  I  E. o  e.  K  x  =  ( `' F "
 o ) } )
 )   &    |-  I  e.  A   &    |-  X  e.  B   &    |-  A. i  e.  I  ( K  e.  Top  /\  F : X --> U. K )   &    |-  I  =/=  (/)   =>    |- 
 A. i  e.  I  F  e.  ( J  Cn  K )
 
Theoremqusp 24941* A quotient space is a topology. (Contributed by FL, 4-Jun-2007.)
 |-  X  =  U. J   &    |-  R  Er  A   =>    |-  ( J  e.  Top  ->  { x  |  ( x  C_  ( X /. R )  /\  U. x  e.  J ) }  e.  Top )
 
Theoremintcont 24942 If  F is continous over two topologies  J and  K then it is continuous over  ( J  i^i  K
). (Contributed by FL, 27-Nov-2011.)
 |-  (
 ( ( J  e.  Top  /\  K  e.  Top  /\  L  e.  Top )  /\  U. J  =  U. K  /\  ( F  e.  ( J  Cn  L )  /\  F  e.  ( K  Cn  L ) ) ) 
 ->  F  e.  ( ( J  i^i  K )  Cn  L ) )
 
Syntaxctopx 24943 Extend class notation with a function whose value is a product topology.
 class  topX
 
Definitiondf-prtop 24944* The product topology of a family  f of topologies is the coarsest topology over the product of the underlying sets that makes the projections continuous. (Bourbaki TG I.14 ex. 3) Experimental. (Contributed by FL, 4-Dec-2011.)
 |-  topX  =  { <. f ,  y >.  |  ( f : dom  f --> Top  /\  y  =  |^| { t  e.  Top  |  ( U. t  =  X_ x  e. 
 dom  f U. (
 f `  x )  /\  A. i  e.  dom  f ( X_ x  e.  dom  f U. (
 f `  x )  pr  i )  e.  (
 t  Cn  ( f `  i ) ) ) } ) }
 
Theoremusptoplem 24945* Lemma for usptop 24949. (Contributed by FL, 5-Dec-2011.) (Revised by Mario Carneiro, 26-Jan-2015.)
 |-  (
 ( F : I --> Top  /\  I  e.  A )  ->  ~P X_ x  e.  I  U. ( F `  x )  e.  { t  e.  Top  |  ( U. t  =  X_ x  e.  I  U. ( F `
  x )  /\  A. i  e.  I  (
 X_ x  e.  I  U. ( F `  x )  pr  i )  e.  ( t  Cn  ( F `  i ) ) ) } )
 
Theoremistopx 24946* Definition of the product topology of a family of topologies  F. (Contributed by FL, 4-Dec-2011.) (Revised by Mario Carneiro, 26-Jan-2015.)
 |-  (
 ( F : I --> Top  /\  I  e.  A )  ->  (  topX  `  F )  =  |^| { t  e.  Top  |  ( U. t  =  X_ x  e.  I  U. ( F `
  x )  /\  A. i  e.  I  (
 X_ x  e.  I  U. ( F `  x )  pr  i )  e.  ( t  Cn  ( F `  i ) ) ) } )
 
Theoremistopxc 24947* Product of  I topology  J. (Contributed by FL, 15-Sep-2013.)
 |-  X  =  U. J   =>    |-  ( ( F : I
 --> { J }  /\  I  e.  A  /\  J  e.  Top )  ->  (  topX  `  F )  =  |^| { t  e. 
 Top  |  ( U. t  =  ( X  ^m  I )  /\  A. i  e.  I  (
 ( X  ^m  I
 )  pr  i )  e.  ( t  Cn  J ) ) }
 )
 
Theoremprtoptop 24948 The product topology of a family  F of topologies is a topology. (Contributed by FL, 5-Dec-2011.) (Proof shortened by Mario Carneiro, 26-Jan-2015.)
 |-  (
 ( F : I --> Top  /\  I  e.  A )  ->  (  topX  `  F )  e.  Top )
 
Theoremusptop 24949* The underlying set of a product topology. (Contributed by FL, 5-Dec-2011.)
 |-  (
 ( F : I --> Top  /\  I  e.  A )  ->  U. (  topX  `  F )  =  X_ x  e.  I  U. ( F `
  x ) )
 
Theoremprcnt 24950* The projections are continuous. (Contributed by FL, 18-Apr-2012.)
 |-  (
 ( F : I --> Top  /\  I  e.  A )  ->  A. i  e.  I  ( X_ x  e.  I  U. ( F `  x )  pr  i )  e.  ( (  topX  `  F )  Cn  ( F `  i ) ) )
 
18.12.31  Filters
 
Theoremefilcp 24951* A filter containing a set  A exists iff  A has the finite intersection property (i.e. no finite intersection of elements of  A is empty). Bourbaki TG I.37 prop. 1. (Contributed by FL, 20-Nov-2007.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  (
 ( A  C_  ~P B  /\  B  e.  V  /\  B  =/=  (/) )  ->  ( -.  (/)  e.  ( fi
 `  A )  <->  E. x  e.  ( Fil `  B ) A 
 C_  x ) )
 
Theoremfilint2 24952 A filter is closed under taking finite intersections. (Contributed by FL, 27-Apr-2008.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  (
 ( A  C_  F  /\  A  =/=  (/)  /\  A  e.  Fin )  ->  |^| A  e.  F ) )
 
Theoremfisub 24953* If a set has the finite intersection property, its subsets have also this property. (Contributed by FL, 27-Apr-2008.)
 |-  B  =  { z  |  E. y ( y  C_  A  /\  y  e.  Fin  /\  z  =  |^| y
 ) }   &    |-  D  =  {
 z  |  E. y
 ( y  C_  C  /\  y  e.  Fin  /\  z  =  |^| y
 ) }   =>    |-  ( C  C_  A  ->  ( -.  (/)  e.  B  ->  -.  (/)  e.  D ) )
 
Theoremfgsb2 24954* Filter generated by a subbasis  A. Bourbaki TG I.37 paragraph above prop. 1. (The theorem has been slightly modified because the definitions of the empty set are different in Bourbaki and Metamath.) (Contributed by FL, 27-Apr-2008.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  (
 ( A  C_  ~P X  /\  X  e.  _V  /\  A  =/=  (/) )  ->  ( -.  (/)  e.  ( fi
 `  A )  ->  { x  e.  ~P X  |  E. y  e.  ( fi `  A ) y  C_  x }  e.  ( Fil `  X ) ) )
 
Theoremcnfilca 24955* Condition to have a filter finer than a given filter and containing a set  A. Bourbaki T.G. I.37 cor. 1 (Contributed by FL, 27-Apr-2008.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  (
 ( F  e.  ( Fil `  X )  /\  A  C_  X  /\  A  =/= 
 (/) )  ->  ( E. g  e.  ( Fil `  X ) ( A  e.  g  /\  F  C_  g )  <->  A. x  e.  F  ( x  i^i  A )  =/=  (/) ) )
 
Theoremfil2ss 24956* A condition for a filter to be finer than another filter. Compare fgss2 17563. (Contributed by FL, 8-Aug-2011.) (Revised by Mario Carneiro, 6-Aug-2015.)
 |-  (
 ( F  e.  ( Fil `  X )  /\  G  e.  ( Fil `  X ) )  ->  ( F  C_  G  <->  A. x  e.  F  E. y  e.  G  y  C_  x ) )
 
18.12.32  Limits
 
Theoremplimfil 24957 The predicate "is a limit of a filter". (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 7-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  X )  /\  L  e.  X )  ->  ( L  e.  ( J  fLim  F )  <->  ( ( nei `  J ) `  { L } )  C_  F ) )
 
Theoremlimvinlv 24958 The limit value of a convergent function whose values are in a Hausdorff space belongs to the set of the limit values. (Contributed by FL, 14-Nov-2010.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  X  =  U. J   &    |-  A  =  U. ( ( J  fLimf  L ) `  F )   =>    |-  ( ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X ) 
 /\  ( ( J 
 fLimf  L ) `  F )  =/=  (/) )  ->  A  e.  ( ( J  fLimf  L ) `  F ) )
 
Theoremflfneih 24959* A neighborhood of the limit value of a convergent function whose values are in a Hausdorff space contains the image of a filter element. (Contributed by FL, 18-Nov-2010.) (Revised by Mario Carneiro, 6-Aug-2015.)
 |-  X  =  U. J   &    |-  A  =  U. ( ( J  fLimf  L ) `  F )   =>    |-  ( ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X ) 
 /\  ( ( J 
 fLimf  L ) `  F )  =/=  (/)  /\  N  e.  ( ( nei `  J ) `  { A }
 ) )  ->  E. s  e.  L  ( F "
 s )  C_  N )
 
Theoremlimfilnei 24960  A is a limit of the filter of the neighborhoods of  A. (Contributed by FL, 27-May-2011.) (Proof shortened by Mario Carneiro, 9-Apr-2015.)
 |-  X  =  U. J   &    |-  F  =  ( ( nei `  J ) `  { A }
 )   =>    |-  ( ( J  e.  Top  /\  A  e.  X ) 
 ->  A  e.  ( J 
 fLim  F ) )
 
Theoremconttnf2 24961  F is continous at point  A iff  ( F `  A ) is a limit of the image filter of the neighborhoods of  A. (Contributed by FL, 7-Aug-2011.) (Revised by Mario Carneiro, 6-Aug-2015.)
 |-  L  =  ( ( nei `  J ) `  { A }
 )   &    |-  X  =  U. K   &    |-  Y  =  U. J   =>    |-  ( ( J  e.  Top  /\  K  e.  Top  /\  ( A  e.  Y  /\  F : Y --> X ) )  ->  ( F  e.  ( ( J  CnP  K ) `  A )  <-> 
 ( F `  A )  e.  ( K  fLim  ( ( X  FilMap  F ) `  L ) ) ) )
 
Theoremiscnp4 24962* The predicate " F is a continuous function from topology  J to topology  K at point  P." in terms of neighborhoods. (Contributed by FL, 18-Jul-2011.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  (
 ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  P  e.  X ) 
 ->  ( F  e.  (
 ( J  CnP  K ) `  P )  <->  ( F : X
 --> Y  /\  A. y  e.  ( ( nei `  K ) `  { ( F `
  P ) }
 ) E. x  e.  ( ( nei `  J ) `  { P }
 ) ( F " x )  C_  y ) ) )
 
Theoremcnpflf4 24963 If  G is continuous at point  A, and the filter base  F converges to  A then  G ( F ) converges to  G ( A ). Bourbaki TG I.50 cor 1. (Contributed by FL, 19-Sep-2011.) (Revised by Stefan O'Rear, 7-Aug-2015.)
 |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  ( ( ( K  e.  Top  /\  F  e.  ( fBas `  X )
 )  /\  ( G  e.  ( ( J  CnP  K ) `  A ) 
 /\  A  e.  ( J  fLim  ( X filGen F ) ) ) ) 
 ->  ( G `  A )  e.  ( K  fLim  ( ( Y  FilMap  G ) `  F ) ) )
 
Theoremlimfn 24964 The limits of a function are elements of its range. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan O'Rear, 7-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  ( ( J  fLimf  L ) `  F ) 
 C_  X )
 
Theoremlimfn2 24965 If  A is a limit of a function  F,  A is an element of the range of  F. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 29-Jul-2015.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  ( A  e.  (
 ( J  fLimf  L ) `
  F )  ->  A  e.  X )
 )
 
Theoremlimfn3 24966 If  A is the limit of a convergent function in a Hausdorff space,  A is an element of the range of the function. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 7-Aug-2015.)
 |-  X  =  U. J   &    |-  A  =  U. ( ( J  fLimf  L ) `  F )   =>    |-  ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  ( ( ( J 
 fLimf  L ) `  F )  =/=  (/)  ->  A  e.  X ) )
 
Theoremcmptdst 24967  G  o.  F tends to  G ( A ) if  G is continuous at point 
A and  F tends to A . Bourbaki TG I.50 cor. 2. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  Y  =  U. J   =>    |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  L  e.  ( Fil `  X ) )  /\  ( F : X --> Y  /\  A  e.  ( ( J  fLimf  L ) `  F )  /\  G  e.  ( ( J  CnP  K ) `  A ) ) )  ->  ( G `  A )  e.  ( ( K  fLimf  L ) `  ( G  o.  F ) ) )
 
Theoremunexun 24968* If  A is an element of  B and  B has a unique element,  U. B  =  A. (Contributed by FL, 15-Oct-2012.)
 |-  (
 ( A  e.  B  /\  E! x  x  e.  B )  ->  U. B  =  A )
 
Theoremlimhun 24969 In a Hausdorff space if  A is a limit of a convergent function  F, then  A is the unique limit of  F. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  X  =  U. J   &    |-  L1  =  U. ( ( J  fLimf  L ) `  F )   =>    |-  ( ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X ) 
 /\  A  e.  (
 ( J  fLimf  L ) `
  F ) ) 
 ->  L1  =  A )
 
Theoremcmptdst2 24970  G  o.  F tends to  G ( L1 ) if  G is continuous at point  L1 and  F tends to  L1. (cmptdst 24967 in the Hausdorff case.) Bourbaki TG I.50 cor. 2. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  Y  =  U. J   &    |-  L1  =  U. ( ( J  fLimf  L ) `  F )   &    |-  L 2  =  U. (
 ( K  fLimf  L ) `
  ( G  o.  F ) )   =>    |-  ( ( ( J  e.  Haus  /\  K  e.  Haus  /\  L  e.  ( Fil `  X )
 )  /\  ( F : X --> Y  /\  (
 ( J  fLimf  L ) `
  F )  =/=  (/)  /\  G  e.  (
 ( J  CnP  K ) `  L1 ) ) ) 
 ->  L 2  =  ( G `  L1 )
 )
 
Theoremexopcopn 24971* For every neighborhood  N of  <. A ,  B >. in a product topology, there exist two open sets  u and  v of the component topologies so that  ( u  X.  v ) is an open neighborhood of  <. A ,  B >. and a part of  N. (Use opelxp 4718 to have  A  e.  u and  B  e.  v.) (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  T  =  ( R  tX  S )   =>    |-  ( ( ( R  e.  Top  /\  S  e.  Top )  /\  ( A  e.  C  /\  B  e.  D )  /\  N  e.  ( ( nei `  T ) `  { <. A ,  B >. } ) ) 
 ->  E. u  e.  R  E. v  e.  S  ( <. A ,  B >.  e.  ( u  X.  v )  /\  ( u  X.  v )  C_  N ) )
 
Theoremprdnei 24972 The product of two neighborhoods is a neighborhood. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 10-Jun-2014.)
 |-  T  =  ( R  tX  S )   &    |-  X  =  U. R   &    |-  Y  =  U. S   =>    |-  ( ( ( R  e.  Top  /\  S  e.  Top )  /\  ( A  e.  X  /\  B  e.  Y )  /\  ( U  e.  ( ( nei `  R ) `  { A } )  /\  V  e.  ( ( nei `  S ) `  { B } ) ) )  ->  ( U  X.  V )  e.  (
 ( nei `  T ) `  { <. A ,  B >. } ) )
 
Theoremlimptlimpr2lem1 24973 Lemma for limptlimpr 24975. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 9-Aug-2015.)
 |-  T  =  ( R  tX  S )   &    |-  X  =  U. R   &    |-  Y  =  U. S   &    |-  Z  =  ( X  X.  Y )   =>    |-  ( ( ( R  e.  Top  /\  S  e.  Top  /\  L  e.  ( Fil `  W ) )  /\  ( F : W --> Z  /\  L 2  e.  A ) )  ->  ( <. L1 ,  L 2 >.  e.  ( ( T  fLimf  L ) `  F ) 
 ->  ( L1  e.  (
 ( R  fLimf  L ) `
  ( 1st  o.  F ) )  /\  L 2  e.  ( ( S  fLimf  L ) `  ( 2nd  o.  F ) ) ) ) )
 
Theoremlimptlimpr2lem2 24974 Lemma for limptlimpr 24975. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 9-Aug-2015.)
 |-  T  =  ( R  tX  S )   &    |-  X  =  U. R   &    |-  Y  =  U. S   &    |-  Z  =  ( X  X.  Y )   =>    |-  ( ( ( R  e.  Top  /\  S  e.  Top  /\  L  e.  ( Fil `  W ) )  /\  F : W --> Z ) 
 ->  ( ( L1  e.  ( ( R  fLimf  L ) `  ( 1st 
 o.  F ) ) 
 /\  L 2  e.  ( ( S  fLimf  L ) `  ( 2nd 
 o.  F ) ) )  ->  <. L1 ,  L 2 >.  e.  ( ( T  fLimf  L ) `  F ) ) )
 
Theoremlimptlimpr 24975 A limit in a product topology exists iff the limits of the projections exist. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 9-Aug-2015.)
 |-  T  =  ( R  tX  S )   &    |-  X  =  U. R   &    |-  Y  =  U. S   &    |-  Z  =  ( X  X.  Y )   =>    |-  ( ( ( R  e.  Top  /\  S  e.  Top  /\  L  e.  ( Fil `  W ) )  /\  ( F : W --> Z  /\  L 2  e.  A ) )  ->  ( <. L1 ,  L 2 >.  e.  ( ( T  fLimf  L ) `  F )  <-> 
 ( L1  e.  (
 ( R  fLimf  L ) `
  ( 1st  o.  F ) )  /\  L 2  e.  ( ( S  fLimf  L ) `  ( 2nd  o.  F ) ) ) ) )
 
Theoremflfnei2 24976* The property of being a limit point of a function in terms of filter and of preimage of a neighborhood. (Contributed by FL, 13-Dec-2013.) (Revised by Mario Carneiro, 8-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  ( A  e.  (
 ( J  fLimf  L ) `
  F )  <->  ( A  e.  X  /\  A. n  e.  ( ( nei `  J ) `  { A }
 ) ( `' F " n )  e.  L ) ) )
 
Syntaxcflimfrs 24977 Extend the definition of a class to include the limit of a function relatively to a subspace.
 class  fLimfrs
 
Definitiondf-flimfrs 24978* Gives the limits of a function  f : a --> U. j at a point  p  e.  ( ( cls `  k
) `  a ) relatively to a subspace  a of a topology  k. ( The condition  p  e.  ( ( cls `  k
) `  a ) ensures the traces of the neighborhoods of  p over  a is a filter ( see trnei 17581). The set  a can't be empty since its closure is not empty ( see cldifemp 24923). Experimental. (Contributed by FL, 15-Sep-2013.)
 |-  fLimfrs  =  ( j  e.  Top ,  k  e.  Top  |->  ( a  e.  ~P U. k  |->  ( p  e.  (
 ( cls `  k ) `  a ) ,  f  e.  ( U. j  ^m  a )  |->  ( ( j  fLimf  ( ( ( nei `  k ) `  { p } )t  a
 ) ) `  f
 ) ) ) )
 
Theoremislimrs 24979 The limits of  F at point  P when one only considers the traces of the neighborhoods of  P over  A.  F is a function whose domain is  A. The point  P must belong to  ( ( cls `  K ) `  A
) (see also the comments under df-flimfrs 24978) . (Contributed by FL, 15-Sep-2013.)
 |-  L  =  ( ( ( nei `  K ) `  { P } )t  A )   &    |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  (
 ( ( J  e.  Top  /\  K  e.  Top )  /\  ( A  C_  Y  /\  P  e.  ( ( cls `  K ) `  A ) )  /\  F  e.  ( X  ^m  A ) )  ->  ( ( ( J 
 fLimfrs  K ) `  A ) `  <. P ,  F >. )  =  ( ( J  fLimf  L ) `  F ) )
 
Theoremislimrs3 24980 The limits of  F at point  P relatively to  A is a limit of  F at point  P relatively to  B  C_  A. The opposite direction doesn't hold. (Contributed by FL, 13-Dec-2013.)
 |-  L  =  ( ( ( nei `  K ) `  { P } )t  A )   &    |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  (
 ( ( J  e.  Top  /\  K  e.  Top  /\  B  C_  A )  /\  ( A  C_  Y  /\  P  e.  ( ( cls `  K ) `  B ) )  /\  F  e.  ( X  ^m  A ) )  ->  ( ( ( J 
 fLimfrs  K ) `  A ) `  <. P ,  F >. )  C_  ( (
 ( J  fLimfrs  K ) `
  B ) `  <. P ,  ( F  |`  B ) >. ) )
 
Theoremislimrs4 24981 The limits of  F at point  P relatively to  A is a limit of  F at point  P relatively to  B. (Contributed by FL, 13-Dec-2013.)
 |-  L  =  ( ( ( nei `  K ) `  { P } )t  A )   &    |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  (
 ( ( J  e.  Top  /\  K  e.  Top  /\  N  e.  ( ( nei `  K ) `  { P } ) ) 
 /\  ( A  C_  Y  /\  P  e.  (
 ( cls `  K ) `  A ) )  /\  F  e.  ( X  ^m  A ) )  ->  ( ( ( J 
 fLimfrs  K ) `  A ) `  <. P ,  F >. )  =  ( ( ( J  fLimfrs  K ) `
  ( A  i^i  N ) ) `  <. P ,  ( F  |`  ( A  i^i  N ) )
 >. ) )
 
Syntaxcisopt 24982 Extend class notation to include isolated points.
 class  IsolatedPt
 
Definitiondf-islpt 24983* Definition of an isolated point. Experimental. (Contributed by FL, 16-Sep-2013.)
 |-  IsolatedPt  =  (
 j  e.  Top  |->  ( a  e.  ~P U. j  |->  { x  e.  a  |  E. v  e.  (
 ( nei `  j ) `  { x } )
 ( v  i^i  a
 )  =  { x } } ) )
 
18.12.33  Uniform spaces
 
Syntaxcunifsp 24984 Extend class notation with the class of all uniform spaces.
 class  UnifSp
 
Definitiondf-unifsp 24985* Definition of a uniform space. Bourbaki TG II.1 def. 1. A uniform structure is used to give a generalization of the idea of Cauchy's sequence. We consider the space is equipped with the topology induced by the uniform structure. (Contributed by FL, 29-May-2014.)
 |-  UnifSp  =  {
 f  |  E. y E. x E. t ( ( y  =  (
 Base `  f )  /\  x  =  ( Unif `  f )  /\  t  =  ( TopOpen `  f )
 )  /\  ( x  C_ 
 ~P ( y  X.  y )  /\  ( y  X.  y )  e.  x  /\  A. v  e.  x  ( A. u  e.  ~P  (
 y  X.  y )
 ( v  C_  u  ->  u  e.  x ) 
 /\  A. w  e.  x  ( v  i^i  w )  e.  x  /\  (
 (  _I  |`  y ) 
 C_  v  /\  `' v  e.  x  /\  E. w  e.  x  ( w  o.  w ) 
 C_  v ) ) )  /\  t  =  ( iota_ s  e.  Top ( U. s  =  y 
 /\  A. a  e.  y  ( ( nei `  s
 ) `  { a } )  =  {
 z  |  E. u  e.  x  z  =  ( u " { a } ) } )
 ) ) }
 
18.12.34  Separated spaces: T0, T1, T2 (Hausdorff) ...
 
Theoremhst1 24986 A Hausdorff space is a T1 space. (Contributed by FL, 18-Jun-2007.)
 |-  Haus  C_ 
 Fre
 
Theoremdtt1 24987 A discrete topology is T1. Morris, Topology without tears. (Contributed by FL, 8-Jun-2007.)
 |-  A  e.  _V   =>    |- 
 ~P A  e.  Fre
 
18.12.35  Compactness
 
Theoremindcomp 24988 The indiscrete topology is compact. (Contributed by FL, 2-Aug-2009.)
 |-  { (/) ,  A }  e.  Comp
 
Theoremtopunfincomp 24989 A topology whose underlying set is finite is compact. (Contributed by FL, 22-Dec-2008.)
 |-  (
 ( J  e.  Top  /\ 
 U. J  e.  Fin )  ->  J  e.  Comp )
 
Theoremstfincomp 24990 The subspace topology induced by a finite part of the underlying set of a topology is compact. (Contributed by FL, 2-Aug-2009.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
 |-  (
 ( J  e.  Top  /\  A  e.  Fin )  ->  ( Jt  A )  e.  Comp )
 
Theorembwt2 24991* The glorious Bolzano-Weierstrass theorem. Certainly the first general topology theorem ever proved. In his course Weierstrass called it a lemma. He certainly didn't know how famous this theorem would be. He used an euclidian space instead of a general compact space. And he was not conscious of the Heine-Borel property. Cantor was one of his students. He used the concept of neighborhood and limit point invented by his master when he studied the linear point sets and the rest of the general topology followed from that. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Comp  /\  A  C_  X  /\  -.  A  e.  Fin )  ->  E. x  e.  X  x  e.  ( ( limPt `  J ) `  A ) )
 
18.12.36  Connectedness
 
Theoremsingempcon 24992 The singleton of the empty set is a connected topology. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  { (/) }  e.  Con
 
Theoremusinuniopb 24993 If a topology is connected, its underlying set can't be partitioned into two non empty non-overlapping open sets. (Contributed by FL, 29-May-2014.) (Revised by Mario Carneiro, 8-Apr-2015.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Con  /\  A  e.  J  /\  B  e.  J )  ->  ( ( A  =/=  (/)  /\  B  =/=  (/)  /\  ( A  i^i  B )  =  (/) )  ->  X  =/=  ( A  u.  B ) ) )
 
Syntaxcopfn 24994 Extend class notation with an operator that derives an operation on functions from an operation on the elements of the common range of those functions.
 class  opfn
 
Definitiondf-opfn 24995* Multiplication or addition of two functions  x and  y derived from the operation  g on the elements of the common range of 
x and  y. The functions  x and  y must also have the same domain  i. (Contributed by FL, 15-Oct-2012.)
 |-  opfn  =  ( g  e.  _V ,  i  e.  _V  |->  ( x  e.  ( dom  dom  g  ^m  i
 ) ,  y  e.  ( dom  dom  g  ^m  i )  |->  ( a  e.  dom  x  |->  ( ( x `  a
 ) g ( y `
  a ) ) ) ) )
 
18.12.37  Topological fields
 
Syntaxctopfld 24996 Extend class notation to include TopFld.
 class  TopFld
 
Definitiondf-topfld 24997* A topological field is a field whose addition, multiplication and inverse are continuous. (Contributed by FL, 21-May-2012.)
 |-  TopFld  =  { <.
 <. g ,  h >. ,  j >.  |  ( <. g ,  h >.  e. 
 Fld  /\  <. <. g ,  h >. ,  j >.  e.  TopRing  /\  ( h  |`  ( ( ran  g  \  {
 (GId `  g ) } )  X.  ( ran  g  \  { (GId `  g ) } )
 ) )  e.  (
 ( j  tX  j
 )  Cn  j )
 ) }
 
18.12.38  Standard topology on RR
 
Theoremintrn 24998 Condition for an interval to belong to the range of  (,) (Contributed by FL, 5-Jan-2009.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  e.  ran  (,) )
 
Theoremaltretop 24999* Alternate definition of the standard topology of the reals. (Morris. Def. 2.1.1 p. 34). Morris calls the standard topology of the reals the euclidean topology. (Contributed by FL, 26-Jan-2009.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
 |-  ( A  e.  ( topGen `  ran  (,) )  <->  ( A  C_  RR  /\  A. y  e.  A  E. a  e. 
 RR  E. b  e.  RR  ( y  e.  (
 a (,) b )  /\  ( a (,) b
 )  C_  A )
 ) )
 
18.12.39  Standard topology of intervals of RR
 
Theoremstoi 25000 The underlying set of the standard topology on an open interval is the open interval itself. (Contributed by FL, 31-May-2007.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  { <. (
 Base `  ndx ) ,  ( A (,) B ) >. ,  <. (TopSet `  ndx ) ,  ( ( topGen `
  ran  (,) )t  ( A (,) B ) )
 >. }  e.  TopSp
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