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Theorem List for Metamath Proof Explorer - 16901-17000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrestcldr 16901 A set which is closed in the subspace topology induced by a closed set is closed in the original topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( A  e.  ( Clsd `  J )  /\  B  e.  ( Clsd `  ( Jt  A ) ) ) 
 ->  B  e.  ( Clsd `  J ) )
 
Theoremrestopnb 16902 If  B is an open subset of the subspace base set  A, then any subset of  B is open iff it is open in  A. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B ) )  ->  ( C  e.  J  <->  C  e.  ( Jt  A ) ) )
 
Theoremssrest 16903 If  K is a finer topology than  J, then the subspace topologies induced by  A maintain this relationship. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 1-May-2015.)
 |-  ( ( K  e.  V  /\  J  C_  K )  ->  ( Jt  A ) 
 C_  ( Kt  A ) )
 
Theoremrestopn2 16904 The if  A is open, then  B is open in  A iff it is an open subset of  A. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( ( J  e.  Top  /\  A  e.  J ) 
 ->  ( B  e.  ( Jt  A )  <->  ( B  e.  J  /\  B  C_  A ) ) )
 
Theoremrestdis 16905 A subspace of a discrete topology is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  ( ( A  e.  V  /\  B  C_  A )  ->  ( ~P At  B )  =  ~P B )
 
Theoremrestfpw 16906 The restriction of the set of finite subsets of  A is the set of finite subsets of  B. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |-  ( ( A  e.  V  /\  B  C_  A )  ->  ( ( ~P A  i^i  Fin )t  B )  =  ( ~P B  i^i  Fin ) )
 
Theoremrestcls 16907 A closure in a subspace topology. (Contributed by Jeff Hankins, 22-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  X  =  U. J   &    |-  K  =  ( Jt  Y )   =>    |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  ( ( cls `  K ) `  S )  =  ( ( ( cls `  J ) `  S )  i^i  Y ) )
 
Theoremrestntr 16908 An interior in a subspace topology. Willard in General Topology says that there is no analog of restcls 16907 for interiors. In some sense, that is true. (Contributed by Jeff Hankins, 23-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  X  =  U. J   &    |-  K  =  ( Jt  Y )   =>    |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  ( ( int `  K ) `  S )  =  ( ( ( int `  J ) `  ( S  u.  ( X  \  Y ) ) )  i^i  Y ) )
 
Theoremrestlp 16909 The limit points of a subset restrict naturally in a subspace. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  X  =  U. J   &    |-  K  =  ( Jt  Y )   =>    |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  ( ( limPt `  K ) `  S )  =  ( ( ( limPt `  J ) `  S )  i^i  Y ) )
 
Theoremrestperf 16910 Perfection of a subspace. Note that the term "perfect set" is reserved for closed sets which are perfect in the subspace topology. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  X  =  U. J   &    |-  K  =  ( Jt  Y )   =>    |-  ( ( J  e.  Top  /\  Y  C_  X )  ->  ( K  e. Perf  <->  Y  C_  ( (
 limPt `  J ) `  Y ) ) )
 
Theoremperfopn 16911 An open subset of a perfect space is perfect. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  X  =  U. J   &    |-  K  =  ( Jt  Y )   =>    |-  ( ( J  e. Perf  /\  Y  e.  J ) 
 ->  K  e. Perf )
 
Theoremresstopn 16912 The topology of a restricted structure. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  H  =  ( Ks  A )   &    |-  J  =  (
 TopOpen `  K )   =>    |-  ( Jt  A )  =  ( TopOpen `  H )
 
Theoremresstps 16913 A restricted topological space is a topological space. Note that this theorem would not be true if  TopSp was defined directly in terms of the TopSet slot instead of the  TopOpen derived function. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( ( K  e.  TopSp  /\  A  e.  V ) 
 ->  ( Ks  A )  e.  TopSp )
 
11.1.8  Order topology
 
Theoremordtbaslem 16914* Lemma for ordtbas 16918. In a total order, unbounded-above intervals are closed under intersection. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  X  =  dom  R   &    |-  A  =  ran  (  x  e.  X  |->  { y  e.  X  |  -.  y R x } )   =>    |-  ( R  e.  TosetRel  ->  ( fi `  A )  =  A )
 
Theoremordtval 16915* Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  X  =  dom  R   &    |-  A  =  ran  (  x  e.  X  |->  { y  e.  X  |  -.  y R x } )   &    |-  B  =  ran  (  x  e.  X  |->  { y  e.  X  |  -.  x R y }
 )   =>    |-  ( R  e.  V  ->  (ordTop `  R )  =  ( topGen `  ( fi `  ( { X }  u.  ( A  u.  B ) ) ) ) )
 
Theoremordtuni 16916* Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  X  =  dom  R   &    |-  A  =  ran  (  x  e.  X  |->  { y  e.  X  |  -.  y R x } )   &    |-  B  =  ran  (  x  e.  X  |->  { y  e.  X  |  -.  x R y }
 )   =>    |-  ( R  e.  V  ->  X  =  U. ( { X }  u.  ( A  u.  B ) ) )
 
Theoremordtbas2 16917* Lemma for ordtbas 16918. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  X  =  dom  R   &    |-  A  =  ran  (  x  e.  X  |->  { y  e.  X  |  -.  y R x } )   &    |-  B  =  ran  (  x  e.  X  |->  { y  e.  X  |  -.  x R y }
 )   &    |-  C  =  ran  (  a  e.  X ,  b  e.  X  |->  { y  e.  X  |  ( -.  y R a  /\  -.  b R y ) } )   =>    |-  ( R  e.  TosetRel  ->  ( fi `  ( A  u.  B ) )  =  ( ( A  u.  B )  u.  C ) )
 
Theoremordtbas 16918* In a total order, the finite intersections of the open rays generates the set of open intervals, but no more - these four collections form a subbasis for the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  X  =  dom  R   &    |-  A  =  ran  (  x  e.  X  |->  { y  e.  X  |  -.  y R x } )   &    |-  B  =  ran  (  x  e.  X  |->  { y  e.  X  |  -.  x R y }
 )   &    |-  C  =  ran  (  a  e.  X ,  b  e.  X  |->  { y  e.  X  |  ( -.  y R a  /\  -.  b R y ) } )   =>    |-  ( R  e.  TosetRel  ->  ( fi `  ( { X }  u.  ( A  u.  B ) ) )  =  ( ( { X }  u.  ( A  u.  B ) )  u.  C ) )
 
Theoremordttopon 16919 Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  X  =  dom  R   =>    |-  ( R  e.  V  ->  (ordTop `  R )  e.  (TopOn `  X ) )
 
Theoremordtopn1 16920* An upward ray  ( P ,  +oo ) is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  V  /\  P  e.  X ) 
 ->  { x  e.  X  |  -.  x R P }  e.  (ordTop `  R ) )
 
Theoremordtopn2 16921* A downward ray  (  -oo ,  P ) is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  V  /\  P  e.  X ) 
 ->  { x  e.  X  |  -.  P R x }  e.  (ordTop `  R ) )
 
Theoremordtopn3 16922* An open interval  ( A ,  B ) is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  { x  e.  X  |  ( -.  x R A  /\  -.  B R x ) }  e.  (ordTop `  R ) )
 
Theoremordtcld1 16923* A downward ray  (  -oo ,  P ] is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  V  /\  P  e.  X ) 
 ->  { x  e.  X  |  x R P }  e.  ( Clsd `  (ordTop `  R ) ) )
 
Theoremordtcld2 16924* An upward ray  [ P ,  +oo ) is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  V  /\  P  e.  X ) 
 ->  { x  e.  X  |  P R x }  e.  ( Clsd `  (ordTop `  R ) ) )
 
Theoremordtcld3 16925* An closed interval  [ A ,  B ] is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  { x  e.  X  |  ( A R x 
 /\  x R B ) }  e.  ( Clsd `  (ordTop `  R ) ) )
 
Theoremordttop 16926 The order topology is a topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( R  e.  V  ->  (ordTop `  R )  e.  Top )
 
Theoremordtcnv 16927 The order dual generates the same topology as the original order. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( R  e.  PosetRel  ->  (ordTop `  `' R )  =  (ordTop `  R ) )
 
Theoremordtrest 16928 The subspace topology of an order topology is in general finer than the topology generated by the restricted order, but we do have inclusion in one direction. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( ( R  e.  PosetRel  /\  A  e.  V ) 
 ->  (ordTop `  ( R  i^i  ( A  X.  A ) ) )  C_  ( (ordTop `  R )t  A ) )
 
Theoremordtrest2lem 16929* Lemma for ordtrest2 16930. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  X  =  dom  R   &    |-  ( ph  ->  R  e.  TosetRel  )   &    |-  ( ph  ->  A  C_  X )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A )
 )  ->  { z  e.  X  |  ( x R z  /\  z R y ) }  C_  A )   =>    |-  ( ph  ->  A. v  e.  ran  (  z  e.  X  |->  { w  e.  X  |  -.  w R z } ) ( v  i^i  A )  e.  (ordTop `  ( R  i^i  ( A  X.  A ) ) ) )
 
Theoremordtrest2 16930* An interval-closed set  A in a total order has the same subspace topology as the restricted order topology. (An interval-closed set is the same thing as an open or half-open or closed interval in  RR, but in other sets like  QQ there are interval-closed sets like  ( pi ,  +oo )  i^i  QQ that are not intervals.) (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  X  =  dom  R   &    |-  ( ph  ->  R  e.  TosetRel  )   &    |-  ( ph  ->  A  C_  X )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A )
 )  ->  { z  e.  X  |  ( x R z  /\  z R y ) }  C_  A )   =>    |-  ( ph  ->  (ordTop `  ( R  i^i  ( A  X.  A ) ) )  =  ( (ordTop `  R )t  A ) )
 
Theoremletopon 16931 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  (ordTop `  <_  )  e.  (TopOn `  RR* )
 
Theoremletop 16932 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  (ordTop `  <_  )  e. 
 Top
 
Theoremletopuni 16933 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  RR*  =  U. (ordTop `  <_  )
 
Theoremxrstopn 16934 The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  (ordTop `  <_  )  =  ( TopOpen `  RR* s )
 
Theoremxrstps 16935 The extended real number structure is a topological space. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  RR* s  e.  TopSp
 
Theoremleordtvallem1 16936* Lemma for leordtval 16939. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  A  =  ran  (  x  e.  RR*  |->  ( x (,]  +oo ) )   =>    |-  A  =  ran  (  x  e.  RR*  |->  { y  e.  RR*  |  -.  y  <_  x } )
 
Theoremleordtvallem2 16937* Lemma for leordtval 16939. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  A  =  ran  (  x  e.  RR*  |->  ( x (,]  +oo ) )   &    |-  B  =  ran  (  x  e.  RR*  |->  (  -oo [,) x ) )   =>    |-  B  =  ran  (  x  e.  RR*  |->  { y  e.  RR*  |  -.  x  <_  y } )
 
Theoremleordtval2 16938 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  A  =  ran  (  x  e.  RR*  |->  ( x (,]  +oo ) )   &    |-  B  =  ran  (  x  e.  RR*  |->  (  -oo [,) x ) )   =>    |-  (ordTop `  <_  )  =  ( topGen `  ( fi `  ( A  u.  B ) ) )
 
Theoremleordtval 16939 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  A  =  ran  (  x  e.  RR*  |->  ( x (,]  +oo ) )   &    |-  B  =  ran  (  x  e.  RR*  |->  (  -oo [,) x ) )   &    |-  C  =  ran  (,)   =>    |-  (ordTop `  <_  )  =  ( topGen `  ( ( A  u.  B )  u.  C ) )
 
Theoremiccordt 16940 A closed interval is closed in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( A [,] B )  e.  ( Clsd `  (ordTop `  <_  ) )
 
Theoremiocpnfordt 16941 An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( A (,]  +oo )  e.  (ordTop `  <_  )
 
Theoremicomnfordt 16942 An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  (  -oo [,) A )  e.  (ordTop `  <_  )
 
Theoremiooordt 16943 An open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( A (,) B )  e.  (ordTop `  <_  )
 
Theoremreordt 16944 The real numbers are an open set in the topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |- 
 RR  e.  (ordTop `  <_  )
 
Theoremlecldbas 16945 The set of closed intervals forms a closed subbasis for the topology on the extended reals. Since our definition of a basis is in terms of open sets, we express this by showing that the complements of closed intervals form an open subbasis for the topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  F  =  ( x  e.  ran  [,]  |->  ( RR*  \  x ) )   =>    |-  (ordTop `  <_  )  =  ( topGen `  ( fi `  ran  F ) )
 
Theorempnfnei 16946* A neighborhood of  +oo contains an unbounded interval based at a real number. Together with xrtgioo 18308 (which describes neighborhoods of  RR) and mnfnei 16947, this gives all "negative" topological information ensuring that it is not too fine (and of course iooordt 16943 and similar ensure that it has all the sets we want). (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( ( A  e.  (ordTop `  <_  )  /\  +oo 
 e.  A )  ->  E. x  e.  RR  ( x (,]  +oo )  C_  A )
 
Theoremmnfnei 16947* A neighborhood of  -oo contains an unbounded interval based at a real number. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( ( A  e.  (ordTop `  <_  )  /\  -oo 
 e.  A )  ->  E. x  e.  RR  (  -oo [,) x ) 
 C_  A )
 
Theoremordtrestixx 16948* The restriction of the less than order to an interval gives the same topology as the subspace topology. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  A  C_  RR*   &    |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x [,] y )  C_  A )   =>    |-  ( (ordTop `  <_  )t  A )  =  (ordTop `  (  <_  i^i  ( A  X.  A ) ) )
 
Theoremordtresticc 16949 The restriction of the less than order to a closed interval gives the same topology as the subspace topology. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( (ordTop `  <_  )t  ( A [,] B ) )  =  (ordTop `  (  <_  i^i  ( ( A [,] B )  X.  ( A [,] B ) ) ) )
 
11.1.9  Limits and Continuity in topological spaces
 
Syntaxccn 16950 Extend class notation with the set of continuous functions between topologies.
 class  Cn
 
Syntaxccnp 16951 Extend class notation with the set of functions between topologies continuous at a point.
 class  CnP
 
Syntaxclm 16952 Extend class notation with a function on topological spaces whose value is the convergence relation for limit sequences in the space.
 class  ~~> t
 
Definitiondf-cn 16953* Define a function on two topologies whose value is the set of continuous mappings from the first topology to the second. Based on definition of continuous function in [Munkres] p. 102. See iscn 16961 for the predicate form. (Contributed by NM, 17-Oct-2006.)
 |- 
 Cn  =  ( j  e.  Top ,  k  e.  Top  |->  { f  e.  ( U. k  ^m  U. j
 )  |  A. y  e.  k  ( `' f " y )  e.  j } )
 
Definitiondf-cnp 16954* Define a function on two topologies whose value is the set of continuous mappings at a specified point in the first topology. Based on Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 17-Oct-2006.)
 |- 
 CnP  =  ( j  e.  Top ,  k  e. 
 Top  |->  ( x  e. 
 U. j  |->  { f  e.  ( U. k  ^m  U. j )  |  A. y  e.  k  (
 ( f `  x )  e.  y  ->  E. g  e.  j  ( x  e.  g  /\  ( f " g
 )  C_  y )
 ) } ) )
 
Definitiondf-lm 16955* Define a function on topologies whose value is the convergence relation for the space. Although  f is typically a function from upper integers to the topological space, it doesn't have to be. Unfortunately, the value of the function must exist to use fvmpt 5564, and we use the otherwise unnecessary conjunct  dom  f  C_  CC to ensure that. (Contributed by NM, 7-Sep-2006.)
 |-  ~~> t  =  ( j  e.  Top  |->  { <. f ,  x >.  |  ( f  e.  ( U. j  ^pm  CC )  /\  x  e. 
 U. j  /\  A. u  e.  j  ( x  e.  u  ->  E. y  e.  ran  ZZ>= ( f  |`  y ) : y --> u ) ) }
 )
 
Theoremlmrel 16956 The topological space convergence relation is a relation. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)
 |- 
 Rel  ( ~~> t `  J )
 
Theoremlmrcl 16957 Reverse closure for the convergence relation. (Contributed by Mario Carneiro, 7-Sep-2015.)
 |-  ( F ( ~~> t `  J ) P  ->  J  e.  Top )
 
Theoremlmfval 16958* The relation "sequence  f converges to point  y " in a metric space. (Contributed by NM, 7-Sep-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
 |-  ( J  e.  (TopOn `  X )  ->  ( ~~> t `  J )  =  { <. f ,  x >.  |  ( f  e.  ( X  ^pm  CC )  /\  x  e.  X  /\  A. u  e.  J  ( x  e.  u  ->  E. y  e.  ran  ZZ>= ( f  |`  y ) : y --> u ) ) } )
 
Theoremcnfval 16959* The set of all continuous functions from topology  J to topology  K. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  ->  ( J  Cn  K )  =  { f  e.  ( Y  ^m  X )  | 
 A. y  e.  K  ( `' f " y )  e.  J } )
 
Theoremcnpfval 16960* The function mapping the points in a topology  J to the set of all functions from  J to topology  K continuous at that point. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  ->  ( J 
 CnP  K )  =  ( x  e.  X  |->  { f  e.  ( Y 
 ^m  X )  | 
 A. w  e.  K  ( ( f `  x )  e.  w  ->  E. v  e.  J  ( x  e.  v  /\  ( f " v
 )  C_  w )
 ) } ) )
 
Theoremiscn 16961* The predicate " F is a continuous function from topology  J to topology  K." Definition of continuous function in [Munkres] p. 102. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  ->  ( F  e.  ( J  Cn  K )  <->  ( F : X
 --> Y  /\  A. y  e.  K  ( `' F " y )  e.  J ) ) )
 
Theoremcnpval 16962* The set of all functions from topology  J to topology  K that are continuous at a point  P. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  P  e.  X )  ->  ( ( J 
 CnP  K ) `  P )  =  { f  e.  ( Y  ^m  X )  |  A. y  e.  K  ( ( f `
  P )  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  ( f " x )  C_  y ) ) } )
 
Theoremiscnp 16963* The predicate " F is a continuous function from topology  J to topology  K at point  P." Based on Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  P  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <-> 
 ( F : X --> Y  /\  A. y  e.  K  ( ( F `
  P )  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x )  C_  y ) ) ) ) )
 
Theoremiscn2 16964* The predicate " F is a continuous function from topology  J to topology  K." Definition of continuous function in [Munkres] p. 102. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  ( F  e.  ( J  Cn  K )  <->  ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : X --> Y  /\  A. y  e.  K  ( `' F " y )  e.  J ) ) )
 
Theoremiscnp2 16965* The predicate " F is a continuous function from topology  J to topology  K at point  P." Based on Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  ( F  e.  (
 ( J  CnP  K ) `  P )  <->  ( ( J  e.  Top  /\  K  e.  Top  /\  P  e.  X ) 
 /\  ( F : X
 --> Y  /\  A. y  e.  K  ( ( F `
  P )  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x )  C_  y ) ) ) ) )
 
Theoremcntop1 16966 Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
 
Theoremcntop2 16967 Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
 
Theoremcnptop1 16968 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( F  e.  (
 ( J  CnP  K ) `  P )  ->  J  e.  Top )
 
Theoremcnptop2 16969 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( F  e.  (
 ( J  CnP  K ) `  P )  ->  K  e.  Top )
 
Theoremiscnp3 16970* The predicate " F is a continuous function from topology  J to topology  K at point  P." (Contributed by NM, 15-May-2007.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  P  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <-> 
 ( F : X --> Y  /\  A. y  e.  K  ( ( F `
  P )  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  x  C_  ( `' F " y ) ) ) ) ) )
 
Theoremcnprcl 16971 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( F  e.  ( ( J  CnP  K ) `  P )  ->  P  e.  X )
 
Theoremcnf 16972 A continuous function is a mapping. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
 |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  ( F  e.  ( J  Cn  K )  ->  F : X --> Y )
 
Theoremcnpf 16973 A continuous function at point  P is a mapping. (Contributed by FL, 17-Nov-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
 |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  ( F  e.  (
 ( J  CnP  K ) `  P )  ->  F : X --> Y )
 
Theoremcnpcl 16974 The value of a continuous function from  J to  K at point  P belongs to the underlying set of topology  K. (Contributed by FL, 27-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
 |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  ( ( F  e.  ( ( J  CnP  K ) `  P ) 
 /\  A  e.  X )  ->  ( F `  A )  e.  Y )
 
Theoremcnf2 16975 A continuous function is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Cn  K ) ) 
 ->  F : X --> Y )
 
Theoremcnpf2 16976 A continuous function at point  P is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  (
 ( J  CnP  K ) `  P ) ) 
 ->  F : X --> Y )
 
Theoremcnprcl2 16977 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  P  e.  X )
 
Theoremtgcn 16978* The contininuity predicate when the range is given by a basis for a topology. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  =  ( topGen `  B )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   =>    |-  ( ph  ->  ( F  e.  ( J  Cn  K )  <->  ( F : X
 --> Y  /\  A. y  e.  B  ( `' F " y )  e.  J ) ) )
 
Theoremtgcnp 16979* The "continuous at a point" predicate when the range is given by a basis for a topology. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  =  ( topGen `  B )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  P  e.  X )   =>    |-  ( ph  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <->  ( F : X
 --> Y  /\  A. y  e.  B  ( ( F `
  P )  e.  y  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x )  C_  y ) ) ) ) )
 
Theoremsubbascn 16980* The contininuity predicate when the range is given by a subbasis for a topology. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  K  =  ( topGen `  ( fi `  B ) ) )   &    |-  ( ph  ->  K  e.  (TopOn `  Y ) )   =>    |-  ( ph  ->  ( F  e.  ( J  Cn  K )  <->  ( F : X
 --> Y  /\  A. y  e.  B  ( `' F " y )  e.  J ) ) )
 
Theoremssidcn 16981 The identity function is a continuous function from one topology to another topology on the same set iff the domain is finer than the codomain. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  ->  ( (  _I  |`  X )  e.  ( J  Cn  K ) 
 <->  K  C_  J )
 )
 
Theoremcnpimaex 16982* Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.)
 |-  ( ( F  e.  ( ( J  CnP  K ) `  P ) 
 /\  A  e.  K  /\  ( F `  P )  e.  A )  ->  E. x  e.  J  ( P  e.  x  /\  ( F " x )  C_  A ) )
 
Theoremidcn 16983 A restricted identity function is a continuous function. (Contributed by FL, 27-Dec-2006.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
 |-  ( J  e.  (TopOn `  X )  ->  (  _I  |`  X )  e.  ( J  Cn  J ) )
 
Theoremlmbr 16984* Express the binary relation "sequence  F converges to point  P " in a topological space. Definition 1.4-1 of [Kreyszig] p. 25. The condition  F  C_  ( CC 
X.  X ) allows us to use objects more general than sequences when convenient; see the comment in df-lm 16955. (Contributed by Mario Carneiro, 14-Nov-2013.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   =>    |-  ( ph  ->  ( F ( ~~> t `  J ) P  <->  ( F  e.  ( X  ^pm  CC )  /\  P  e.  X  /\  A. u  e.  J  ( P  e.  u  ->  E. y  e.  ran  ZZ>= ( F  |`  y ) : y --> u ) ) ) )
 
Theoremlmbr2 16985* Express the binary relation "sequence  F converges to point  P " in a metric space using an arbitrary set of upper integers. (Contributed by Mario Carneiro, 14-Nov-2013.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   =>    |-  ( ph  ->  ( F ( ~~> t `  J ) P  <->  ( F  e.  ( X  ^pm  CC )  /\  P  e.  X  /\  A. u  e.  J  ( P  e.  u  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( k  e.  dom  F  /\  ( F `  k )  e.  u ) ) ) ) )
 
Theoremlmbrf 16986* Express the binary relation "sequence  F converges to point  P " in a metric space using an arbitrary set of upper integers. This version of lmbr2 16985 presupposes that  F is a function. (Contributed by Mario Carneiro, 14-Nov-2013.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F : Z --> X )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  =  A )   =>    |-  ( ph  ->  ( F (
 ~~> t `  J ) P  <->  ( P  e.  X  /\  A. u  e.  J  ( P  e.  u  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) A  e.  u ) ) ) )
 
Theoremlmconst 16987 A constant sequence converges to its value. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  P  e.  X  /\  M  e.  ZZ )  ->  ( Z  X.  { P } ) ( ~~> t `  J ) P )
 
Theoremlmcvg 16988* Convergence property of a converging sequence. (Contributed by Mario Carneiro, 14-Nov-2013.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  P  e.  U )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F (
 ~~> t `  J ) P )   &    |-  ( ph  ->  U  e.  J )   =>    |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( F `
  k )  e.  U )
 
Theoremcnpnei 16989* A condition for continuity at a point in terms of neighborhoods. (Contributed by Jeff Hankins, 7-Sep-2009.)
 |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  F : X --> Y ) 
 /\  A  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  A )  <->  A. y  e.  (
 ( nei `  K ) `  { ( F `  A ) } )
 ( `' F "
 y )  e.  (
 ( nei `  J ) `  { A } )
 ) )
 
Theoremcnima 16990 An open subset of the codomain of a continuous function has an open preimage. (Contributed by FL, 15-Dec-2006.)
 |-  ( ( F  e.  ( J  Cn  K ) 
 /\  A  e.  K )  ->  ( `' F " A )  e.  J )
 
Theoremcnco 16991 The composition of two continuous functions is a continuous function. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
 |-  ( ( F  e.  ( J  Cn  K ) 
 /\  G  e.  ( K  Cn  L ) ) 
 ->  ( G  o.  F )  e.  ( J  Cn  L ) )
 
Theoremcnpco 16992 The composition of two continuous functions at point  P is a continuous function at point 
P. Proposition of [BourbakiTop1] p. I.9. (Contributed by FL, 16-Nov-2006.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
 |-  ( ( F  e.  ( ( J  CnP  K ) `  P ) 
 /\  G  e.  (
 ( K  CnP  L ) `  ( F `  P ) ) ) 
 ->  ( G  o.  F )  e.  ( ( J  CnP  L ) `  P ) )
 
Theoremcnclima 16993 A closed subset of the codomain of a continuous function has a closed preimage. (Contributed by NM, 15-Mar-2007.) (Revised by Mario Carneiro, 21-Aug-2015.)
 |-  ( ( F  e.  ( J  Cn  K ) 
 /\  A  e.  ( Clsd `  K ) ) 
 ->  ( `' F " A )  e.  ( Clsd `  J ) )
 
Theoremiscncl 16994* A definition of a continuous function using closed sets. Theorem 1 (d) of [BourbakiTop1] p. I.9. (Contributed by FL, 19-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  ->  ( F  e.  ( J  Cn  K )  <->  ( F : X
 --> Y  /\  A. y  e.  ( Clsd `  K )
 ( `' F "
 y )  e.  ( Clsd `  J ) ) ) )
 
Theoremcncls2i 16995 Property of the preimage of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  Y  =  U. K   =>    |-  (
 ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  ->  ( ( cls `  J ) `  ( `' F " S ) )  C_  ( `' F " ( ( cls `  K ) `  S ) ) )
 
Theoremcnntri 16996 Property of the preimage of an interior. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  Y  =  U. K   =>    |-  (
 ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  ->  ( `' F " ( ( int `  K ) `  S ) )  C_  ( ( int `  J ) `  ( `' F " S ) ) )
 
Theoremcnclsi 16997 Property of the image of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  ( F " ( ( cls `  J ) `  S ) )  C_  ( ( cls `  K ) `  ( F " S ) ) )
 
Theoremcncls2 16998* Continuity in terms of closure. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  ->  ( F  e.  ( J  Cn  K )  <->  ( F : X
 --> Y  /\  A. x  e.  ~P  Y ( ( cls `  J ) `  ( `' F " x ) )  C_  ( `' F " ( ( cls `  K ) `  x ) ) ) ) )
 
Theoremcncls 16999* Continuity in terms of closure. (Contributed by Jeff Hankins, 1-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  ->  ( F  e.  ( J  Cn  K )  <->  ( F : X
 --> Y  /\  A. x  e.  ~P  X ( F
 " ( ( cls `  J ) `  x ) )  C_  ( ( cls `  K ) `  ( F " x ) ) ) ) )
 
Theoremcnntr 17000* Continuity in terms of interior. (Contributed by Jeff Hankins, 2-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  ->  ( F  e.  ( J  Cn  K )  <->  ( F : X
 --> Y  /\  A. x  e.  ~P  Y ( `' F " ( ( int `  K ) `  x ) )  C_  ( ( int `  J ) `  ( `' F " x ) ) ) ) )
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