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Theorem List for Intuitionistic Logic Explorer - 12301-12400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremclsss 12301 Subset relationship for closure. (Contributed by NM, 10-Feb-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  ( ( cls `  J ) `  T )  C_  ( ( cls `  J ) `  S ) )
 
Theoremntrss 12302 Subset relationship for interior. (Contributed by NM, 3-Oct-2007.) (Revised by Jim Kingdon, 11-Mar-2023.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  ( ( int `  J ) `  T )  C_  ( ( int `  J ) `  S ) )
 
Theoremsscls 12303 A subset of a topology's underlying set is included in its closure. (Contributed by NM, 22-Feb-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  S  C_  ( ( cls `  J ) `  S ) )
 
Theoremntrss2 12304 A subset includes its interior. (Contributed by NM, 3-Oct-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( ( int `  J ) `  S )  C_  S )
 
Theoremssntr 12305 An open subset of a set is a subset of the set's interior. (Contributed by Jeff Hankins, 31-Aug-2009.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  (
 ( ( J  e.  Top  /\  S  C_  X )  /\  ( O  e.  J  /\  O  C_  S )
 )  ->  O  C_  (
 ( int `  J ) `  S ) )
 
Theoremntrss3 12306 The interior of a subset of a topological space is included in the space. (Contributed by NM, 1-Oct-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( ( int `  J ) `  S )  C_  X )
 
Theoremntrin 12307 A pairwise intersection of interiors is the interior of the intersection. This does not always hold for arbitrary intersections. (Contributed by Jeff Hankins, 31-Aug-2009.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  ( ( int `  J ) `  ( A  i^i  B ) )  =  ( ( ( int `  J ) `  A )  i^i  ( ( int `  J ) `  B ) ) )
 
Theoremisopn3 12308 A subset is open iff it equals its own interior. (Contributed by NM, 9-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( S  e.  J  <->  ( ( int `  J ) `  S )  =  S ) )
 
Theoremntridm 12309 The interior operation is idempotent. (Contributed by NM, 2-Oct-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( ( int `  J ) `  ( ( int `  J ) `  S ) )  =  (
 ( int `  J ) `  S ) )
 
Theoremclstop 12310 The closure of a topology's underlying set is the entire set. (Contributed by NM, 5-Oct-2007.) (Proof shortened by Jim Kingdon, 11-Mar-2023.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  (
 ( cls `  J ) `  X )  =  X )
 
Theoremntrtop 12311 The interior of a topology's underlying set is the entire set. (Contributed by NM, 12-Sep-2006.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  (
 ( int `  J ) `  X )  =  X )
 
Theoremclsss2 12312 If a subset is included in a closed set, so is the subset's closure. (Contributed by NM, 22-Feb-2007.)
 |-  X  =  U. J   =>    |-  (
 ( C  e.  ( Clsd `  J )  /\  S  C_  C )  ->  ( ( cls `  J ) `  S )  C_  C )
 
Theoremclsss3 12313 The closure of a subset of a topological space is included in the space. (Contributed by NM, 26-Feb-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( ( cls `  J ) `  S )  C_  X )
 
Theoremntrcls0 12314 A subset whose closure has an empty interior also has an empty interior. (Contributed by NM, 4-Oct-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X  /\  ( ( int `  J ) `  ( ( cls `  J ) `  S ) )  =  (/) )  ->  ( ( int `  J ) `  S )  =  (/) )
 
Theoremntreq0 12315* Two ways to say that a subset has an empty interior. (Contributed by NM, 3-Oct-2007.) (Revised by Jim Kingdon, 11-Mar-2023.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( ( ( int `  J ) `  S )  =  (/)  <->  A. x  e.  J  ( x  C_  S  ->  x  =  (/) ) ) )
 
Theoremcls0 12316 The closure of the empty set. (Contributed by NM, 2-Oct-2007.) (Proof shortened by Jim Kingdon, 12-Mar-2023.)
 |-  ( J  e.  Top  ->  ( ( cls `  J ) `  (/) )  =  (/) )
 
Theoremntr0 12317 The interior of the empty set. (Contributed by NM, 2-Oct-2007.)
 |-  ( J  e.  Top  ->  ( ( int `  J ) `  (/) )  =  (/) )
 
Theoremisopn3i 12318 An open subset equals its own interior. (Contributed by Mario Carneiro, 30-Dec-2016.)
 |-  ( ( J  e.  Top  /\  S  e.  J ) 
 ->  ( ( int `  J ) `  S )  =  S )
 
Theoremdiscld 12319 The open sets of a discrete topology are closed and its closed sets are open. (Contributed by FL, 7-Jun-2007.) (Revised by Mario Carneiro, 7-Apr-2015.)
 |-  ( A  e.  V  ->  ( Clsd `  ~P A )  =  ~P A )
 
Theoremsn0cld 12320 The closed sets of the topology 
{ (/) }. (Contributed by FL, 5-Jan-2009.)
 |-  ( Clsd `  { (/) } )  =  { (/) }
 
7.1.5  Neighborhoods
 
Syntaxcnei 12321 Extend class notation with neighborhood relation for topologies.
 class  nei
 
Definitiondf-nei 12322* Define a function on topologies whose value is a map from a subset to its neighborhoods. (Contributed by NM, 11-Feb-2007.)
 |- 
 nei  =  ( j  e.  Top  |->  ( x  e. 
 ~P U. j  |->  { y  e.  ~P U. j  | 
 E. g  e.  j  ( x  C_  g  /\  g  C_  y ) }
 ) )
 
Theoremneifval 12323* Value of the neighborhood function on the subsets of the base set of a topology. (Contributed by NM, 11-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  ( nei `  J )  =  ( x  e.  ~P X  |->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v
 ) } ) )
 
Theoremneif 12324 The neighborhood function is a function from the set of the subsets of the base set of a topology. (Contributed by NM, 12-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  ( nei `  J )  Fn 
 ~P X )
 
Theoremneiss2 12325 A set with a neighborhood is a subset of the base set of a topology. (This theorem depends on a function's value being empty outside of its domain, but it will make later theorems simpler to state.) (Contributed by NM, 12-Feb-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  X )
 
Theoremneival 12326* Value of the set of neighborhoods of a subset of the base set of a topology. (Contributed by NM, 11-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( ( nei `  J ) `  S )  =  { v  e.  ~P X  |  E. g  e.  J  ( S  C_  g  /\  g  C_  v
 ) } )
 
Theoremisnei 12327* The predicate "the class  N is a neighborhood of  S". (Contributed by FL, 25-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( N  e.  (
 ( nei `  J ) `  S )  <->  ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) ) )
 
Theoremneiint 12328 An intuitive definition of a neighborhood in terms of interior. (Contributed by Szymon Jaroszewicz, 18-Dec-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  ->  ( N  e.  (
 ( nei `  J ) `  S )  <->  S  C_  ( ( int `  J ) `  N ) ) )
 
Theoremisneip 12329* The predicate "the class  N is a neighborhood of point  P". (Contributed by NM, 26-Feb-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  P  e.  X ) 
 ->  ( N  e.  (
 ( nei `  J ) `  { P } )  <->  ( N  C_  X  /\  E. g  e.  J  ( P  e.  g  /\  g  C_  N ) ) ) )
 
Theoremneii1 12330 A neighborhood is included in the topology's base set. (Contributed by NM, 12-Feb-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  N  C_  X )
 
Theoremneisspw 12331 The neighborhoods of any set are subsets of the base set. (Contributed by Stefan O'Rear, 6-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  (
 ( nei `  J ) `  S )  C_  ~P X )
 
Theoremneii2 12332* Property of a neighborhood. (Contributed by NM, 12-Feb-2007.)
 |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
 
Theoremneiss 12333 Any neighborhood of a set  S is also a neighborhood of any subset  R  C_  S. Similar to Proposition 1 of [BourbakiTop1] p. I.2. (Contributed by FL, 25-Sep-2006.)
 |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  N  e.  ( ( nei `  J ) `  R ) )
 
Theoremssnei 12334 A set is included in any of its neighborhoods. Generalization to subsets of elnei 12335. (Contributed by FL, 16-Nov-2006.)
 |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  N )
 
Theoremelnei 12335 A point belongs to any of its neighborhoods. Property Viii of [BourbakiTop1] p. I.3. (Contributed by FL, 28-Sep-2006.)
 |-  ( ( J  e.  Top  /\  P  e.  A  /\  N  e.  ( ( nei `  J ) `  { P } ) ) 
 ->  P  e.  N )
 
Theorem0nnei 12336 The empty set is not a neighborhood of a nonempty set. (Contributed by FL, 18-Sep-2007.)
 |-  ( ( J  e.  Top  /\  S  =/=  (/) )  ->  -.  (/)  e.  ( ( nei `  J ) `  S ) )
 
Theoremneipsm 12337* A neighborhood of a set is a neighborhood of every point in the set. Proposition 1 of [BourbakiTop1] p. I.2. (Contributed by FL, 16-Nov-2006.) (Revised by Jim Kingdon, 22-Mar-2023.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X  /\  E. x  x  e.  S )  ->  ( N  e.  ( ( nei `  J ) `  S )  <->  A. p  e.  S  N  e.  ( ( nei `  J ) `  { p } ) ) )
 
Theoremopnneissb 12338 An open set is a neighborhood of any of its subsets. (Contributed by FL, 2-Oct-2006.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  N  e.  J  /\  S  C_  X )  ->  ( S  C_  N  <->  N  e.  (
 ( nei `  J ) `  S ) ) )
 
Theoremopnssneib 12339 Any superset of an open set is a neighborhood of it. (Contributed by NM, 14-Feb-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  e.  J  /\  N  C_  X )  ->  ( S  C_  N  <->  N  e.  (
 ( nei `  J ) `  S ) ) )
 
Theoremssnei2 12340 Any subset  M of  X containing a neighborhood  N of a set  S is a neighborhood of this set. Generalization to subsets of Property Vi of [BourbakiTop1] p. I.3. (Contributed by FL, 2-Oct-2006.)
 |-  X  =  U. J   =>    |-  (
 ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  /\  ( N  C_  M  /\  M  C_  X ) ) 
 ->  M  e.  ( ( nei `  J ) `  S ) )
 
Theoremopnneiss 12341 An open set is a neighborhood of any of its subsets. (Contributed by NM, 13-Feb-2007.)
 |-  ( ( J  e.  Top  /\  N  e.  J  /\  S  C_  N )  ->  N  e.  ( ( nei `  J ) `  S ) )
 
Theoremopnneip 12342 An open set is a neighborhood of any of its members. (Contributed by NM, 8-Mar-2007.)
 |-  ( ( J  e.  Top  /\  N  e.  J  /\  P  e.  N )  ->  N  e.  ( ( nei `  J ) `  { P } )
 )
 
Theoremtpnei 12343 The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 12341. (Contributed by FL, 2-Oct-2006.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  ( S  C_  X  <->  X  e.  (
 ( nei `  J ) `  S ) ) )
 
Theoremneiuni 12344 The union of the neighborhoods of a set equals the topology's underlying set. (Contributed by FL, 18-Sep-2007.) (Revised by Mario Carneiro, 9-Apr-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  X  =  U. (
 ( nei `  J ) `  S ) )
 
Theoremtopssnei 12345 A finer topology has more neighborhoods. (Contributed by Mario Carneiro, 9-Apr-2015.)
 |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y ) 
 /\  J  C_  K )  ->  ( ( nei `  J ) `  S )  C_  ( ( nei `  K ) `  S ) )
 
Theoreminnei 12346 The intersection of two neighborhoods of a set is also a neighborhood of the set. Generalization to subsets of Property Vii of [BourbakiTop1] p. I.3 for binary intersections. (Contributed by FL, 28-Sep-2006.)
 |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  M  e.  ( ( nei `  J ) `  S ) ) 
 ->  ( N  i^i  M )  e.  ( ( nei `  J ) `  S ) )
 
Theoremopnneiid 12347 Only an open set is a neighborhood of itself. (Contributed by FL, 2-Oct-2006.)
 |-  ( J  e.  Top  ->  ( N  e.  (
 ( nei `  J ) `  N )  <->  N  e.  J ) )
 
Theoremneissex 12348* For any neighborhood  N of  S, there is a neighborhood  x of  S such that  N is a neighborhood of all subsets of  x. Generalization to subsets of Property Viv of [BourbakiTop1] p. I.3. (Contributed by FL, 2-Oct-2006.)
 |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  E. x  e.  (
 ( nei `  J ) `  S ) A. y
 ( y  C_  x  ->  N  e.  ( ( nei `  J ) `  y ) ) )
 
Theorem0nei 12349 The empty set is a neighborhood of itself. (Contributed by FL, 10-Dec-2006.)
 |-  ( J  e.  Top  ->  (/) 
 e.  ( ( nei `  J ) `  (/) ) )
 
7.1.6  Subspace topologies
 
Theoremrestrcl 12350 Reverse closure for the subspace topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Proof shortened by Jim Kingdon, 23-Mar-2023.)
 |-  ( ( Jt  A )  e.  Top  ->  ( J  e.  _V  /\  A  e.  _V ) )
 
Theoremrestbasg 12351 A subspace topology basis is a basis. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  ( ( B  e.  TopBases  /\  A  e.  V ) 
 ->  ( Bt  A )  e.  TopBases )
 
Theoremtgrest 12352 A subspace can be generated by restricted sets from a basis for the original topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Proof shortened by Mario Carneiro, 30-Aug-2015.)
 |-  ( ( B  e.  V  /\  A  e.  W )  ->  ( topGen `  ( Bt  A ) )  =  ( ( topGen `  B )t  A ) )
 
Theoremresttop 12353 A subspace topology is a topology. Definition of subspace topology in [Munkres] p. 89.  A is normally a subset of the base set of  J. (Contributed by FL, 15-Apr-2007.) (Revised by Mario Carneiro, 1-May-2015.)
 |-  ( ( J  e.  Top  /\  A  e.  V ) 
 ->  ( Jt  A )  e.  Top )
 
Theoremresttopon 12354 A subspace topology is a topology on the base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( Jt  A )  e.  (TopOn `  A ) )
 
Theoremrestuni 12355 The underlying set of a subspace topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  A  C_  X )  ->  A  =  U. ( Jt  A ) )
 
Theoremstoig 12356 The topological space built with a subspace topology. (Contributed by FL, 5-Jan-2009.) (Proof shortened by Mario Carneiro, 1-May-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  A  C_  X )  ->  { <. ( Base `  ndx ) ,  A >. , 
 <. (TopSet `  ndx ) ,  ( Jt  A ) >. }  e.  TopSp
 )
 
Theoremrestco 12357 Composition of subspaces. (Contributed by Mario Carneiro, 15-Dec-2013.) (Revised by Mario Carneiro, 1-May-2015.)
 |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X ) 
 ->  ( ( Jt  A )t  B )  =  ( Jt  ( A  i^i  B ) ) )
 
Theoremrestabs 12358 Equivalence of being a subspace of a subspace and being a subspace of the original. (Contributed by Jeff Hankins, 11-Jul-2009.) (Proof shortened by Mario Carneiro, 1-May-2015.)
 |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W ) 
 ->  ( ( Jt  T )t  S )  =  ( Jt  S ) )
 
Theoremrestin 12359 When the subspace region is not a subset of the base of the topology, the resulting set is the same as the subspace restricted to the base. (Contributed by Mario Carneiro, 15-Dec-2013.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  V  /\  A  e.  W ) 
 ->  ( Jt  A )  =  ( Jt  ( A  i^i  X ) ) )
 
Theoremrestuni2 12360 The underlying set of a subspace topology. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  A  e.  V ) 
 ->  ( A  i^i  X )  =  U. ( Jt  A ) )
 
Theoremresttopon2 12361 The underlying set of a subspace topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  ->  ( Jt  A )  e.  (TopOn `  ( A  i^i  X ) ) )
 
Theoremrest0 12362 The subspace topology induced by the topology  J on the empty set. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 1-May-2015.)
 |-  ( J  e.  Top  ->  ( Jt  (/) )  =  { (/)
 } )
 
Theoremrestsn 12363 The only subspace topology induced by the topology  { (/)
}. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  ( A  e.  V  ->  ( { (/) }t  A )  =  { (/) } )
 
Theoremrestopnb 12364 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 12365 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 12366 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 12367 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 )
 
7.1.7  Limits and continuity in topological spaces
 
Syntaxccn 12368 Extend class notation with the class of continuous functions between topologies.
 class  Cn
 
Syntaxccnp 12369 Extend class notation with the class of functions between topologies continuous at a given point.
 class  CnP
 
Syntaxclm 12370 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 12371* 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 12380 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 12372* 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 12373* Define a function on topologies whose value is the convergence relation for sequences into the given topological space. Although  f is typically a sequence (a function from an upperset of integers) with values in the topological space, it need not be. Note, however, that the limit property concerns only values at integers, so that the real-valued function  ( x  e.  RR  |->  ( sin `  ( pi  x.  x ) ) ) converges to zero (in the standard topology on the reals) with this definition. (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 ) ) }
 )
 
Theoremlmrcl 12374 Reverse closure for the convergence relation. (Contributed by Mario Carneiro, 7-Sep-2015.)
 |-  ( F ( ~~> t `  J ) P  ->  J  e.  Top )
 
Theoremlmfval 12375* 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 ) ) } )
 
Theoremlmreltop 12376 The topological space convergence relation is a relation. (Contributed by Jim Kingdon, 25-Mar-2023.)
 |-  ( J  e.  Top  ->  Rel  ( ~~> t `  J ) )
 
Theoremcnfval 12377* 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 12378* 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 )
 ) } ) )
 
Theoremcnovex 12379 The class of all continuous functions from a topology to another is a set. (Contributed by Jim Kingdon, 14-Dec-2023.)
 |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J  Cn  K )  e.  _V )
 
Theoremiscn 12380* The predicate "the class  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 12381* 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 12382* The predicate "the class  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 12383* The predicate "the class  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 ) ) )
 
Theoremcntop1 12384 Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
 
Theoremcntop2 12385 Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
 
Theoremiscnp3 12386* The predicate "the class  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 ) ) ) ) ) )
 
Theoremcnf 12387 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 )
 
Theoremcnf2 12388 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 )
 
Theoremcnprcl2k 12389 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.) (Revised by Jim Kingdon, 28-Mar-2023.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  Top  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  P  e.  X )
 
Theoremcnpf2 12390 A continuous function at point  P is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.) (Revised by Jim Kingdon, 28-Mar-2023.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  (
 ( J  CnP  K ) `  P ) ) 
 ->  F : X --> Y )
 
Theoremtgcn 12391* The continuity 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 12392* 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 ) ) ) ) )
 
Theoremssidcn 12393 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 )
 )
 
Theoremicnpimaex 12394* Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.) (Revised by Jim Kingdon, 28-Mar-2023.)
 |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  P  e.  X )  /\  ( 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 12395 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 12396* 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 12373. (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 12397* Express the binary relation "sequence  F converges to point  P " in a metric space using an arbitrary upper set of 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 12398* Express the binary relation "sequence  F converges to point  P " in a metric space using an arbitrary upper set of integers. This version of lmbr2 12397 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 12399 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 12400* 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 )
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