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Theorem List for Metamath Proof Explorer - 16801-16900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiscnp2 16801* 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 16802 Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
 
Theoremcntop2 16803 Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
 
Theoremcnptop1 16804 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 16805 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 16806* 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 16807 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 16808 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 16809 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 16810 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 16811 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 16812 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 16813 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 16814* 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 16815* 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 16816* 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 16817 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 16818* 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 16819 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 16820* 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 16791. (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 16821* 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 16822* 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 16821 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 16823 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 16824* 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 16825* 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 16826 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 16827 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 16828 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 16829 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 16830* 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 16831 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 16832 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 16833 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 16834* 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 16835* 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 16836* 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 ) ) ) ) )
 
Theoremcnss1 16837 If the topology  K is finer than  J, then there are more continuous functions from  K than from  J. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( K  e.  (TopOn `  X )  /\  J  C_  K )  ->  ( J  Cn  L )  C_  ( K  Cn  L ) )
 
Theoremcnss2 16838 If the topology  K is finer than  J, then there are fewer continuous functions into  K than into  J from some other space. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
 |-  Y  =  U. K   =>    |-  (
 ( L  e.  (TopOn `  Y )  /\  L  C_  K )  ->  ( J  Cn  K )  C_  ( J  Cn  L ) )
 
Theoremcncnpi 16839 A continuous function is continuous at all points. One direction of Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( F  e.  ( J  Cn  K )  /\  A  e.  X )  ->  F  e.  ( ( J  CnP  K ) `
  A ) )
 
Theoremcnsscnp 16840 The set of continuous functions is a subset of the set of continuous functions at a point. (Contributed by Raph Levien, 21-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( P  e.  X  ->  ( J  Cn  K ) 
 C_  ( ( J 
 CnP  K ) `  P ) )
 
Theoremcncnp 16841* A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 15-May-2007.) (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. x  e.  X  F  e.  (
 ( J  CnP  K ) `  x ) ) ) )
 
Theoremcncnp2 16842* A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
 |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  ( X  =/=  (/)  ->  ( F  e.  ( J  Cn  K )  <->  A. x  e.  X  F  e.  ( ( J  CnP  K ) `  x ) ) )
 
Theoremcnconst2 16843 A constant function is continuous. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  B  e.  Y )  ->  ( X  X.  { B } )  e.  ( J  Cn  K ) )
 
Theoremcnconst 16844 A constant function is continuous. (Contributed by FL, 15-Jan-2007.) (Proof shortened by Mario Carneiro, 19-Mar-2015.)
 |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  /\  ( B  e.  Y  /\  F : X --> { B } ) )  ->  F  e.  ( J  Cn  K ) )
 
Theoremcnrest 16845 Continuity of a restriction from a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 21-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )
 
Theoremcnrest2 16846 Equivalence of continuity in the parent topology and continuity in a subspace. (Contributed by Jeff Hankins, 10-Jul-2009.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
 |-  ( ( K  e.  (TopOn `  Y )  /\  ran 
 F  C_  B  /\  B  C_  Y )  ->  ( F  e.  ( J  Cn  K )  <->  F  e.  ( J  Cn  ( Kt  B ) ) ) )
 
Theoremcnrest2r 16847 Equivalence of continuity in the parent topology and continuity in a subspace. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Jun-2014.)
 |-  ( K  e.  Top  ->  ( J  Cn  ( Kt  B ) )  C_  ( J  Cn  K ) )
 
Theoremcnpresti 16848 One direction of cnprest 16849 under the weaker condition that the point is in the subset rather than the interior of the subset. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 1-May-2015.)
 |-  X  =  U. J   =>    |-  (
 ( A  C_  X  /\  P  e.  A  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  ( F  |`  A )  e.  ( ( ( Jt  A )  CnP  K ) `  P ) )
 
Theoremcnprest 16849 Equivalence of continuity at a point and continuity of the restricted function at a point. (Contributed by Mario Carneiro, 8-Aug-2014.)
 |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  ( P  e.  ( ( int `  J ) `  A )  /\  F : X --> Y ) )  ->  ( F  e.  (
 ( J  CnP  K ) `  P )  <->  ( F  |`  A )  e.  ( ( ( Jt  A )  CnP  K ) `  P ) ) )
 
Theoremcnprest2 16850 Equivalence of point-continuity in the parent topology and point-continuity in a subspace. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 21-Aug-2015.)
 |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  ( ( K  e.  Top  /\  F : X --> B  /\  B  C_  Y )  ->  ( F  e.  (
 ( J  CnP  K ) `  P )  <->  F  e.  (
 ( J  CnP  ( Kt  B ) ) `  P ) ) )
 
Theoremcndis 16851 Every function is continuous when the domain is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
 |-  ( ( A  e.  V  /\  J  e.  (TopOn `  X ) )  ->  ( ~P A  Cn  J )  =  ( X  ^m  A ) )
 
Theoremcnindis 16852 Every function is continuous when the codomain is indiscrete (trivial). (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  ->  ( J  Cn  { (/)
 ,  A } )  =  ( A  ^m  X ) )
 
Theoremcnpdis 16853 If  A is an isolated point in  X (or equivalently, the singleton  { A } is open in  X), then every function is continuous at  A. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X )  /\  { A }  e.  J )  ->  ( ( J 
 CnP  K ) `  A )  =  ( Y  ^m  X ) )
 
Theorempaste 16854 Pasting lemma. If  A and  B are closed sets in  X with  A  u.  B  =  X, then any function whose restrictions to  A and  B are continuous is continuous on all of  X. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
 |-  X  =  U. J   &    |-  Y  =  U. K   &    |-  ( ph  ->  A  e.  ( Clsd `  J ) )   &    |-  ( ph  ->  B  e.  ( Clsd `  J ) )   &    |-  ( ph  ->  ( A  u.  B )  =  X )   &    |-  ( ph  ->  F : X --> Y )   &    |-  ( ph  ->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )   &    |-  ( ph  ->  ( F  |`  B )  e.  (
 ( Jt  B )  Cn  K ) )   =>    |-  ( ph  ->  F  e.  ( J  Cn  K ) )
 
Theoremlmfpm 16855 If  F converges, then  F is a partial function. (Contributed by Mario Carneiro, 23-Dec-2013.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F ( ~~> t `  J ) P ) 
 ->  F  e.  ( X 
 ^pm  CC ) )
 
Theoremlmfss 16856 Inclusion of a function having a limit (used to ensure the limit relation is a set, under our definition). (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F ( ~~> t `  J ) P ) 
 ->  F  C_  ( CC  X.  X ) )
 
Theoremlmcl 16857 Closure of a limit. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F ( ~~> t `  J ) P ) 
 ->  P  e.  X )
 
Theoremlmss 16858 Limit on a subspace. (Contributed by NM, 30-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)
 |-  K  =  ( Jt  Y )   &    |-  Z  =  (
 ZZ>= `  M )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  J  e.  Top )   &    |-  ( ph  ->  P  e.  Y )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F : Z --> Y )   =>    |-  ( ph  ->  ( F ( ~~> t `  J ) P  <->  F ( ~~> t `  K ) P ) )
 
Theoremsslm 16859 A finer topology has fewer convergent sequences (but the sequences that do converge converge to the same value). (Contributed by Mario Carneiro, 15-Sep-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  ( ~~> t `  K )  C_  ( ~~> t `  J ) )
 
Theoremlmres 16860 A function converges iff its restriction to an upper integers set converges. (Contributed by Mario Carneiro, 31-Dec-2013.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  F  e.  ( X  ^pm  CC ) )   &    |-  ( ph  ->  M  e.  ZZ )   =>    |-  ( ph  ->  ( F ( ~~> t `  J ) P  <->  ( F  |`  ( ZZ>= `  M ) ) ( ~~> t `  J ) P ) )
 
Theoremlmff 16861* If  F converges, there is some upper integer set on which 
F is a total function. (Contributed by Mario Carneiro, 31-Dec-2013.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  e.  dom  (
 ~~> t `  J ) )   =>    |-  ( ph  ->  E. j  e.  Z  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X )
 
Theoremlmcls 16862* Any convergent sequence of points in a subset of a topological space converges to a point in the closure of the subset. (Contributed by Mario Carneiro, 30-Dec-2013.) (Revised by Mario Carneiro, 1-May-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F ( ~~> t `  J ) P )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e.  S )   &    |-  ( ph  ->  S 
 C_  X )   =>    |-  ( ph  ->  P  e.  ( ( cls `  J ) `  S ) )
 
Theoremlmcld 16863* Any convergent sequence of points in a closed subset of a topological space converges to a point in the set. (Contributed by Mario Carneiro, 30-Dec-2013.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F ( ~~> t `  J ) P )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e.  S )   &    |-  ( ph  ->  S  e.  ( Clsd `  J ) )   =>    |-  ( ph  ->  P  e.  S )
 
Theoremlmcnp 16864 The image of a convergent sequence under a continuous map is convergent to the image of the original point. (Contributed by Mario Carneiro, 3-May-2014.)
 |-  ( ph  ->  F (
 ~~> t `  J ) P )   &    |-  ( ph  ->  G  e.  ( ( J 
 CnP  K ) `  P ) )   =>    |-  ( ph  ->  ( G  o.  F ) ( ~~> t `  K ) ( G `  P ) )
 
Theoremlmcn 16865 The image of a convergent sequence under a continuous map is convergent to the image of the original point. (Contributed by Mario Carneiro, 3-May-2014.)
 |-  ( ph  ->  F (
 ~~> t `  J ) P )   &    |-  ( ph  ->  G  e.  ( J  Cn  K ) )   =>    |-  ( ph  ->  ( G  o.  F ) ( ~~> t `  K ) ( G `  P ) )
 
11.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...
 
Syntaxct0 16866 Extend class notation with the class of all T0 spaces.
 class  Kol2
 
Syntaxct1 16867 Extend class notation to include T1 spaces (also called Fréchet spaces).
 class  Fre
 
Syntaxcha 16868 Extend class notation with the class of all Hausdorff spaces.
 class  Haus
 
Syntaxcreg 16869 Extend class notation with the class of all regular topologies.
 class  Reg
 
Syntaxcnrm 16870 Extend class notation with the class of all normal topologies.
 class  Nrm
 
Syntaxccnrm 16871 Extend class notation with the class of all completely normal topologies.
 class CNrm
 
Syntaxcpnrm 16872 Extend class notation with the class of all perfectly normal topologies.
 class PNrm
 
Definitiondf-t0 16873* Define T0 or Kolmogorov spaces. A T0 space satisfies a kind of "topological extensionality" principle (compare ax-ext 2234): any two points which are members of the same open sets are equal, or in contraposition, for any two distinct points there is an open set which contains one point but not the other. This differs from T1 spaces (see ist1-2 16907) in that in a T1 space you can choose which point will be in the open set and which outside; in a T0 space you only know that one of the two points is in the set. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |- 
 Kol2  =  { j  e.  Top  |  A. x  e.  U. j A. y  e.  U. j ( A. o  e.  j  ( x  e.  o  <->  y  e.  o
 )  ->  x  =  y ) }
 
Definitiondf-t1 16874* The class of all T1 spaces, also called Fréchet spaces. Morris, Topology without tears, p. 30 ex. 3. (Contributed by FL, 18-Jun-2007.)
 |- 
 Fre  =  { x  e.  Top  |  A. a  e.  U. x { a }  e.  ( Clsd `  x ) }
 
Definitiondf-haus 16875* Define the class of all Hausdorff spaces. A Hausdorff space is a topology in which distinct points have disjoint open neighborhoods. Definition of Hausdorff space in [Munkres] p. 98. (Contributed by NM, 8-Mar-2007.)
 |- 
 Haus  =  { j  e.  Top  |  A. x  e.  U. j A. y  e.  U. j ( x  =/=  y  ->  E. n  e.  j  E. m  e.  j  ( x  e.  n  /\  y  e.  m  /\  ( n  i^i  m )  =  (/) ) ) }
 
Definitiondf-reg 16876* Define regular spaces. A space is regular if a point and a closed set can be separated by neighborhoods. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |- 
 Reg  =  { j  e.  Top  |  A. x  e.  j  A. y  e.  x  E. z  e.  j  ( y  e.  z  /\  ( ( cls `  j ) `  z )  C_  x ) }
 
Definitiondf-nrm 16877* Define normal spaces. A space is normal if disjoint closed sets can be separated by neighborhoods. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |- 
 Nrm  =  { j  e.  Top  |  A. x  e.  j  A. y  e.  ( ( Clsd `  j
 )  i^i  ~P x ) E. z  e.  j  ( y  C_  z  /\  ( ( cls `  j
 ) `  z )  C_  x ) }
 
Definitiondf-cnrm 16878* Define completely normal spaces. A space is completely normal if all its subspaces are normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |- CNrm  =  { j  e.  Top  | 
 A. x  e.  ~P  U. j ( jt  x )  e.  Nrm }
 
Definitiondf-pnrm 16879* Define perfectly normal spaces. A space is perfectly normal if it is normal and every closed set is a G&delta; set, meaning that it is a countable intersection of open sets. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |- PNrm  =  { j  e.  Nrm  |  ( Clsd `  j )  C_ 
 ran  (  f  e.  ( j  ^m  NN )  |->  |^| ran  f ) }
 
Theoremist0 16880* The predicate "is a T0 space." Every pair of distinct points is topologically distinguishable. For the way this definition is usually encountered, see ist0-3 16905. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |-  X  =  U. J   =>    |-  ( J  e.  Kol2  <->  ( J  e.  Top  /\  A. x  e.  X  A. y  e.  X  (
 A. o  e.  J  ( x  e.  o  <->  y  e.  o )  ->  x  =  y )
 ) )
 
Theoremist1 16881* The predicate  J is T1. (Contributed by FL, 18-Jun-2007.)
 |-  X  =  U. J   =>    |-  ( J  e.  Fre  <->  ( J  e.  Top  /\  A. a  e.  X  { a }  e.  ( Clsd `  J )
 ) )
 
Theoremishaus 16882* Express the predicate " J is a Hausdorff space." (Contributed by NM, 8-Mar-2007.)
 |-  X  =  U. J   =>    |-  ( J  e.  Haus  <->  ( J  e.  Top  /\  A. x  e.  X  A. y  e.  X  ( x  =/=  y  ->  E. n  e.  J  E. m  e.  J  ( x  e.  n  /\  y  e.  m  /\  ( n  i^i  m )  =  (/) ) ) ) )
 
Theoremiscnrm 16883* The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( J  e. CNrm  <->  ( J  e.  Top  /\  A. x  e.  ~P  X ( Jt  x )  e.  Nrm ) )
 
Theoremt0sep 16884* Any two topologically indistinguishable points in a T0 space are identical. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Kol2  /\  ( A  e.  X  /\  B  e.  X ) )  ->  ( A. x  e.  J  ( A  e.  x  <->  B  e.  x )  ->  A  =  B ) )
 
Theoremt0dist 16885* Any two distinct points in a T0 space are topologically distinguishable. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Kol2  /\  ( A  e.  X  /\  B  e.  X  /\  A  =/=  B ) ) 
 ->  E. o  e.  J  -.  ( A  e.  o  <->  B  e.  o ) )
 
Theoremt1sncld 16886 In a T1 space, one-point sets are closed. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Fre  /\  A  e.  X ) 
 ->  { A }  e.  ( Clsd `  J )
 )
 
Theoremt1ficld 16887 In a T1 space, finite sets are closed. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Fre  /\  A  C_  X  /\  A  e.  Fin )  ->  A  e.  ( Clsd `  J ) )
 
Theoremhausnei 16888* Neighborhood property of a Hausdorff space. (Contributed by NM, 8-Mar-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Haus  /\  ( P  e.  X  /\  Q  e.  X  /\  P  =/=  Q ) ) 
 ->  E. n  e.  J  E. m  e.  J  ( P  e.  n  /\  Q  e.  m  /\  ( n  i^i  m )  =  (/) ) )
 
Theoremt0top 16889 A T0 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |-  ( J  e.  Kol2  ->  J  e.  Top )
 
Theoremt1top 16890 A T1 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |-  ( J  e.  Fre  ->  J  e.  Top )
 
Theoremhaustop 16891 A Hausdorff space is a topology. (Contributed by NM, 5-Mar-2007.)
 |-  ( J  e.  Haus  ->  J  e.  Top )
 
Theoremisreg 16892* The predicate "is a regular space." In a regular space, any open neighborhood has a closed subneighborhood. Note that some authors require the space to be Hausdorff (which would make it the same as T3), but we reserve the phrase "regular Hausdorff" for that as many topologists do. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  e.  Reg  <->  ( J  e.  Top  /\  A. x  e.  J  A. y  e.  x  E. z  e.  J  ( y  e.  z  /\  ( ( cls `  J ) `  z )  C_  x ) ) )
 
Theoremregtop 16893 A regular space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |-  ( J  e.  Reg  ->  J  e.  Top )
 
Theoremregsep 16894* In a regular space, every neighborhood of a point contains a closed subneighborhood. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( ( J  e.  Reg  /\  U  e.  J  /\  A  e.  U )  ->  E. x  e.  J  ( A  e.  x  /\  ( ( cls `  J ) `  x )  C_  U ) )
 
Theoremisnrm 16895* The predicate "is a normal space." Much like the case for regular spaces, normal does not imply Hausdorff or even regular. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 24-Aug-2015.)
 |-  ( J  e.  Nrm  <->  ( J  e.  Top  /\  A. x  e.  J  A. y  e.  ( ( Clsd `  J )  i^i  ~P x ) E. z  e.  J  ( y  C_  z  /\  ( ( cls `  J ) `  z )  C_  x ) ) )
 
Theoremnrmtop 16896 A normal space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |-  ( J  e.  Nrm  ->  J  e.  Top )
 
Theoremcnrmtop 16897 A completely normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( J  e. CNrm  ->  J  e.  Top )
 
Theoremiscnrm2 16898* The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( J  e.  (TopOn `  X )  ->  ( J  e. CNrm  <->  A. x  e.  ~P  X ( Jt  x )  e.  Nrm ) )
 
Theoremispnrm 16899* The property of being perfectly normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( J  e. PNrm  <->  ( J  e.  Nrm  /\  ( Clsd `  J )  C_ 
 ran  (  f  e.  ( J  ^m  NN )  |->  |^| ran  f )
 ) )
 
Theorempnrmnrm 16900 A perfectly normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( J  e. PNrm  ->  J  e.  Nrm )
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