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Theorem List for Metamath Proof Explorer - 20101-20200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremptcnp 20101* If every projection of a function is continuous at  D, then the function itself is continuous at  D into the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  K  =  ( Xt_ `  F )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  F : I --> Top )   &    |-  ( ph  ->  D  e.  X )   &    |-  (
 ( ph  /\  k  e.  I )  ->  ( x  e.  X  |->  A )  e.  ( ( J 
 CnP  ( F `  k ) ) `  D ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( k  e.  I  |->  A ) )  e.  (
 ( J  CnP  K ) `  D ) )
 
Theoremupxp 20102* Universal property of the Cartesian product considered as a categorical product in the category of sets. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  P  =  ( 1st  |`  ( B  X.  C ) )   &    |-  Q  =  ( 2nd  |`  ( B  X.  C ) )   =>    |-  ( ( A  e.  D  /\  F : A --> B  /\  G : A --> C )  ->  E! h ( h : A
 --> ( B  X.  C )  /\  F  =  ( P  o.  h ) 
 /\  G  =  ( Q  o.  h ) ) )
 
Theoremtxcnmpt 20103* A map into the product of two topological spaces is continuous if both of its projections are continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  W  =  U. U   &    |-  H  =  ( x  e.  W  |->  <.
 ( F `  x ) ,  ( G `  x ) >. )   =>    |-  ( ( F  e.  ( U  Cn  R )  /\  G  e.  ( U  Cn  S ) )  ->  H  e.  ( U  Cn  ( R  tX  S ) ) )
 
Theoremuptx 20104* Universal property of the binary topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
 |-  T  =  ( R 
 tX  S )   &    |-  X  =  U. R   &    |-  Y  =  U. S   &    |-  Z  =  ( X  X.  Y )   &    |-  P  =  ( 1st  |`  Z )   &    |-  Q  =  ( 2nd  |`  Z )   =>    |-  ( ( F  e.  ( U  Cn  R ) 
 /\  G  e.  ( U  Cn  S ) ) 
 ->  E! h  e.  ( U  Cn  T ) ( F  =  ( P  o.  h )  /\  G  =  ( Q  o.  h ) ) )
 
Theoremtxcn 20105 A map into the product of two topological spaces is continuous iff both of its projections are continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
 |-  X  =  U. R   &    |-  Y  =  U. S   &    |-  Z  =  ( X  X.  Y )   &    |-  W  =  U. U   &    |-  P  =  ( 1st  |`  Z )   &    |-  Q  =  ( 2nd  |`  Z )   =>    |-  ( ( R  e.  Top  /\  S  e.  Top  /\  F : W --> Z ) 
 ->  ( F  e.  ( U  Cn  ( R  tX  S ) )  <->  ( ( P  o.  F )  e.  ( U  Cn  R )  /\  ( Q  o.  F )  e.  ( U  Cn  S ) ) ) )
 
Theoremptcn 20106* If every projection of a function is continuous, then the function itself is continuous into the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  K  =  ( Xt_ `  F )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  F : I --> Top )   &    |-  ( ( ph  /\  k  e.  I ) 
 ->  ( x  e.  X  |->  A )  e.  ( J  Cn  ( F `  k ) ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( k  e.  I  |->  A ) )  e.  ( J  Cn  K ) )
 
Theoremprdstopn 20107 Topology of a structure product. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  Fn  I )   &    |-  O  =  (
 TopOpen `  Y )   =>    |-  ( ph  ->  O  =  ( Xt_ `  ( TopOpen  o.  R ) ) )
 
Theoremprdstps 20108 A structure product of topologies is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R : I --> TopSp )   =>    |-  ( ph  ->  Y  e.  TopSp )
 
Theorempwstps 20109 A structure product of topologies is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  Y  =  ( R 
 ^s  I )   =>    |-  ( ( R  e.  TopSp  /\  I  e.  V )  ->  Y  e.  TopSp )
 
Theoremtxrest 20110 The subspace of a topological product space induced by a subset with a Cartesian product representation is a topological product of the subspaces induced by the subspaces of the terms of the products. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
 |-  ( ( ( R  e.  V  /\  S  e.  W )  /\  ( A  e.  X  /\  B  e.  Y )
 )  ->  ( ( R  tX  S )t  ( A  X.  B ) )  =  ( ( Rt  A )  tX  ( St  B ) ) )
 
Theoremtxdis 20111 The topological product of discrete spaces is discrete. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ~P A  tX 
 ~P B )  =  ~P ( A  X.  B ) )
 
Theoremtxindislem 20112 Lemma for txindis 20113. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( (  _I  `  A )  X.  (  _I  `  B ) )  =  (  _I  `  ( A  X.  B ) )
 
Theoremtxindis 20113 The topological product of indiscrete spaces is indiscrete. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( { (/) ,  A }  tX  { (/) ,  B } )  =  { (/)
 ,  ( A  X.  B ) }
 
Theoremtxdis1cn 20114* A function is jointly continuous on a discrete left topology iff it is continuous as a function of its right argument, for each fixed left value. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  J  e.  (TopOn `  Y ) )   &    |-  ( ph  ->  K  e.  Top )   &    |-  ( ph  ->  F  Fn  ( X  X.  Y ) )   &    |-  ( ( ph  /\  x  e.  X )  ->  (
 y  e.  Y  |->  ( x F y ) )  e.  ( J  Cn  K ) )   =>    |-  ( ph  ->  F  e.  ( ( ~P X  tX  J )  Cn  K ) )
 
Theoremtxlly 20115* If the property  A is preserved under topological products, then so is the property of being locally  A. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  ( ( j  e.  A  /\  k  e.  A )  ->  (
 j  tX  k )  e.  A )   =>    |-  ( ( R  e. Locally  A 
 /\  S  e. Locally  A ) 
 ->  ( R  tX  S )  e. Locally  A )
 
Theoremtxnlly 20116* If the property  A is preserved under topological products, then so is the property of being n-locally  A. (Contributed by Mario Carneiro, 13-Apr-2015.)
 |-  ( ( j  e.  A  /\  k  e.  A )  ->  (
 j  tX  k )  e.  A )   =>    |-  ( ( R  e. 𝑛Locally  A  /\  S  e. 𝑛Locally  A )  ->  ( R  tX  S )  e. 𝑛Locally  A )
 
Theorempthaus 20117 The product of a collection of Hausdorff spaces is Hausdorff. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  ( ( A  e.  V  /\  F : A --> Haus )  ->  ( Xt_ `  F )  e.  Haus )
 
Theoremptrescn 20118* Restriction is a continuous function on product topologies. (Contributed by Mario Carneiro, 7-Feb-2015.)
 |-  X  =  U. J   &    |-  J  =  ( Xt_ `  F )   &    |-  K  =  ( Xt_ `  ( F  |`  B ) )   =>    |-  ( ( A  e.  V  /\  F : A --> Top  /\  B  C_  A )  ->  ( x  e.  X  |->  ( x  |`  B ) )  e.  ( J  Cn  K ) )
 
Theoremtxtube 20119* The "tube lemma". If  X is compact and there is an open set  U containing the line  X  X.  { A }, then there is a "tube"  X  X.  u for some neighborhood  u of  A which is entirely contained within  U. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  X  =  U. R   &    |-  Y  =  U. S   &    |-  ( ph  ->  R  e.  Comp )   &    |-  ( ph  ->  S  e.  Top )   &    |-  ( ph  ->  U  e.  ( R  tX  S ) )   &    |-  ( ph  ->  ( X  X.  { A } )  C_  U )   &    |-  ( ph  ->  A  e.  Y )   =>    |-  ( ph  ->  E. u  e.  S  ( A  e.  u  /\  ( X  X.  u )  C_  U ) )
 
Theoremtxcmplem1 20120* Lemma for txcmp 20122. (Contributed by Mario Carneiro, 14-Sep-2014.)
 |-  X  =  U. R   &    |-  Y  =  U. S   &    |-  ( ph  ->  R  e.  Comp )   &    |-  ( ph  ->  S  e.  Comp )   &    |-  ( ph  ->  W 
 C_  ( R  tX  S ) )   &    |-  ( ph  ->  ( X  X.  Y )  =  U. W )   &    |-  ( ph  ->  A  e.  Y )   =>    |-  ( ph  ->  E. u  e.  S  ( A  e.  u  /\  E. v  e.  ( ~P W  i^i  Fin )
 ( X  X.  u )  C_  U. v ) )
 
Theoremtxcmplem2 20121* Lemma for txcmp 20122. (Contributed by Mario Carneiro, 14-Sep-2014.)
 |-  X  =  U. R   &    |-  Y  =  U. S   &    |-  ( ph  ->  R  e.  Comp )   &    |-  ( ph  ->  S  e.  Comp )   &    |-  ( ph  ->  W 
 C_  ( R  tX  S ) )   &    |-  ( ph  ->  ( X  X.  Y )  =  U. W )   =>    |-  ( ph  ->  E. v  e.  ( ~P W  i^i  Fin ) ( X  X.  Y )  =  U. v )
 
Theoremtxcmp 20122 The topological product of two compact spaces is compact. (Contributed by Mario Carneiro, 14-Sep-2014.) (Proof shortened 21-Mar-2015.)
 |-  ( ( R  e.  Comp  /\  S  e.  Comp )  ->  ( R  tX  S )  e.  Comp )
 
Theoremtxcmpb 20123 The topological product of two nonempty topologies is compact iff the component topologies are both compact. (Contributed by Mario Carneiro, 14-Sep-2014.)
 |-  X  =  U. R   &    |-  Y  =  U. S   =>    |-  ( ( ( R  e.  Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  ->  (
 ( R  tX  S )  e.  Comp  <->  ( R  e.  Comp  /\  S  e.  Comp )
 ) )
 
Theoremhausdiag 20124 A topology is Hausdorff iff the diagonal set is closed in the topology's product with itself. EDITORIAL: very clumsy proof, can probably be shortened substantially. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  X  =  U. J   =>    |-  ( J  e.  Haus  <->  ( J  e.  Top  /\  (  _I  |`  X )  e.  ( Clsd `  ( J  tX  J ) ) ) )
 
Theoremhauseqlcld 20125 In a Hausdorff topology, the equalizer of two continuous functions is closed (thus, two continuous functions which agree on a dense set agree everywhere). (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  K  e.  Haus )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  G  e.  ( J  Cn  K ) )   =>    |-  ( ph  ->  dom  ( F  i^i  G )  e.  ( Clsd `  J )
 )
 
Theoremtxhaus 20126 The topological product of two Hausdorff spaces is Hausdorff. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  ( ( R  e.  Haus  /\  S  e.  Haus )  ->  ( R  tX  S )  e.  Haus )
 
Theoremtxlm 20127* Two sequences converge iff the sequence of their ordered pairs converges. Proposition 14-2.6 of [Gleason] p. 230. (Contributed by NM, 16-Jul-2007.) (Revised by Mario Carneiro, 5-May-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  K  e.  (TopOn `  Y ) )   &    |-  ( ph  ->  F : Z --> X )   &    |-  ( ph  ->  G : Z --> Y )   &    |-  H  =  ( n  e.  Z  |->  <. ( F `  n ) ,  ( G `  n ) >. )   =>    |-  ( ph  ->  ( ( F ( ~~> t `  J ) R  /\  G ( ~~> t `  K ) S )  <->  H ( ~~> t `  ( J  tX  K ) ) <. R ,  S >. ) )
 
Theoremlmcn2 20128* The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 15-May-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  K  e.  (TopOn `  Y ) )   &    |-  ( ph  ->  F : Z --> X )   &    |-  ( ph  ->  G : Z --> Y )   &    |-  ( ph  ->  F ( ~~> t `  J ) R )   &    |-  ( ph  ->  G ( ~~> t `  K ) S )   &    |-  ( ph  ->  O  e.  ( ( J 
 tX  K )  Cn  N ) )   &    |-  H  =  ( n  e.  Z  |->  ( ( F `  n ) O ( G `  n ) ) )   =>    |-  ( ph  ->  H (
 ~~> t `  N ) ( R O S ) )
 
Theoremtx1stc 20129 The topological product of two first-countable spaces is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( ( R  e.  1stc  /\  S  e.  1stc )  ->  ( R  tX  S )  e.  1stc )
 
Theoremtx2ndc 20130 The topological product of two second-countable spaces is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( ( R  e.  2ndc  /\  S  e.  2ndc )  ->  ( R  tX  S )  e.  2ndc )
 
Theoremtxkgen 20131 The topological product of a locally compact space and a compactly generated Hausdorff space is compactly generated. (The condition on  S can also be replaced with either "compactly generated weak Hausdorff (CGWH)" or "compact Hausdorff-ly generated (CHG)", where WH means that all images of compact Hausdorff spaces are closed and CHG means that a set is open iff it is open in all compact Hausdorff spaces.) (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  ( ran 𝑘Gen  i^i  Haus ) )  ->  ( R  tX  S )  e. 
 ran 𝑘Gen )
 
Theoremxkohaus 20132 If the codomain space is Hausdorff, then the compact-open topology of continuous functions is also Hausdorff. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  ( ( R  e.  Top  /\  S  e.  Haus )  ->  ( S  ^ko  R )  e.  Haus )
 
Theoremxkoptsub 20133 The compact-open topology is finer than the product topology restricted to continuous functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  X  =  U. R   &    |-  J  =  ( Xt_ `  ( X  X.  { S }
 ) )   =>    |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( Jt  ( R  Cn  S ) )  C_  ( S 
 ^ko  R ) )
 
Theoremxkopt 20134 The compact-open topology on a discrete set coincides with the product topology where all the factors are the same. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 12-Sep-2015.)
 |-  ( ( R  e.  Top  /\  A  e.  V ) 
 ->  ( R  ^ko  ~P A )  =  ( Xt_ `  ( A  X.  { R }
 ) ) )
 
Theoremxkopjcn 20135* Continuity of a projection map from the space of continuous functions. (This theorem can be strengthened, to joint continuity in both  f and  A as a function on  ( S  ^ko  R )  tX  R, but not without stronger assumptions on  R; see xkofvcn 20163.) (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  X  =  U. R   =>    |-  (
 ( R  e.  Top  /\  S  e.  Top  /\  A  e.  X )  ->  ( f  e.  ( R  Cn  S )  |->  ( f `  A ) )  e.  ( ( S  ^ko  R )  Cn  S ) )
 
Theoremxkoco1cn 20136* If  F is a continuous function, then  g  |->  g  o.  F is a continuous function on function spaces. (The reason we prove this and xkoco2cn 20137 independently of the more general xkococn 20139 is because that requires some inconvenient extra assumptions on  S.) (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  ( ph  ->  T  e.  Top )   &    |-  ( ph  ->  F  e.  ( R  Cn  S ) )   =>    |-  ( ph  ->  ( g  e.  ( S  Cn  T )  |->  ( g  o.  F ) )  e.  ( ( T  ^ko  S )  Cn  ( T  ^ko  R ) ) )
 
Theoremxkoco2cn 20137* If  F is a continuous function, then  g  |->  F  o.  g is a continuous function on function spaces. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  ( ph  ->  R  e.  Top )   &    |-  ( ph  ->  F  e.  ( S  Cn  T ) )   =>    |-  ( ph  ->  ( g  e.  ( R  Cn  S )  |->  ( F  o.  g ) )  e.  ( ( S  ^ko  R )  Cn  ( T  ^ko  R ) ) )
 
Theoremxkococnlem 20138* Continuity of the composition operation as a function on continuous function spaces. (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  F  =  ( f  e.  ( S  Cn  T ) ,  g  e.  ( R  Cn  S )  |->  ( f  o.  g ) )   &    |-  ( ph  ->  S  e. 𝑛Locally  Comp )   &    |-  ( ph  ->  K  C_  U. R )   &    |-  ( ph  ->  ( Rt  K )  e.  Comp )   &    |-  ( ph  ->  V  e.  T )   &    |-  ( ph  ->  A  e.  ( S  Cn  T ) )   &    |-  ( ph  ->  B  e.  ( R  Cn  S ) )   &    |-  ( ph  ->  ( ( A  o.  B ) " K )  C_  V )   =>    |-  ( ph  ->  E. z  e.  ( ( T  ^ko  S )  tX  ( S  ^ko  R ) ) (
 <. A ,  B >.  e.  z  /\  z  C_  ( `' F " { h  e.  ( R  Cn  T )  |  ( h " K )  C_  V } ) ) )
 
Theoremxkococn 20139* Continuity of the composition operation as a function on continuous function spaces. (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  F  =  ( f  e.  ( S  Cn  T ) ,  g  e.  ( R  Cn  S )  |->  ( f  o.  g ) )   =>    |-  ( ( R  e.  Top  /\  S  e. 𝑛Locally  Comp  /\  T  e.  Top )  ->  F  e.  ( ( ( T  ^ko  S )  tX  ( S  ^ko  R ) )  Cn  ( T  ^ko  R ) ) )
 
12.1.19  Continuous function-builders
 
Theoremcnmptid 20140* The identity function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   =>    |-  ( ph  ->  ( x  e.  X  |->  x )  e.  ( J  Cn  J ) )
 
Theoremcnmptc 20141* A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  P  e.  Y )   =>    |-  ( ph  ->  ( x  e.  X  |->  P )  e.  ( J  Cn  K ) )
 
Theoremcnmpt11 20142* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  K  e.  (TopOn `  Y ) )   &    |-  ( ph  ->  ( y  e.  Y  |->  B )  e.  ( K  Cn  L ) )   &    |-  ( y  =  A  ->  B  =  C )   =>    |-  ( ph  ->  ( x  e.  X  |->  C )  e.  ( J  Cn  L ) )
 
Theoremcnmpt11f 20143* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  F  e.  ( K  Cn  L ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( F `  A ) )  e.  ( J  Cn  L ) )
 
Theoremcnmpt1t 20144* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( J  Cn  L ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  <. A ,  B >. )  e.  ( J  Cn  ( K  tX  L ) ) )
 
Theoremcnmpt12f 20145* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( J  Cn  L ) )   &    |-  ( ph  ->  F  e.  ( ( K 
 tX  L )  Cn  M ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( A F B ) )  e.  ( J  Cn  M ) )
 
Theoremcnmpt12 20146* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 12-Jun-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( J  Cn  L ) )   &    |-  ( ph  ->  K  e.  (TopOn `  Y ) )   &    |-  ( ph  ->  L  e.  (TopOn `  Z ) )   &    |-  ( ph  ->  ( y  e.  Y ,  z  e.  Z  |->  C )  e.  ( ( K 
 tX  L )  Cn  M ) )   &    |-  (
 ( y  =  A  /\  z  =  B )  ->  C  =  D )   =>    |-  ( ph  ->  ( x  e.  X  |->  D )  e.  ( J  Cn  M ) )
 
Theoremcnmpt1st 20147* The projection onto the first coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  x )  e.  ( ( J 
 tX  K )  Cn  J ) )
 
Theoremcnmpt2nd 20148* The projection onto the second coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  y )  e.  ( ( J 
 tX  K )  Cn  K ) )
 
Theoremcnmpt2c 20149* A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  (TopOn `  Z )
 )   &    |-  ( ph  ->  P  e.  Z )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  P )  e.  ( ( J 
 tX  K )  Cn  L ) )
 
Theoremcnmpt21 20150* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( J 
 tX  K )  Cn  L ) )   &    |-  ( ph  ->  L  e.  (TopOn `  Z ) )   &    |-  ( ph  ->  ( z  e.  Z  |->  B )  e.  ( L  Cn  M ) )   &    |-  ( z  =  A  ->  B  =  C )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  C )  e.  ( ( J 
 tX  K )  Cn  M ) )
 
Theoremcnmpt21f 20151* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( J 
 tX  K )  Cn  L ) )   &    |-  ( ph  ->  F  e.  ( L  Cn  M ) )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( F `  A ) )  e.  ( ( J  tX  K )  Cn  M ) )
 
Theoremcnmpt2t 20152* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( J 
 tX  K )  Cn  L ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  B )  e.  ( ( J  tX  K )  Cn  M ) )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  <. A ,  B >. )  e.  (
 ( J  tX  K )  Cn  ( L  tX  M ) ) )
 
Theoremcnmpt22 20153* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( J 
 tX  K )  Cn  L ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  B )  e.  ( ( J  tX  K )  Cn  M ) )   &    |-  ( ph  ->  L  e.  (TopOn `  Z ) )   &    |-  ( ph  ->  M  e.  (TopOn `  W ) )   &    |-  ( ph  ->  ( z  e.  Z ,  w  e.  W  |->  C )  e.  ( ( L 
 tX  M )  Cn  N ) )   &    |-  (
 ( z  =  A  /\  w  =  B )  ->  C  =  D )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  D )  e.  ( ( J 
 tX  K )  Cn  N ) )
 
Theoremcnmpt22f 20154* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( J 
 tX  K )  Cn  L ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  B )  e.  ( ( J  tX  K )  Cn  M ) )   &    |-  ( ph  ->  F  e.  ( ( L 
 tX  M )  Cn  N ) )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( A F B ) )  e.  ( ( J 
 tX  K )  Cn  N ) )
 
Theoremcnmpt1res 20155* The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 5-Jun-2014.)
 |-  K  =  ( Jt  Y )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y 
 C_  X )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  L ) )   =>    |-  ( ph  ->  ( x  e.  Y  |->  A )  e.  ( K  Cn  L ) )
 
Theoremcnmpt2res 20156* The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
 |-  K  =  ( Jt  Y )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y 
 C_  X )   &    |-  N  =  ( Mt  W )   &    |-  ( ph  ->  M  e.  (TopOn `  Z ) )   &    |-  ( ph  ->  W 
 C_  Z )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Z  |->  A )  e.  ( ( J  tX  M )  Cn  L ) )   =>    |-  ( ph  ->  ( x  e.  Y ,  y  e.  W  |->  A )  e.  ( ( K 
 tX  N )  Cn  L ) )
 
Theoremcnmptcom 20157* The argument converse of a continuous function is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( J 
 tX  K )  Cn  L ) )   =>    |-  ( ph  ->  ( y  e.  Y ,  x  e.  X  |->  A )  e.  ( ( K 
 tX  J )  Cn  L ) )
 
Theoremcnmptkc 20158* The curried first projection function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( y  e.  Y  |->  x ) )  e.  ( J  Cn  ( J  ^ko  K )
 ) )
 
Theoremcnmptkp 20159* The evaluation of the inner function in a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  (TopOn `  Z )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  ( y  e.  Y  |->  A ) )  e.  ( J  Cn  ( L  ^ko  K )
 ) )   &    |-  ( ph  ->  B  e.  Y )   &    |-  (
 y  =  B  ->  A  =  C )   =>    |-  ( ph  ->  ( x  e.  X  |->  C )  e.  ( J  Cn  L ) )
 
Theoremcnmptk1 20160* The composition of a curried function with a one-arg function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  (TopOn `  Z )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  ( y  e.  Y  |->  A ) )  e.  ( J  Cn  ( L  ^ko  K )
 ) )   &    |-  ( ph  ->  ( z  e.  Z  |->  B )  e.  ( L  Cn  M ) )   &    |-  ( z  =  A  ->  B  =  C )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( y  e.  Y  |->  C ) )  e.  ( J  Cn  ( M  ^ko  K ) ) )
 
Theoremcnmpt1k 20161* The composition of a one-arg function with a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  (TopOn `  Z )
 )   &    |-  ( ph  ->  M  e.  (TopOn `  W )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( J  Cn  L ) )   &    |-  ( ph  ->  ( y  e.  Y  |->  ( z  e.  Z  |->  B ) )  e.  ( K  Cn  ( M  ^ko  L ) ) )   &    |-  ( z  =  A  ->  B  =  C )   =>    |-  ( ph  ->  ( y  e.  Y  |->  ( x  e.  X  |->  C ) )  e.  ( K  Cn  ( M  ^ko  J ) ) )
 
Theoremcnmptkk 20162* The composition of two curried functions is jointly continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  (TopOn `  Z )
 )   &    |-  ( ph  ->  M  e.  (TopOn `  W )
 )   &    |-  ( ph  ->  L  e. 𝑛Locally  Comp )   &    |-  ( ph  ->  ( x  e.  X  |->  ( y  e.  Y  |->  A ) )  e.  ( J  Cn  ( L  ^ko  K )
 ) )   &    |-  ( ph  ->  ( x  e.  X  |->  ( z  e.  Z  |->  B ) )  e.  ( J  Cn  ( M  ^ko  L )
 ) )   &    |-  ( z  =  A  ->  B  =  C )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( y  e.  Y  |->  C ) )  e.  ( J  Cn  ( M  ^ko  K )
 ) )
 
Theoremxkofvcn 20163* Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 20135.) (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  X  =  U. R   &    |-  F  =  ( f  e.  ( R  Cn  S ) ,  x  e.  X  |->  ( f `  x ) )   =>    |-  ( ( R  e. 𝑛Locally  Comp  /\  S  e.  Top )  ->  F  e.  ( ( ( S  ^ko  R )  tX  R )  Cn  S ) )
 
Theoremcnmptk1p 20164* The evaluation of a curried function by a one-arg function is jointly continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  (TopOn `  Z )
 )   &    |-  ( ph  ->  K  e. 𝑛Locally  Comp )   &    |-  ( ph  ->  ( x  e.  X  |->  ( y  e.  Y  |->  A ) )  e.  ( J  Cn  ( L  ^ko  K )
 ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( J  Cn  K ) )   &    |-  ( y  =  B  ->  A  =  C )   =>    |-  ( ph  ->  ( x  e.  X  |->  C )  e.  ( J  Cn  L ) )
 
Theoremcnmptk2 20165* The uncurrying of a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  (TopOn `  Z )
 )   &    |-  ( ph  ->  K  e. 𝑛Locally  Comp )   &    |-  ( ph  ->  ( x  e.  X  |->  ( y  e.  Y  |->  A ) )  e.  ( J  Cn  ( L  ^ko  K )
 ) )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( J 
 tX  K )  Cn  L ) )
 
Theoremxkoinjcn 20166* Continuity of "injection", i.e. currying, as a function on continuous function spaces. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  F  =  ( x  e.  X  |->  ( y  e.  Y  |->  <. y ,  x >. ) )   =>    |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y ) )  ->  F  e.  ( R  Cn  ( ( S  tX  R )  ^ko  S ) ) )
 
Theoremcnmpt2k 20167* The currying of a two-argument function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( J 
 tX  K )  Cn  L ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( y  e.  Y  |->  A ) )  e.  ( J  Cn  ( L  ^ko  K )
 ) )
 
Theoremtxcon 20168 The topological product of two connected spaces is connected. (Contributed by Mario Carneiro, 29-Mar-2015.)
 |-  ( ( R  e.  Con  /\  S  e.  Con )  ->  ( R  tX  S )  e.  Con )
 
Theoremimasnopn 20169 If a relation graph is open, then an image set of a singleton is also open. Corollary of proposition 4 of [BourbakiTop1] p. I.26. (Contributed by Thierry Arnoux, 14-Jan-2018.)
 |-  X  =  U. J   =>    |-  (
 ( ( J  e.  Top  /\  K  e.  Top )  /\  ( R  e.  ( J  tX  K )  /\  A  e.  X )
 )  ->  ( R " { A } )  e.  K )
 
Theoremimasncld 20170 If a relation graph is closed, then an image set of a singleton is also closed. Corollary of proposition 4 of [BourbakiTop1] p. I.26. (Contributed by Thierry Arnoux, 14-Jan-2018.)
 |-  X  =  U. J   =>    |-  (
 ( ( J  e.  Top  /\  K  e.  Top )  /\  ( R  e.  ( Clsd `  ( J  tX  K ) )  /\  A  e.  X )
 )  ->  ( R " { A } )  e.  ( Clsd `  K )
 )
 
Theoremimasncls 20171 If a relation graph is closed, then an image set of a singleton is also closed. Corollary of proposition 4 of [BourbakiTop1] p. I.26. (Contributed by Thierry Arnoux, 14-Jan-2018.)
 |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( R 
 C_  ( X  X.  Y )  /\  A  e.  X ) )  ->  ( ( cls `  K ) `  ( R " { A } ) ) 
 C_  ( ( ( cls `  ( J  tX  K ) ) `  R ) " { A } ) )
 
12.1.20  Quotient maps and quotient topology
 
Syntaxckq 20172 Extend class notation with the Kolmogorov quotient function.
 class KQ
 
Definitiondf-kq 20173* Define the Kolmogorov quotient. This is a function on topologies which maps a topology to its quotient under the topological distinguishability map, which takes a point to the set of open sets that contain it. Two points are mapped to the same image under this function iff they are topologically indistinguishable. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |- KQ 
 =  ( j  e. 
 Top  |->  ( j qTop  ( x  e.  U. j  |->  { y  e.  j  |  x  e.  y }
 ) ) )
 
Theoremqtopval 20174* Value of the quotient topology function. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  V  /\  F  e.  W ) 
 ->  ( J qTop  F )  =  { s  e. 
 ~P ( F " X )  |  (
 ( `' F "
 s )  i^i  X )  e.  J }
 )
 
Theoremqtopval2 20175* Value of the quotient topology function when  F is a function on the base set. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  ( J qTop  F )  =  { s  e. 
 ~P Y  |  ( `' F " s )  e.  J } )
 
Theoremelqtop 20176 Value of the quotient topology function. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  ( A  e.  ( J qTop  F )  <->  ( A  C_  Y  /\  ( `' F " A )  e.  J ) ) )
 
Theoremqtopres 20177 The quotient topology is unaffected by restriction to the base set. This property makes it slightly more convenient to use, since we don't have to require that  F be a function with domain  X. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  X  =  U. J   =>    |-  ( F  e.  V  ->  ( J qTop  F )  =  ( J qTop  ( F  |`  X ) ) )
 
Theoremqtoptop2 20178 The quotient topology is a topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  ( ( J  e.  Top  /\  F  e.  V  /\  Fun 
 F )  ->  ( J qTop  F )  e.  Top )
 
Theoremqtoptop 20179 The quotient topology is a topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  F  Fn  X ) 
 ->  ( J qTop  F )  e.  Top )
 
Theoremelqtop2 20180 Value of the quotient topology function. (Contributed by Mario Carneiro, 9-Apr-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  V  /\  F : X -onto-> Y )  ->  ( A  e.  ( J qTop  F )  <->  ( A  C_  Y  /\  ( `' F " A )  e.  J ) ) )
 
Theoremqtopuni 20181 The base set of the quotient topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  F : X -onto-> Y )  ->  Y  =  U. ( J qTop  F ) )
 
Theoremelqtop3 20182 Value of the quotient topology function. (Contributed by Mario Carneiro, 9-Apr-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F : X -onto-> Y ) 
 ->  ( A  e.  ( J qTop  F )  <->  ( A  C_  Y  /\  ( `' F " A )  e.  J ) ) )
 
Theoremqtoptopon 20183 The base set of the quotient topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F : X -onto-> Y ) 
 ->  ( J qTop  F )  e.  (TopOn `  Y ) )
 
Theoremqtopid 20184 A quotient map is a continuous function into its quotient topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F  e.  ( J  Cn  ( J qTop  F ) ) )
 
Theoremidqtop 20185 The quotient topology induced by the identity function is the original topology. (Contributed by Mario Carneiro, 30-Aug-2015.)
 |-  ( J  e.  (TopOn `  X )  ->  ( J qTop  (  _I  |`  X ) )  =  J )
 
Theoremqtopcmplem 20186 Lemma for qtopcmp 20187 and qtopcon 20188. (Contributed by Mario Carneiro, 24-Mar-2015.)
 |-  X  =  U. J   &    |-  ( J  e.  A  ->  J  e.  Top )   &    |-  (
 ( J  e.  A  /\  F : X -onto-> U. ( J qTop  F )  /\  F  e.  ( J  Cn  ( J qTop  F ) ) )  ->  ( J qTop  F )  e.  A )   =>    |-  ( ( J  e.  A  /\  F  Fn  X )  ->  ( J qTop  F )  e.  A )
 
Theoremqtopcmp 20187 A quotient of a compact space is compact. (Contributed by Mario Carneiro, 24-Mar-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Comp  /\  F  Fn  X ) 
 ->  ( J qTop  F )  e.  Comp )
 
Theoremqtopcon 20188 A quotient of a connected space is connected. (Contributed by Mario Carneiro, 24-Mar-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Con  /\  F  Fn  X ) 
 ->  ( J qTop  F )  e.  Con )
 
Theoremqtopkgen 20189 A quotient of a compactly generated space is compactly generated. (Contributed by Mario Carneiro, 9-Apr-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  ->  ( J qTop  F )  e. 
 ran 𝑘Gen )
 
Theorembasqtop 20190 An injection maps bases to bases. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  TopBases  /\  F : X -1-1-onto-> Y )  ->  ( J qTop  F )  e.  TopBases )
 
Theoremtgqtop 20191 An injection maps generated topologies to each other. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  TopBases  /\  F : X -1-1-onto-> Y )  ->  (
 ( topGen `  J ) qTop  F )  =  ( topGen `  ( J qTop  F ) ) )
 
Theoremqtopcld 20192 The property of being a closed set in the quotient topology. (Contributed by Mario Carneiro, 24-Mar-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F : X -onto-> Y ) 
 ->  ( A  e.  ( Clsd `  ( J qTop  F ) )  <->  ( A  C_  Y  /\  ( `' F " A )  e.  ( Clsd `  J ) ) ) )
 
Theoremqtopcn 20193 Universal property of a quotient map. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z ) )  /\  ( F : X -onto-> Y  /\  G : Y --> Z ) )  ->  ( G  e.  ( ( J qTop  F )  Cn  K )  <->  ( G  o.  F )  e.  ( J  Cn  K ) ) )
 
Theoremqtopss 20194 A surjective continuous function from  J to  K induces a topology  J qTop  F on the base set of  K. This topology is in general finer than  K. Together with qtopid 20184, this implies that  J qTop  F is the finest topology making  F continuous, i.e. the final topology with respect to the family  { F }. (Contributed by Mario Carneiro, 24-Mar-2015.)
 |-  ( ( F  e.  ( J  Cn  K ) 
 /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  ->  K  C_  ( J qTop  F ) )
 
Theoremqtopeu 20195* Universal property of the quotient topology. If  G is a function from  J to  K which is equal on all equivalent elements under  F, then there is a unique continuous map  f : ( J  /  F ) --> K such that  G  =  f  o.  F, and we say that  G "passes to the quotient". (Contributed by Mario Carneiro, 24-Mar-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  F : X -onto-> Y )   &    |-  ( ph  ->  G  e.  ( J  Cn  K ) )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  ( F `
  x )  =  ( F `  y
 ) ) )  ->  ( G `  x )  =  ( G `  y ) )   =>    |-  ( ph  ->  E! f  e.  ( ( J qTop  F )  Cn  K ) G  =  ( f  o.  F ) )
 
Theoremqtoprest 20196 If  A is a saturated open or closed set (where saturated means that  A  =  ( `' F " U ) for some  U), then the restriction of the quotient map  F to  A is a quotient map. (Contributed by Mario Carneiro, 24-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  F : X -onto-> Y )   &    |-  ( ph  ->  U 
 C_  Y )   &    |-  ( ph  ->  A  =  ( `' F " U ) )   &    |-  ( ph  ->  ( A  e.  J  \/  A  e.  ( Clsd `  J ) ) )   =>    |-  ( ph  ->  ( ( J qTop  F )t  U )  =  ( ( Jt  A ) qTop  ( F  |`  A ) ) )
 
Theoremqtopomap 20197* If  F is a surjective continuous open map, then it is a quotient map. (An open map is a function that maps open sets to open sets.) (Contributed by Mario Carneiro, 24-Mar-2015.)
 |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  ran 
 F  =  Y )   &    |-  ( ( ph  /\  x  e.  J )  ->  ( F " x )  e.  K )   =>    |-  ( ph  ->  K  =  ( J qTop  F ) )
 
Theoremqtopcmap 20198* If  F is a surjective continuous closed map, then it is a quotient map. (A closed map is a function that maps closed sets to closed sets.) (Contributed by Mario Carneiro, 24-Mar-2015.)
 |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  ran 
 F  =  Y )   &    |-  ( ( ph  /\  x  e.  ( Clsd `  J )
 )  ->  ( F " x )  e.  ( Clsd `  K ) )   =>    |-  ( ph  ->  K  =  ( J qTop  F ) )
 
Theoremimastopn 20199 The topology of an image structure. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ph  ->  R  e.  W )   &    |-  J  =  ( TopOpen `  R )   &    |-  O  =  ( TopOpen `  U )   =>    |-  ( ph  ->  O  =  ( J qTop  F ) )
 
Theoremimastps 20200 The image of a topological space under a function is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ph  ->  R  e.  TopSp )   =>    |-  ( ph  ->  U  e.  TopSp )
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