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Theorem List for Metamath Proof Explorer - 17201-17300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem2ndcsep 17201* A second-countable topology is separable, which is to say it contains a countable dense subset. (Contributed by Mario Carneiro, 13-Apr-2015.)
 |-  X  =  U. J   =>    |-  ( J  e.  2ndc  ->  E. x  e.  ~P  X ( x  ~<_ 
 om  /\  ( ( cls `  J ) `  x )  =  X ) )
 
Theoremdis2ndc 17202 A discrete space is second-countable iff it is countable. (Contributed by Mario Carneiro, 13-Apr-2015.)
 |-  ( X  ~<_  om  <->  ~P X  e.  2ndc )
 
Theorem1stcelcls 17203* A point belongs to the closure of a subset iff there is a sequence in the subset converging to it. Theorem 1.4-6(a) of [Kreyszig] p. 30. This proof uses countable choice ax-cc 8077. A space satisfying the conclusion of this theorem is called a sequential space, so the theorem can also be stated as "every first-countable space is a sequential space". (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  1stc  /\  S  C_  X )  ->  ( P  e.  (
 ( cls `  J ) `  S )  <->  E. f ( f : NN --> S  /\  f ( ~~> t `  J ) P ) ) )
 
Theorem1stccnp 17204* A mapping is continuous at  P in a first-countable space  X iff it is sequentially continuous at  P, meaning that the image under  F of every sequence converging at  P converges to  F ( P ). This proof uses ax-cc 8077, but only via 1stcelcls 17203, so it could be refactored into a proof that continuity and sequential continuity are the same in sequential spaces. (Contributed by Mario Carneiro, 7-Sep-2015.)
 |-  ( ph  ->  J  e.  1stc )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  K  e.  (TopOn `  Y ) )   &    |-  ( ph  ->  P  e.  X )   =>    |-  ( ph  ->  ( F  e.  ( ( J  CnP  K ) `
  P )  <->  ( F : X
 --> Y  /\  A. f
 ( ( f : NN --> X  /\  f
 ( ~~> t `  J ) P )  ->  ( F  o.  f ) ( ~~> t `  K ) ( F `  P ) ) ) ) )
 
Theorem1stccn 17205* A mapping  X --> Y, where  X is first-countable, is continuous iff it is sequentially continuous, meaning that for any sequence  f (
n ) converging to  x, its image under  F converges to  F ( x ). (Contributed by Mario Carneiro, 7-Sep-2015.)
 |-  ( ph  ->  J  e.  1stc )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  K  e.  (TopOn `  Y ) )   &    |-  ( ph  ->  F : X --> Y )   =>    |-  ( ph  ->  ( F  e.  ( J  Cn  K ) 
 <-> 
 A. f ( f : NN --> X  ->  A. x ( f ( ~~> t `  J ) x  ->  ( F  o.  f ) ( ~~> t `  K ) ( F `
  x ) ) ) ) )
 
11.1.14  Local topological properties
 
Syntaxclly 17206 Extend class notation with the "locally  A " predicate of a topological space.
 class Locally  A
 
Syntaxcnlly 17207 Extend class notation with the "N-locally  A " predicate of a topological space.
 class 𝑛Locally  A
 
Definitiondf-lly 17208* Define a space that is locally  A, where  A is a topological property like "compact", "connected", or "path-connected". A topological space is locally 
A if every neighborhood of a point contains an open sub-neighborhood that is  A in the subspace topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |- Locally  A  =  { j  e.  Top  |  A. x  e.  j  A. y  e.  x  E. u  e.  ( j  i^i  ~P x ) ( y  e.  u  /\  (
 jt 
 u )  e.  A ) }
 
Definitiondf-nlly 17209* Define a space that is n-locally  A, where  A is a topological property like "compact", "connected", or "path-connected". A topological space is n-locally  A if every neighborhood of a point contains a sub-neighborhood that is  A in the subspace topology.

The terminology "n-locally", where 'n' stands for "neighborhood", is not standard, although this is sometimes called "weakly locally  A". The reason for the distinction is that some notions only make sense for arbitrary neighborhoods (such as "locally compact", which is actually 𝑛Locally  Comp in our teminology - open compact sets are not very useful), while others such as "locally connected" are strictly weaker notions if the neighborhoods are not required to be open. (Contributed by Mario Carneiro, 2-Mar-2015.)

 |- 𝑛Locally  A  =  { j  e. 
 Top  |  A. x  e.  j  A. y  e.  x  E. u  e.  ( ( ( nei `  j ) `  { y } )  i^i  ~P x ) ( jt  u )  e.  A }
 
Theoremislly 17210* The property of being a locally  A topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( J  e. Locally  A  <->  ( J  e.  Top  /\  A. x  e.  J  A. y  e.  x  E. u  e.  ( J  i^i  ~P x ) ( y  e.  u  /\  ( Jt  u )  e.  A ) ) )
 
Theoremisnlly 17211* The property of being an n-locally 
A topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( J  e. 𝑛Locally  A  <->  ( J  e.  Top  /\  A. x  e.  J  A. y  e.  x  E. u  e.  ( (
 ( nei `  J ) `  { y } )  i^i  ~P x ) ( Jt  u )  e.  A ) )
 
Theoremllyeq 17212 Equality theorem for the Locally  A predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( A  =  B  -> Locally  A  = Locally  B )
 
Theoremnllyeq 17213 Equality theorem for the Locally  A predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( A  =  B  -> 𝑛Locally  A  = 𝑛Locally  B )
 
Theoremllytop 17214 A locally  A space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( J  e. Locally  A  ->  J  e.  Top )
 
Theoremnllytop 17215 A locally  A space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( J  e. 𝑛Locally  A  ->  J  e.  Top )
 
Theoremllyi 17216* The property of a locally  A topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( ( J  e. Locally  A 
 /\  U  e.  J  /\  P  e.  U ) 
 ->  E. u  e.  J  ( u  C_  U  /\  P  e.  u  /\  ( Jt  u )  e.  A ) )
 
Theoremnllyi 17217* The property of an n-locally  A topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( ( J  e. 𝑛Locally  A  /\  U  e.  J  /\  P  e.  U )  ->  E. u  e.  (
 ( nei `  J ) `  { P } )
 ( u  C_  U  /\  ( Jt  u )  e.  A ) )
 
Theoremnlly2i 17218* Eliminate the neighborhood symbol from nllyi 17217. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( ( J  e. 𝑛Locally  A  /\  U  e.  J  /\  P  e.  U )  ->  E. s  e.  ~P  U E. u  e.  J  ( P  e.  u  /\  u  C_  s  /\  ( Jt  s )  e.  A ) )
 
Theoremllynlly 17219 A locally  A space is n-locally  A: the "n-locally" predicate is the weaker notion. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( J  e. Locally  A  ->  J  e. 𝑛Locally  A )
 
Theoremllyssnlly 17220 A locally  A space is n-locally  A. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |- Locally  A  C_ 𝑛Locally  A
 
Theoremllyss 17221 The "locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( A  C_  B  -> Locally  A  C_ Locally  B )
 
Theoremnllyss 17222 The "n-locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( A  C_  B  -> 𝑛Locally  A 
 C_ 𝑛Locally 
 B )
 
Theoremsubislly 17223* The property of a subspace being locally  A. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  ( ( J  e.  Top  /\  B  e.  V ) 
 ->  ( ( Jt  B )  e. Locally  A  <->  A. x  e.  J  A. y  e.  ( x  i^i  B ) E. u  e.  J  (
 ( u  i^i  B )  C_  x  /\  y  e.  u  /\  ( Jt  ( u  i^i  B ) )  e.  A ) ) )
 
Theoremrestnlly 17224* If the property  A passes to open subspaces, then a space is n-locally  A iff it is locally  A. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( ( ph  /\  (
 j  e.  A  /\  x  e.  j )
 )  ->  ( jt  x )  e.  A )   =>    |-  ( ph  -> 𝑛Locally  A  = Locally  A )
 
Theoremrestlly 17225* If the property  A passes to open subspaces, then a space which is  A is also locally  A. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( ( ph  /\  (
 j  e.  A  /\  x  e.  j )
 )  ->  ( jt  x )  e.  A )   &    |-  ( ph  ->  A  C_  Top )   =>    |-  ( ph  ->  A  C_ Locally  A )
 
Theoremislly2 17226* An alternative expression for  J  e. Locally  A when  A passes to open subspaces: A space is locally  A if every point is contained in an open neighborhood with property  A. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( ( ph  /\  (
 j  e.  A  /\  x  e.  j )
 )  ->  ( jt  x )  e.  A )   &    |-  X  =  U. J   =>    |-  ( ph  ->  ( J  e. Locally  A  <->  ( J  e.  Top  /\  A. y  e.  X  E. u  e.  J  ( y  e.  u  /\  ( Jt  u )  e.  A ) ) ) )
 
Theoremllyrest 17227 An open subspace of a locally  A space is also locally  A. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( ( J  e. Locally  A 
 /\  B  e.  J )  ->  ( Jt  B )  e. Locally  A )
 
Theoremnllyrest 17228 An open subspace of an n-locally  A space is also n-locally 
A. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( ( J  e. 𝑛Locally  A  /\  B  e.  J ) 
 ->  ( Jt  B )  e. 𝑛Locally  A )
 
Theoremloclly 17229 If  A is a local property, then both Locally  A and 𝑛Locally  A simplify to  A. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  (Locally  A  =  A  <-> 𝑛Locally  A  =  A )
 
Theoremllyidm 17230 Idempotence of the "locally" predicate, i.e. being "locally  A " is a local property. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |- Locally Locally  A  = Locally  A
 
Theoremnllyidm 17231 Idempotence of the "n-locally" predicate, i.e. being "n-locally  A " is a local property. (Use loclly 17229 to show 𝑛Locally 𝑛Locally  A  = 𝑛Locally  A.) (Contributed by Mario Carneiro, 2-Mar-2015.)
 |- Locally 𝑛Locally  A  = 𝑛Locally  A
 
Theoremtoplly 17232 A topology is locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |- Locally  Top  =  Top
 
Theoremtopnlly 17233 A topology is n-locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |- 𝑛Locally  Top 
 =  Top
 
Theoremhauslly 17234 A Hausdorff space is locally Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( J  e.  Haus  ->  J  e. Locally  Haus )
 
Theoremhausnlly 17235 A Hausdorff space is n-locally Hausdorff iff it is locally Hausdorff (both notions are thus referred to as "locally Hausdorff"). (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( J  e. 𝑛Locally  Haus  <->  J  e. Locally  Haus )
 
Theoremhausllycmp 17236 A compact Hausdorff space is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( ( J  e.  Haus  /\  J  e.  Comp )  ->  J  e. 𝑛Locally  Comp )
 
Theoremcldllycmp 17237 A closed subspace of a locally compact space is also locally compact. (The analogous result for open subspaces follows from the more general nllyrest 17228.) (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( ( J  e. 𝑛Locally  Comp  /\  A  e.  ( Clsd `  J ) )  ->  ( Jt  A )  e. 𝑛Locally  Comp )
 
Theoremlly1stc 17238 First-countability is a local property (unlike second-countability). (Contributed by Mario Carneiro, 21-Mar-2015.)
 |- Locally  1stc 
 =  1stc
 
Theoremdislly 17239* The discrete space  ~P X is locally  A if and only if every singleton space has property 
A. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  ( X  e.  V  ->  ( ~P X  e. Locally  A  <->  A. x  e.  X  ~P { x }  e.  A ) )
 
Theoremdisllycmp 17240 A discrete space is locally compact. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  ( X  e.  V  ->  ~P X  e. Locally  Comp )
 
Theoremdis1stc 17241 A discrete space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( X  e.  V  ->  ~P X  e.  1stc )
 
Theoremhausmapdom 17242 If  X is a first-countable Hausdorff space, then the cardinality of the closure of a set  A is bounded by  NN to the power  A. In particular, a first-countable Hausdorff space with a dense subset  A has cardinality at most  A ^ NN, and a separable first-countable Hausdorff space has cardinality at most  ~P NN. (Compare hauspwpwdom 17699 to see a weaker result if the assumption of first-countability is omitted.) (Contributed by Mario Carneiro, 9-Apr-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  ->  (
 ( cls `  J ) `  A )  ~<_  ( A 
 ^m  NN ) )
 
Theoremhauspwdom 17243 Simplify the cardinal  A ^ NN of hausmapdom 17242 to  ~P B  =  2 ^ B when  B is an infinite cardinal greater than  A. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  X  =  U. J   =>    |-  (
 ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A 
 ~<_  ~P B  /\  NN  ~<_  B ) )  ->  ( ( cls `  J ) `  A )  ~<_  ~P B )
 
11.1.15  Compactly generated spaces
 
Syntaxckgen 17244 Extend class notation with the compact generator operation.
 class 𝑘Gen
 
Definitiondf-kgen 17245* Define the "compact generator", the functor from topological spaces to compactly generated spaces. Also known as the k-ification operation. A set is k-open, i.e.  x  e.  (𝑘Gen `  j
), iff the preimage of 
x is open in all compact Hausdorff spaces. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |- 𝑘Gen  =  ( j  e.  Top  |->  { x  e.  ~P U. j  |  A. k  e. 
 ~P  U. j ( ( jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( jt  k ) ) } )
 
Theoremkgenval 17246* Value of the compact generator. (The "k" in 𝑘Gen comes from the name "k-space" for these spaces, after the German word kompakt.) (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  ( J  e.  (TopOn `  X )  ->  (𝑘Gen `  J )  =  { x  e.  ~P X  |  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) ) } )
 
Theoremelkgen 17247* Value of the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  ( J  e.  (TopOn `  X )  ->  ( A  e.  (𝑘Gen `  J ) 
 <->  ( A  C_  X  /\  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  ->  ( A  i^i  k )  e.  ( Jt  k ) ) ) ) )
 
Theoremkgeni 17248 Property of the open sets in the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  ( ( A  e.  (𝑘Gen
 `  J )  /\  ( Jt  K )  e.  Comp ) 
 ->  ( A  i^i  K )  e.  ( Jt  K ) )
 
Theoremkgentopon 17249 The compact generator generates a topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
 |-  ( J  e.  (TopOn `  X )  ->  (𝑘Gen `  J )  e.  (TopOn `  X ) )
 
Theoremkgenuni 17250 The base set of the compact generator is the same as the original topology. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  X  =  U. (𝑘Gen `  J ) )
 
Theoremkgenftop 17251 The compact generator generates a topology. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  ( J  e.  Top  ->  (𝑘Gen
 `  J )  e. 
 Top )
 
Theoremkgenf 17252 The compact generator is a function on topologies. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |- 𝑘Gen : Top --> Top
 
Theoremkgentop 17253 A compactly generated space is a topology. (Note: henceforth we will use the idiom " J  e.  ran 𝑘Gen " to denote " J is compactly generated", since as we will show a space is compactly generated iff it is in the range of the compact generator.) (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  ( J  e.  ran 𝑘Gen  ->  J  e.  Top )
 
Theoremkgenss 17254 The compact generator generates a finer topology than the original. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  ( J  e.  Top  ->  J  C_  (𝑘Gen `  J ) )
 
Theoremkgenhaus 17255 The compact generator generates another Hausdorff topology given a Hausdorff topology to start from. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( J  e.  Haus  ->  (𝑘Gen
 `  J )  e. 
 Haus )
 
Theoremkgencmp 17256 The compact generator topology is the same as the original topology on compact subspaces. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp ) 
 ->  ( Jt  K )  =  ( (𝑘Gen `  J )t  K ) )
 
Theoremkgencmp2 17257 The compact generator topology has the same compact sets as the original topology. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  ( J  e.  Top  ->  ( ( Jt  K )  e.  Comp  <->  ( (𝑘Gen `  J )t  K )  e.  Comp )
 )
 
Theoremkgenidm 17258 The compact generator is idempotent on compactly generated spaces. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  ( J  e.  ran 𝑘Gen  ->  (𝑘Gen `  J )  =  J )
 
Theoremiskgen2 17259 A space is compactly generated iff it contains its image under the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  ( J  e.  ran 𝑘Gen  <->  ( J  e.  Top  /\  (𝑘Gen `  J )  C_  J ) )
 
Theoremiskgen3 17260* Derive the usual definition of "compactly generated". A topology is compactly generated if every subset of  X that is open in every compact subset is open. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  X  =  U. J   =>    |-  ( J  e.  ran 𝑘Gen  <->  ( J  e.  Top  /\  A. x  e.  ~P  X ( A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  ->  ( x  i^i  k )  e.  ( Jt  k ) )  ->  x  e.  J ) ) )
 
Theoremllycmpkgen2 17261* A locally compact space is compactly generated. (This variant of llycmpkgen 17263 uses the weaker definition of locally compact, "every point has a compact neighborhood", instead of "every point has a local base of compact neighborhoods".) (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  X  =  U. J   &    |-  ( ph  ->  J  e.  Top )   &    |-  ( ( ph  /\  x  e.  X )  ->  E. k  e.  ( ( nei `  J ) `  { x }
 ) ( Jt  k )  e.  Comp )   =>    |-  ( ph  ->  J  e.  ran 𝑘Gen )
 
Theoremcmpkgen 17262 A compact space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( J  e.  Comp  ->  J  e.  ran 𝑘Gen )
 
Theoremllycmpkgen 17263 A locally compact space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( J  e. 𝑛Locally  Comp  ->  J  e.  ran 𝑘Gen )
 
Theorem1stckgenlem 17264 The one-point compactification of 
NN is compact. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  F : NN --> X )   &    |-  ( ph  ->  F ( ~~> t `  J ) A )   =>    |-  ( ph  ->  ( Jt  ( ran  F  u.  { A } ) )  e. 
 Comp )
 
Theorem1stckgen 17265 A first-countable space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( J  e.  1stc  ->  J  e.  ran 𝑘Gen )
 
Theoremkgen2ss 17266 The compact generator preserves the subset (fineness) relationship on topologies. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  (𝑘Gen `  J )  C_  (𝑘Gen
 `  K ) )
 
Theoremkgencn 17267* A function from a compactly generated space is continuous iff it is continuous "on compacta". (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  ->  ( F  e.  ( (𝑘Gen `  J )  Cn  K )  <->  ( F : X
 --> Y  /\  A. k  e.  ~P  X ( ( Jt  k )  e.  Comp  ->  ( F  |`  k )  e.  ( ( Jt  k )  Cn  K ) ) ) ) )
 
Theoremkgencn2 17268* A function  F : J --> K from a compactly generated space is continuous iff for all compact spaces  z and continuous  g : z --> J, the composite  F  o.  g : z --> K is continuous. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  ->  ( F  e.  ( (𝑘Gen `  J )  Cn  K )  <->  ( F : X
 --> Y  /\  A. z  e.  Comp  A. g  e.  (
 z  Cn  J )
 ( F  o.  g
 )  e.  ( z  Cn  K ) ) ) )
 
Theoremkgencn3 17269 The set of continuous functions from  J to  K is unaffected by k-ification of  K, if  J is already compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( ( J  e.  ran 𝑘Gen  /\  K  e.  Top )  ->  ( J  Cn  K )  =  ( J  Cn  (𝑘Gen `  K ) ) )
 
Theoremkgen2cn 17270 A continuous function is also continuous with the domain and codomain replaced by their compact generator topologies. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( F  e.  ( J  Cn  K )  ->  F  e.  ( (𝑘Gen `  J )  Cn  (𝑘Gen `  K ) ) )
 
11.1.16  Product topologies
 
Syntaxctx 17271 Extend class notation with the binary topological product operation.
 class  tX
 
Syntaxcxko 17272 Extend class notation with a function whose value is the compact-open topology.
 class  ^ k o
 
Definitiondf-tx 17273* Define the binary topological product, which is homeomorphic to the general topological product over a two element set, but is more convenient to use. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  tX  =  ( r  e.  _V ,  s  e. 
 _V  |->  ( topGen `  ran  ( x  e.  r ,  y  e.  s  |->  ( x  X.  y
 ) ) ) )
 
Definitiondf-xko 17274* Define the compact-open topology, which is the natural topology on the set of continuous functions between two topological spaces. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |- 
 ^ k o  =  ( s  e.  Top ,  r  e.  Top  |->  ( topGen `  ( fi `  ran  (
 k  e.  { x  e.  ~P U. r  |  ( rt  x )  e.  Comp } ,  v  e.  s  |->  { f  e.  (
 r  Cn  s )  |  ( f " k
 )  C_  v }
 ) ) ) )
 
Theoremtxval 17275* Value of the binary topological product operation. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 30-Aug-2015.)
 |-  B  =  ran  ( x  e.  R ,  y  e.  S  |->  ( x  X.  y ) )   =>    |-  ( ( R  e.  V  /\  S  e.  W )  ->  ( R  tX  S )  =  ( topGen `
  B ) )
 
Theoremtxuni2 17276* The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  B  =  ran  ( x  e.  R ,  y  e.  S  |->  ( x  X.  y ) )   &    |-  X  =  U. R   &    |-  Y  =  U. S   =>    |-  ( X  X.  Y )  =  U. B
 
Theoremtxbasex 17277* The basis for the product topology is a set. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  B  =  ran  ( x  e.  R ,  y  e.  S  |->  ( x  X.  y ) )   =>    |-  ( ( R  e.  V  /\  S  e.  W )  ->  B  e.  _V )
 
Theoremtxbas 17278* The set of Cartesian products of elements from two topological bases is a basis. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  B  =  ran  ( x  e.  R ,  y  e.  S  |->  ( x  X.  y ) )   =>    |-  ( ( R  e.  TopBases  /\  S  e.  TopBases )  ->  B  e.  TopBases )
 
Theoremeltx 17279* A set in a product is open iff each point is surrounded by an open rectangle. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  ( ( J  e.  V  /\  K  e.  W )  ->  ( S  e.  ( J  tX  K )  <->  A. p  e.  S  E. x  e.  J  E. y  e.  K  ( p  e.  ( x  X.  y )  /\  ( x  X.  y
 )  C_  S )
 ) )
 
Theoremtxtop 17280 The product of two topologies is a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( R  tX  S )  e.  Top )
 
Theoremptval 17281* The value of the product topology function. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  B  =  { x  |  E. g ( ( g  Fn  A  /\  A. y  e.  A  ( g `  y )  e.  ( F `  y )  /\  E. z  e.  Fin  A. y  e.  ( A  \  z ) ( g `  y )  =  U. ( F `
  y ) ) 
 /\  x  =  X_ y  e.  A  (
 g `  y )
 ) }   =>    |-  ( ( A  e.  V  /\  F  Fn  A )  ->  ( Xt_ `  F )  =  ( topGen `  B ) )
 
Theoremptpjpre1 17282* The preimage of a projection function can be expressed as an indexed cartesian product. (Contributed by Mario Carneiro, 6-Feb-2015.)
 |-  X  =  X_ k  e.  A  U. ( F `
  k )   =>    |-  ( ( ( A  e.  V  /\  F : A --> Top )  /\  ( I  e.  A  /\  U  e.  ( F `
  I ) ) )  ->  ( `' ( w  e.  X  |->  ( w `  I ) ) " U )  =  X_ k  e.  A  if ( k  =  I ,  U ,  U. ( F `  k ) ) )
 
Theoremelpt 17283* Elementhood in the bases of a product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  B  =  { x  |  E. g ( ( g  Fn  A  /\  A. y  e.  A  ( g `  y )  e.  ( F `  y )  /\  E. z  e.  Fin  A. y  e.  ( A  \  z ) ( g `  y )  =  U. ( F `
  y ) ) 
 /\  x  =  X_ y  e.  A  (
 g `  y )
 ) }   =>    |-  ( S  e.  B  <->  E. h ( ( h  Fn  A  /\  A. y  e.  A  ( h `  y )  e.  ( F `  y
 )  /\  E. w  e.  Fin  A. y  e.  ( A  \  w ) ( h `  y )  =  U. ( F `
  y ) ) 
 /\  S  =  X_ y  e.  A  ( h `  y ) ) )
 
Theoremelptr 17284* A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  B  =  { x  |  E. g ( ( g  Fn  A  /\  A. y  e.  A  ( g `  y )  e.  ( F `  y )  /\  E. z  e.  Fin  A. y  e.  ( A  \  z ) ( g `  y )  =  U. ( F `
  y ) ) 
 /\  x  =  X_ y  e.  A  (
 g `  y )
 ) }   =>    |-  ( ( A  e.  V  /\  ( G  Fn  A  /\  A. y  e.  A  ( G `  y )  e.  ( F `  y ) ) 
 /\  ( W  e.  Fin  /\  A. y  e.  ( A  \  W ) ( G `  y )  =  U. ( F `
  y ) ) )  ->  X_ y  e.  A  ( G `  y )  e.  B )
 
Theoremelptr2 17285* A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  B  =  { x  |  E. g ( ( g  Fn  A  /\  A. y  e.  A  ( g `  y )  e.  ( F `  y )  /\  E. z  e.  Fin  A. y  e.  ( A  \  z ) ( g `  y )  =  U. ( F `
  y ) ) 
 /\  x  =  X_ y  e.  A  (
 g `  y )
 ) }   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  W  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A )  ->  S  e.  ( F `  k
 ) )   &    |-  ( ( ph  /\  k  e.  ( A 
 \  W ) ) 
 ->  S  =  U. ( F `  k ) )   =>    |-  ( ph  ->  X_ k  e.  A  S  e.  B )
 
Theoremptbasid 17286* The base set of the product topology is a basic open set. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  B  =  { x  |  E. g ( ( g  Fn  A  /\  A. y  e.  A  ( g `  y )  e.  ( F `  y )  /\  E. z  e.  Fin  A. y  e.  ( A  \  z ) ( g `  y )  =  U. ( F `
  y ) ) 
 /\  x  =  X_ y  e.  A  (
 g `  y )
 ) }   =>    |-  ( ( A  e.  V  /\  F : A --> Top )  ->  X_ k  e.  A  U. ( F `
  k )  e.  B )
 
Theoremptuni2 17287* The base set for the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  B  =  { x  |  E. g ( ( g  Fn  A  /\  A. y  e.  A  ( g `  y )  e.  ( F `  y )  /\  E. z  e.  Fin  A. y  e.  ( A  \  z ) ( g `  y )  =  U. ( F `
  y ) ) 
 /\  x  =  X_ y  e.  A  (
 g `  y )
 ) }   =>    |-  ( ( A  e.  V  /\  F : A --> Top )  ->  X_ k  e.  A  U. ( F `
  k )  = 
 U. B )
 
Theoremptbasin 17288* The basis for a product topology is closed under intersections. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  B  =  { x  |  E. g ( ( g  Fn  A  /\  A. y  e.  A  ( g `  y )  e.  ( F `  y )  /\  E. z  e.  Fin  A. y  e.  ( A  \  z ) ( g `  y )  =  U. ( F `
  y ) ) 
 /\  x  =  X_ y  e.  A  (
 g `  y )
 ) }   =>    |-  ( ( ( A  e.  V  /\  F : A --> Top )  /\  ( X  e.  B  /\  Y  e.  B )
 )  ->  ( X  i^i  Y )  e.  B )
 
Theoremptbasin2 17289* The basis for a product topology is closed under intersections. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  B  =  { x  |  E. g ( ( g  Fn  A  /\  A. y  e.  A  ( g `  y )  e.  ( F `  y )  /\  E. z  e.  Fin  A. y  e.  ( A  \  z ) ( g `  y )  =  U. ( F `
  y ) ) 
 /\  x  =  X_ y  e.  A  (
 g `  y )
 ) }   =>    |-  ( ( A  e.  V  /\  F : A --> Top )  ->  ( fi
 `  B )  =  B )
 
Theoremptbas 17290* The basis for a product topology is a basis. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  B  =  { x  |  E. g ( ( g  Fn  A  /\  A. y  e.  A  ( g `  y )  e.  ( F `  y )  /\  E. z  e.  Fin  A. y  e.  ( A  \  z ) ( g `  y )  =  U. ( F `
  y ) ) 
 /\  x  =  X_ y  e.  A  (
 g `  y )
 ) }   =>    |-  ( ( A  e.  V  /\  F : A --> Top )  ->  B  e.  TopBases )
 
Theoremptpjpre2 17291* The basis for a product topology is a basis. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  B  =  { x  |  E. g ( ( g  Fn  A  /\  A. y  e.  A  ( g `  y )  e.  ( F `  y )  /\  E. z  e.  Fin  A. y  e.  ( A  \  z ) ( g `  y )  =  U. ( F `
  y ) ) 
 /\  x  =  X_ y  e.  A  (
 g `  y )
 ) }   &    |-  X  =  X_ n  e.  A  U. ( F `  n )   =>    |-  ( ( ( A  e.  V  /\  F : A --> Top )  /\  ( I  e.  A  /\  U  e.  ( F `
  I ) ) )  ->  ( `' ( w  e.  X  |->  ( w `  I ) ) " U )  e.  B )
 
Theoremptbasfi 17292* The basis for the product topology can also be written as the set of finite intersections of "cylinder sets", the preimages of projections into one factor from open sets in the factor. (We have to add  X itself to the list because if  A is empty we get  ( fi `  (/) )  =  (/) while  B  =  { (/) }.) (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  B  =  { x  |  E. g ( ( g  Fn  A  /\  A. y  e.  A  ( g `  y )  e.  ( F `  y )  /\  E. z  e.  Fin  A. y  e.  ( A  \  z ) ( g `  y )  =  U. ( F `
  y ) ) 
 /\  x  =  X_ y  e.  A  (
 g `  y )
 ) }   &    |-  X  =  X_ n  e.  A  U. ( F `  n )   =>    |-  ( ( A  e.  V  /\  F : A --> Top )  ->  B  =  ( fi `  ( { X }  u.  ran  ( k  e.  A ,  u  e.  ( F `  k )  |->  ( `' ( w  e.  X  |->  ( w `  k ) ) " u ) ) ) ) )
 
Theorempttop 17293 The product topology is a topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  ( ( A  e.  V  /\  F : A --> Top )  ->  ( Xt_ `  F )  e.  Top )
 
Theoremptopn 17294* A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> Top )   &    |-  ( ph  ->  W  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A )  ->  S  e.  ( F `  k
 ) )   &    |-  ( ( ph  /\  k  e.  ( A 
 \  W ) ) 
 ->  S  =  U. ( F `  k ) )   =>    |-  ( ph  ->  X_ k  e.  A  S  e.  ( Xt_ `  F ) )
 
Theoremptopn2 17295* A sub-basic open set in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> Top )   &    |-  ( ph  ->  O  e.  ( F `  Y ) )   =>    |-  ( ph  ->  X_ k  e.  A  if ( k  =  Y ,  O ,  U. ( F `  k ) )  e.  ( Xt_ `  F ) )
 
Theoremxkotf 17296* Functionality of function  T. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  X  =  U. R   &    |-  K  =  { x  e.  ~P X  |  ( Rt  x )  e.  Comp }   &    |-  T  =  ( k  e.  K ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f " k
 )  C_  v }
 )   =>    |-  T : ( K  X.  S ) --> ~P ( R  Cn  S )
 
Theoremxkobval 17297* Alternative expression for the subbase of the compact-open topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |-  X  =  U. R   &    |-  K  =  { x  e.  ~P X  |  ( Rt  x )  e.  Comp }   &    |-  T  =  ( k  e.  K ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f " k
 )  C_  v }
 )   =>    |- 
 ran  T  =  {
 s  |  E. k  e.  ~P  X E. v  e.  S  ( ( Rt  k )  e.  Comp  /\  s  =  { f  e.  ( R  Cn  S )  |  ( f " k
 )  C_  v }
 ) }
 
Theoremxkoval 17298* Value of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  X  =  U. R   &    |-  K  =  { x  e.  ~P X  |  ( Rt  x )  e.  Comp }   &    |-  T  =  ( k  e.  K ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f " k
 )  C_  v }
 )   =>    |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( S  ^ k o  R )  =  (
 topGen `  ( fi `  ran  T ) ) )
 
Theoremxkotop 17299 The compact-open topology is a topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( S  ^ k o  R )  e.  Top )
 
Theoremxkoopn 17300* A basic open set of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  X  =  U. R   &    |-  ( ph  ->  R  e.  Top )   &    |-  ( ph  ->  S  e.  Top )   &    |-  ( ph  ->  A 
 C_  X )   &    |-  ( ph  ->  ( Rt  A )  e.  Comp )   &    |-  ( ph  ->  U  e.  S )   =>    |-  ( ph  ->  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  U }  e.  ( S  ^ k o  R ) )
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