HomeHome Metamath Proof Explorer
Theorem List (p. 171 of 309)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21328)
  Hilbert Space Explorer  Hilbert Space Explorer
(21329-22851)
  Users' Mathboxes  Users' Mathboxes
(22852-30843)
 

Theorem List for Metamath Proof Explorer - 17001-17100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem1stctop 17001 A first-countable topology is a topology. (Contributed by Jeff Hankins, 22-Aug-2009.)
 |-  ( J  e.  1stc  ->  J  e.  Top )
 
Theorem1stcclb 17002* A property of points in a first-countable topology. (Contributed by Jeff Hankins, 22-Aug-2009.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  1stc  /\  A  e.  X ) 
 ->  E. x  e.  ~P  J ( x  ~<_  om 
 /\  A. y  e.  J  ( A  e.  y  ->  E. z  e.  x  ( A  e.  z  /\  z  C_  y ) ) ) )
 
Theorem1stcfb 17003* For any point  A in a first-countable topology, there is a function  f : NN --> J enumerating neighborhoods of  A which is decreasing and forms a local base. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  1stc  /\  A  e.  X ) 
 ->  E. f ( f : NN --> J  /\  A. k  e.  NN  ( A  e.  ( f `  k )  /\  (
 f `  ( k  +  1 ) ) 
 C_  ( f `  k ) )  /\  A. y  e.  J  ( A  e.  y  ->  E. k  e.  NN  ( f `  k
 )  C_  y )
 ) )
 
Theoremis2ndc 17004* The property of being second-countable. (Contributed by Jeff Hankins, 17-Jan-2010.) (Revised by Mario Carneiro, 21-Mar-2015.)
 |-  ( J  e.  2ndc  <->  E. x  e.  TopBases  ( x  ~<_  om 
 /\  ( topGen `  x )  =  J )
 )
 
Theorem2ndctop 17005 A second-countable topology is a topology. (Contributed by Jeff Hankins, 17-Jan-2010.) (Revised by Mario Carneiro, 21-Mar-2015.)
 |-  ( J  e.  2ndc  ->  J  e.  Top )
 
Theorem2ndci 17006 A countable basis generates a second-countable topology. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( ( B  e.  TopBases  /\  B  ~<_  om )  ->  ( topGen `
  B )  e. 
 2ndc )
 
Theorem2ndcsb 17007* Having a countable subbase is a sufficient condition for second-countability. (Contributed by Jeff Hankins, 17-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
 |-  ( J  e.  2ndc  <->  E. x ( x  ~<_  om 
 /\  ( topGen `  ( fi `  x ) )  =  J ) )
 
Theorem2ndcredom 17008 A second-countable space has at most the cardinality of the continuum. (Contributed by Mario Carneiro, 9-Apr-2015.)
 |-  ( J  e.  2ndc  ->  J 
 ~<_  RR )
 
Theorem2ndc1stc 17009 A second-countable space is first-countable. (Contributed by Jeff Hankins, 17-Jan-2010.)
 |-  ( J  e.  2ndc  ->  J  e.  1stc )
 
Theorem1stcrestlem 17010* Lemma for 1stcrest 17011. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( B  ~<_  om  ->  ran  (  x  e.  B  |->  C )  ~<_  om )
 
Theorem1stcrest 17011 A subspace of a first-countable space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( ( J  e.  1stc  /\  A  e.  V ) 
 ->  ( Jt  A )  e.  1stc )
 
Theorem2ndcrest 17012 A subspace of a second-countable space is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( ( J  e.  2ndc  /\  A  e.  V ) 
 ->  ( Jt  A )  e.  2ndc )
 
Theorem2ndcctbss 17013* If a topology is second-countable, every base has a countable subset which is a base. Exercise 16B2 in Willard. (Contributed by Jeff Hankins, 28-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
 |-  X  =  U. B   &    |-  J  =  ( topGen `  B )   &    |-  S  =  { <. u ,  v >.  |  ( u  e.  c  /\  v  e.  c  /\  E. w  e.  B  ( u  C_  w  /\  w  C_  v
 ) ) }   =>    |-  ( ( B  e.  TopBases  /\  J  e.  2ndc ) 
 ->  E. b  e.  TopBases  ( b  ~<_  om  /\  b  C_  B  /\  J  =  (
 topGen `  b ) ) )
 
Theorem2ndcdisj 17014* Any disjoint family of open sets in a second-countable space is countable. (The sets are required to be nonempty because otherwise there could be many empty sets in the family.) (Contributed by Mario Carneiro, 21-Mar-2015.) (Proof shortened by Mario Carneiro, 9-Apr-2015.)
 |-  ( ( J  e.  2ndc  /\  A. x  e.  A  B  e.  ( J  \  { (/) } )  /\  A. y E* x ( x  e.  A  /\  y  e.  B )
 )  ->  A  ~<_  om )
 
Theorem2ndcdisj2 17015* Any disjoint collection of open sets in a second-countable space is countable. (Contributed by Mario Carneiro, 21-Mar-2015.) (Proof shortened by Mario Carneiro, 9-Apr-2015.)
 |-  ( ( J  e.  2ndc  /\  A  C_  J  /\  A. y E* x ( x  e.  A  /\  y  e.  x )
 )  ->  A  ~<_  om )
 
Theorem2ndcomap 17016* A surjective continuous open map maps second-countable spaces to second-countable spaces. (Contributed by Mario Carneiro, 9-Apr-2015.)
 |-  Y  =  U. K   &    |-  ( ph  ->  J  e.  2ndc )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  ran 
 F  =  Y )   &    |-  ( ( ph  /\  x  e.  J )  ->  ( F " x )  e.  K )   =>    |-  ( ph  ->  K  e.  2ndc )
 
Theorem2ndcsep 17017* 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 17018 A discrete space is second-countable iff it is countable. (Contributed by Mario Carneiro, 13-Apr-2015.)
 |-  ( X  ~<_  om  <->  ~P X  e.  2ndc )
 
Theorem1stcelcls 17019* 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 7945. 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 17020* 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 7945, but only via 1stcelcls 17019, 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 17021* 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 17022 Extend class notation with the "locally  A " predicate of a topological space.
 class Locally  A
 
Syntaxcnlly 17023 Extend class notation with the "N-locally  A " predicate of a topological space.
 class 𝑛Locally  A
 
Definitiondf-lly 17024* 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 17025* 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 because 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 17026* 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 17027* 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 17028 Equality theorem for the Locally  A predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( A  =  B  -> Locally  A  = Locally  B )
 
Theoremnllyeq 17029 Equality theorem for the Locally  A predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( A  =  B  -> 𝑛Locally  A  = 𝑛Locally  B )
 
Theoremllytop 17030 A locally  A space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( J  e. Locally  A  ->  J  e.  Top )
 
Theoremnllytop 17031 A locally  A space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( J  e. 𝑛Locally  A  ->  J  e.  Top )
 
Theoremllyi 17032* 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 17033* 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 17034* Eliminate the neighborhood symbol from nllyi 17033. (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 17035 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 17036 A locally  A space is n-locally  A. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |- Locally  A  C_ 𝑛Locally  A
 
Theoremllyss 17037 The "locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( A  C_  B  -> Locally  A  C_ Locally  B )
 
Theoremnllyss 17038 The "n-locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( A  C_  B  -> 𝑛Locally  A 
 C_ 𝑛Locally 
 B )
 
Theoremsubislly 17039* 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 17040* 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 17041* 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 17042* 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 17043 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 17044 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 17045 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 17046 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 17047 Idempotence of the "n-locally" predicate, i.e. being "n-locally  A " is a local property. (Use loclly 17045 to show 𝑛Locally 𝑛Locally  A  = 𝑛Locally  A.) (Contributed by Mario Carneiro, 2-Mar-2015.)
 |- Locally 𝑛Locally  A  = 𝑛Locally  A
 
Theoremtoplly 17048 A topology is locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |- Locally  Top  =  Top
 
Theoremtopnlly 17049 A topology is n-locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |- 𝑛Locally  Top 
 =  Top
 
Theoremhauslly 17050 A Hausdorff space is locally Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( J  e.  Haus  ->  J  e. Locally  Haus )
 
Theoremhausnlly 17051 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 17052 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 17053 A closed subspace of a locally compact space is also locally compact. (The analogous result for open subspaces follows from the more general nllyrest 17044.) (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( ( J  e. 𝑛Locally  Comp  /\  A  e.  ( Clsd `  J ) )  ->  ( Jt  A )  e. 𝑛Locally  Comp )
 
Theoremlly1stc 17054 First-countability is a local property (unlike second-countability). (Contributed by Mario Carneiro, 21-Mar-2015.)
 |- Locally  1stc 
 =  1stc
 
Theoremdislly 17055* 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 17056 A discrete space is locally compact. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  ( X  e.  V  ->  ~P X  e. Locally  Comp )
 
Theoremdis1stc 17057 A discrete space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( X  e.  V  ->  ~P X  e.  1stc )
 
Theoremhausmapdom 17058 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 17515 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 17059 Simplify the cardinal  A ^ NN of hausmapdom 17058 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 17060 Extend class notation with the compact generator operation.
 class 𝑘Gen
 
Definitiondf-kgen 17061* 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 17062* 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 17063* 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 17064 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 17065 The compact generator generates a topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
 |-  ( J  e.  (TopOn `  X )  ->  (𝑘Gen `  J )  e.  (TopOn `  X ) )
 
Theoremkgenuni 17066 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 17067 The compact generator generates a topology. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  ( J  e.  Top  ->  (𝑘Gen
 `  J )  e. 
 Top )
 
Theoremkgenf 17068 The compact generator is a function on topologies. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |- 𝑘Gen : Top --> Top
 
Theoremkgentop 17069 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 17070 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 17071 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 17072 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 17073 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 17074 The compact generator is idempotent on compactly generated spaces. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  ( J  e.  ran 𝑘Gen  ->  (𝑘Gen `  J )  =  J )
 
Theoremiskgen2 17075 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 17076* 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 17077* A locally compact space is compactly generated. (This variant of llycmpkgen 17079 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 17078 A compact space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( J  e.  Comp  ->  J  e.  ran 𝑘Gen )
 
Theoremllycmpkgen 17079 A locally compact space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( J  e. 𝑛Locally  Comp  ->  J  e.  ran 𝑘Gen )
 
Theorem1stckgenlem 17080 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 17081 A first-countable space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( J  e.  1stc  ->  J  e.  ran 𝑘Gen )
 
Theoremkgen2ss 17082 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 17083* 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 17084* 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 17085 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 17086 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 17087 Extend class notation with the binary topological product operation.
 class  tX
 
Syntaxcxko 17088 Extend class notation with a function whose value is the compact-open topology.
 class  ^ k o
 
Definitiondf-tx 17089* 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 17090* 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 17091* 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 17092* 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 17093* 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 17094* 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 17095* 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 17096 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 17097* 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 17098* 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 17099* 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 17100* 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 )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30843
  Copyright terms: Public domain < Previous  Next >