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Theorem List for Metamath Proof Explorer - 17101-17200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiscon 17101 The predicate  J is a connected topology . (Contributed by FL, 17-Nov-2008.)
 |-  X  =  U. J   =>    |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J ) )  =  { (/) ,  X } ) )
 
Theoremiscon2 17102 The predicate  J is a connected topology . (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  X  =  U. J   =>    |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J ) ) 
 C_  { (/) ,  X }
 ) )
 
Theoremconclo 17103 The only nonempty clopen set of a connected topology is the whole space. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  X  =  U. J   &    |-  ( ph  ->  J  e.  Con )   &    |-  ( ph  ->  A  e.  J )   &    |-  ( ph  ->  A  =/=  (/) )   &    |-  ( ph  ->  A  e.  ( Clsd `  J ) )   =>    |-  ( ph  ->  A  =  X )
 
Theoremconndisj 17104 If a topology is connected, its underlying set can't be partitioned into two non empty non-overlapping open sets. (Contributed by FL, 16-Nov-2008.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
 |-  X  =  U. J   &    |-  ( ph  ->  J  e.  Con )   &    |-  ( ph  ->  A  e.  J )   &    |-  ( ph  ->  A  =/=  (/) )   &    |-  ( ph  ->  B  e.  J )   &    |-  ( ph  ->  B  =/=  (/) )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   =>    |-  ( ph  ->  ( A  u.  B )  =/=  X )
 
Theoremcontop 17105 A connected topology is a topology. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 14-Dec-2013.)
 |-  ( J  e.  Con  ->  J  e.  Top )
 
Theoremindiscon 17106 The indiscrete topology (or trivial topology) on any set is connected. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 { (/) ,  A }  e.  Con
 
Theoremdfcon2 17107* An alternate definition of connectedness. (Contributed by Jeff Hankins, 9-Jul-2009.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
 |-  ( J  e.  (TopOn `  X )  ->  ( J  e.  Con  <->  A. x  e.  J  A. y  e.  J  ( ( x  =/=  (/)  /\  y  =/= 
 (/)  /\  ( x  i^i  y )  =  (/) )  ->  ( x  u.  y
 )  =/=  X )
 ) )
 
Theoremconsuba 17108* Connectedness for a subspace. See connsub 17109. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( ( Jt  A )  e.  Con  <->  A. x  e.  J  A. y  e.  J  ( ( ( x  i^i  A )  =/=  (/)  /\  (
 y  i^i  A )  =/= 
 (/)  /\  ( ( x  i^i  y )  i^i 
 A )  =  (/) )  ->  ( ( x  u.  y )  i^i 
 A )  =/=  A ) ) )
 
Theoremconnsub 17109* Two equivalent ways of saying that a subspace topology is connected. (Contributed by Jeff Hankins, 9-Jul-2009.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  ( ( Jt  S )  e.  Con  <->  A. x  e.  J  A. y  e.  J  ( ( ( x  i^i  S )  =/=  (/)  /\  (
 y  i^i  S )  =/= 
 (/)  /\  ( x  i^i  y )  C_  ( X 
 \  S ) ) 
 ->  -.  S  C_  ( x  u.  y ) ) ) )
 
Theoremcnconn 17110 Connectedness is respected by a continuous onto map. (Contributed by Jeff Hankins, 12-Jul-2009.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
 |-  Y  =  U. K   =>    |-  (
 ( J  e.  Con  /\  F : X -onto-> Y  /\  F  e.  ( J  Cn  K ) ) 
 ->  K  e.  Con )
 
Theoremnconsubb 17111 Disconnectedness for a subspace. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  A  C_  X )   &    |-  ( ph  ->  U  e.  J )   &    |-  ( ph  ->  V  e.  J )   &    |-  ( ph  ->  ( U  i^i  A )  =/=  (/) )   &    |-  ( ph  ->  ( V  i^i  A )  =/=  (/) )   &    |-  ( ph  ->  ( ( U  i^i  V )  i^i  A )  =  (/) )   &    |-  ( ph  ->  A 
 C_  ( U  u.  V ) )   =>    |-  ( ph  ->  -.  ( Jt  A )  e.  Con )
 
Theoremconsubclo 17112 If a clopen set meets a connected subspace, it must contain the entire subspace. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  X  =  U. J   &    |-  ( ph  ->  A  C_  X )   &    |-  ( ph  ->  ( Jt  A )  e.  Con )   &    |-  ( ph  ->  B  e.  J )   &    |-  ( ph  ->  ( B  i^i  A )  =/=  (/) )   &    |-  ( ph  ->  B  e.  ( Clsd `  J ) )   =>    |-  ( ph  ->  A  C_  B )
 
Theoremconima 17113 The image of a connected set is connected. (Contributed by Mario Carneiro, 7-Jul-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  X  =  U. J   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  A  C_  X )   &    |-  ( ph  ->  ( Jt  A )  e.  Con )   =>    |-  ( ph  ->  ( Kt  ( F " A ) )  e.  Con )
 
Theoremconcn 17114 A continuous function from a connected topology with one point in a clopen set must lie entirely within the set. (Contributed by Mario Carneiro, 16-Feb-2015.)
 |-  X  =  U. J   &    |-  ( ph  ->  J  e.  Con )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  U  e.  K )   &    |-  ( ph  ->  U  e.  ( Clsd `  K ) )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  ( F `  A )  e.  U )   =>    |-  ( ph  ->  F : X --> U )
 
Theoremiunconlem 17115* Lemma for iuncon 17116. (Contributed by Mario Carneiro, 11-Jun-2014.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  C_  X )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  P  e.  B )   &    |-  ( ( ph  /\  k  e.  A )  ->  ( Jt  B )  e.  Con )   &    |-  ( ph  ->  U  e.  J )   &    |-  ( ph  ->  V  e.  J )   &    |-  ( ph  ->  ( V  i^i  U_ k  e.  A  B )  =/=  (/) )   &    |-  ( ph  ->  ( U  i^i  V ) 
 C_  ( X  \  U_ k  e.  A  B ) )   &    |-  ( ph  ->  U_ k  e.  A  B  C_  ( U  u.  V ) )   &    |-  F/ k ph   =>    |-  ( ph  ->  -.  P  e.  U )
 
Theoremiuncon 17116* The indexed union of connected overlapping subspaces sharing a common point is connected. (Contributed by Mario Carneiro, 11-Jun-2014.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  C_  X )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  P  e.  B )   &    |-  ( ( ph  /\  k  e.  A )  ->  ( Jt  B )  e.  Con )   =>    |-  ( ph  ->  ( Jt  U_ k  e.  A  B )  e.  Con )
 
Theoremuncon 17117 The union of two connected overlapping subspaces is connected. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 11-Jun-2014.)
 |-  ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X )  /\  ( A  i^i  B )  =/=  (/) )  ->  (
 ( ( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con )  ->  ( Jt  ( A  u.  B ) )  e.  Con ) )
 
Theoremclscon 17118 The closure of a connected set is connected. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X  /\  ( Jt  A )  e.  Con )  ->  ( Jt  ( ( cls `  J ) `  A ) )  e. 
 Con )
 
Theoremconcompid 17119* The connected component containing 
A contains  A. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  S  =  U. { x  e.  ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }   =>    |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  A  e.  S )
 
Theoremconcompcon 17120* The connected component containing 
A is connected. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  S  =  U. { x  e.  ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }   =>    |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  ( Jt  S )  e.  Con )
 
Theoremconcompss 17121* The connected component containing 
A is a superset of any other connected set containing  A. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  S  =  U. { x  e.  ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }   =>    |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  T  C_  S )
 
Theoremconcompcld 17122* The connected component containing 
A is a closed set. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  S  =  U. { x  e.  ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }   =>    |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  S  e.  ( Clsd `  J ) )
 
Theoremconcompclo 17123* The connected component containing 
A is a subset of any clopen set containing  A. (Contributed by Mario Carneiro, 20-Sep-2015.)
 |-  S  =  U. { x  e.  ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }   =>    |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J )
 )  /\  A  e.  T )  ->  S  C_  T )
 
Theoremt1conperf 17124 A connected T1 space is perfect, unless it is the topology of a singleton. (Contributed by Mario Carneiro, 26-Dec-2016.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Fre  /\  J  e.  Con  /\  -.  X  ~~  1o )  ->  J  e. Perf )
 
11.1.13  First- and second-countability
 
Syntaxc1stc 17125 Extend class definition to include the class of all first-countable topologies.
 class  1stc
 
Syntaxc2ndc 17126 Extend class definition to include the class of all second-countable topologies.
 class  2ndc
 
Definitiondf-1stc 17127* Define the class of all first-countable topologies. (Contributed by Jeff Hankins, 22-Aug-2009.)
 |- 
 1stc  =  { j  e.  Top  |  A. x  e.  U. j E. y  e.  ~P  j ( y  ~<_ 
 om  /\  A. z  e.  j  ( x  e.  z  ->  x  e.  U. ( y  i^i  ~P z ) ) ) }
 
Definitiondf-2ndc 17128* Define the class of all second-countable topologies. (Contributed by Jeff Hankins, 17-Jan-2010.)
 |- 
 2ndc  =  { j  |  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  j
 ) }
 
Theoremis1stc 17129* The predicate "is a first-countable topology." This can be described as "every point has a countable local basis" - that is, every point has a countable collection of open sets containing it such that every open set containing the point has an open set from this collection as a subset. (Contributed by Jeff Hankins, 22-Aug-2009.)
 |-  X  =  U. J   =>    |-  ( J  e.  1stc  <->  ( J  e.  Top  /\  A. x  e.  X  E. y  e.  ~P  J ( y  ~<_  om 
 /\  A. z  e.  J  ( x  e.  z  ->  x  e.  U. (
 y  i^i  ~P z
 ) ) ) ) )
 
Theoremis1stc2 17130* An equivalent way of saying "is a first-countable topology." (Contributed by Jeff Hankins, 22-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.)
 |-  X  =  U. J   =>    |-  ( J  e.  1stc  <->  ( J  e.  Top  /\  A. x  e.  X  E. y  e.  ~P  J ( y  ~<_  om 
 /\  A. z  e.  J  ( x  e.  z  ->  E. w  e.  y  ( x  e.  w  /\  w  C_  z ) ) ) ) )
 
Theorem1stctop 17131 A first-countable topology is a topology. (Contributed by Jeff Hankins, 22-Aug-2009.)
 |-  ( J  e.  1stc  ->  J  e.  Top )
 
Theorem1stcclb 17132* 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 17133* 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 17134* 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 17135 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 17136 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 17137* 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 17138 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 17139 A second-countable space is first-countable. (Contributed by Jeff Hankins, 17-Jan-2010.)
 |-  ( J  e.  2ndc  ->  J  e.  1stc )
 
Theorem1stcrestlem 17140* Lemma for 1stcrest 17141. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( B  ~<_  om  ->  ran  (  x  e.  B  |->  C )  ~<_  om )
 
Theorem1stcrest 17141 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 17142 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 17143* 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 17144* 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.) (Revised by NM, 17-Jun-2017.)
 |-  ( ( J  e.  2ndc  /\  A. x  e.  A  B  e.  ( J  \  { (/) } )  /\  A. y E* x  e.  A y  e.  B )  ->  A  ~<_  om )
 
Theorem2ndcdisj2 17145* 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.) (Revised by NM, 17-Jun-2017.)
 |-  ( ( J  e.  2ndc  /\  A  C_  J  /\  A. y E* x  e.  A y  e.  x )  ->  A  ~<_  om )
 
Theorem2ndcomap 17146* 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 17147* 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 17148 A discrete space is second-countable iff it is countable. (Contributed by Mario Carneiro, 13-Apr-2015.)
 |-  ( X  ~<_  om  <->  ~P X  e.  2ndc )
 
Theorem1stcelcls 17149* 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 8029. 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 17150* 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 8029, but only via 1stcelcls 17149, 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 17151* 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 17152 Extend class notation with the "locally  A " predicate of a topological space.
 class Locally  A
 
Syntaxcnlly 17153 Extend class notation with the "N-locally  A " predicate of a topological space.
 class 𝑛Locally  A
 
Definitiondf-lly 17154* 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 17155* 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 17156* 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 17157* 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 17158 Equality theorem for the Locally  A predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( A  =  B  -> Locally  A  = Locally  B )
 
Theoremnllyeq 17159 Equality theorem for the Locally  A predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( A  =  B  -> 𝑛Locally  A  = 𝑛Locally  B )
 
Theoremllytop 17160 A locally  A space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( J  e. Locally  A  ->  J  e.  Top )
 
Theoremnllytop 17161 A locally  A space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( J  e. 𝑛Locally  A  ->  J  e.  Top )
 
Theoremllyi 17162* 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 17163* 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 17164* Eliminate the neighborhood symbol from nllyi 17163. (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 17165 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 17166 A locally  A space is n-locally  A. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |- Locally  A  C_ 𝑛Locally  A
 
Theoremllyss 17167 The "locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( A  C_  B  -> Locally  A  C_ Locally  B )
 
Theoremnllyss 17168 The "n-locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( A  C_  B  -> 𝑛Locally  A 
 C_ 𝑛Locally 
 B )
 
Theoremsubislly 17169* 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 17170* 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 17171* 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 17172* 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 17173 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 17174 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 17175 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 17176 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 17177 Idempotence of the "n-locally" predicate, i.e. being "n-locally  A " is a local property. (Use loclly 17175 to show 𝑛Locally 𝑛Locally  A  = 𝑛Locally  A.) (Contributed by Mario Carneiro, 2-Mar-2015.)
 |- Locally 𝑛Locally  A  = 𝑛Locally  A
 
Theoremtoplly 17178 A topology is locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |- Locally  Top  =  Top
 
Theoremtopnlly 17179 A topology is n-locally a topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |- 𝑛Locally  Top 
 =  Top
 
Theoremhauslly 17180 A Hausdorff space is locally Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( J  e.  Haus  ->  J  e. Locally  Haus )
 
Theoremhausnlly 17181 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 17182 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 17183 A closed subspace of a locally compact space is also locally compact. (The analogous result for open subspaces follows from the more general nllyrest 17174.) (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( ( J  e. 𝑛Locally  Comp  /\  A  e.  ( Clsd `  J ) )  ->  ( Jt  A )  e. 𝑛Locally  Comp )
 
Theoremlly1stc 17184 First-countability is a local property (unlike second-countability). (Contributed by Mario Carneiro, 21-Mar-2015.)
 |- Locally  1stc 
 =  1stc
 
Theoremdislly 17185* 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 17186 A discrete space is locally compact. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  ( X  e.  V  ->  ~P X  e. Locally  Comp )
 
Theoremdis1stc 17187 A discrete space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( X  e.  V  ->  ~P X  e.  1stc )
 
Theoremhausmapdom 17188 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 17645 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 17189 Simplify the cardinal  A ^ NN of hausmapdom 17188 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 17190 Extend class notation with the compact generator operation.
 class 𝑘Gen
 
Definitiondf-kgen 17191* 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 17192* 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 17193* 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 17194 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 17195 The compact generator generates a topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
 |-  ( J  e.  (TopOn `  X )  ->  (𝑘Gen `  J )  e.  (TopOn `  X ) )
 
Theoremkgenuni 17196 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 17197 The compact generator generates a topology. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  ( J  e.  Top  ->  (𝑘Gen
 `  J )  e. 
 Top )
 
Theoremkgenf 17198 The compact generator is a function on topologies. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |- 𝑘Gen : Top --> Top
 
Theoremkgentop 17199 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 17200 The compact generator generates a finer topology than the original. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  ( J  e.  Top  ->  J  C_  (𝑘Gen `  J ) )
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