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Theorem List for Metamath Proof Explorer - 17101-17200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdnsconst 17101 If a continuous mapping to a T1 space is constant on a dense subset, it is constant on the entire space. Note that  ( ( cls `  J ) `  A )  =  X means " A is dense in  X " and  A  C_  ( `' F " { P } ) means " F is constant on  A " (see funconstss 5605). (Contributed by NM, 15-Mar-2007.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
 |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  ( ( ( K  e.  Fre  /\  F  e.  ( J  Cn  K ) )  /\  ( P  e.  Y  /\  A  C_  ( `' F " { P } )  /\  ( ( cls `  J ) `  A )  =  X ) )  ->  F : X --> { P } )
 
Theoremordtt1 17102 The order topology is T1 for any poset. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( R  e.  PosetRel  ->  (ordTop `  R )  e.  Fre )
 
Theoremlmmo 17103 A sequence in a Hausdorff space converges to at most one limit. Part of Lemma 1.4-2(a) of [Kreyszig] p. 26. (Contributed by NM, 31-Jan-2008.) (Proof shortened by Mario Carneiro, 1-May-2014.)
 |-  ( ph  ->  J  e.  Haus )   &    |-  ( ph  ->  F ( ~~> t `  J ) A )   &    |-  ( ph  ->  F ( ~~> t `  J ) B )   =>    |-  ( ph  ->  A  =  B )
 
Theoremlmfun 17104 The convergence relation is function-like in a Hausdorff space. (Contributed by Mario Carneiro, 26-Dec-2013.)
 |-  ( J  e.  Haus  ->  Fun  ( ~~> t `  J ) )
 
Theoremdishaus 17105 A discrete topology is Hausdorff. Morris, Topology without tears, p.72, ex. 13. (Contributed by FL, 24-Jun-2007.) (Proof shortened by Mario Carneiro, 8-Apr-2015.)
 |-  ( A  e.  V  ->  ~P A  e.  Haus )
 
Theoremordthauslem 17106* Lemma for ordthaus 17107. (Contributed by Mario Carneiro, 13-Sep-2015.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  TosetRel  /\  A  e.  X  /\  B  e.  X )  ->  ( A R B  ->  ( A  =/=  B  ->  E. m  e.  (ordTop `  R ) E. n  e.  (ordTop `  R )
 ( A  e.  m  /\  B  e.  n  /\  ( m  i^i  n )  =  (/) ) ) ) )
 
Theoremordthaus 17107 The order topology of a total order is Hausdorff. (Contributed by Mario Carneiro, 13-Sep-2015.)
 |-  ( R  e.  TosetRel  ->  (ordTop `  R )  e.  Haus )
 
11.1.11  Compactness
 
Syntaxccmp 17108 Extend class notation with the class of all compact spaces.
 class  Comp
 
Definitiondf-cmp 17109* Definition of a compact topology. A topology is compact iff any open covering of its underlying set contains a finite sub-covering (Heine-Borel property). Definition C''' of [BourbakiTop1] p. I.59. Note: Bourbaki uses the term quasi-compact topology but it is not the modern usage (which we follow). (Contributed by FL, 22-Dec-2008.)
 |- 
 Comp  =  { x  e.  Top  |  A. y  e.  ~P  x ( U. x  =  U. y  ->  E. z  e.  ( ~P y  i^i  Fin ) U. x  =  U. z ) }
 
Theoremiscmp 17110* The predicate "is a compact topology". (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  X  =  U. J   =>    |-  ( J  e.  Comp  <->  ( J  e.  Top  /\  A. y  e.  ~P  J ( X  =  U. y  ->  E. z  e.  ( ~P y  i^i 
 Fin ) X  =  U. z ) ) )
 
Theoremcmpcov 17111* An open cover of a compact topology has a finite subcover. (Contributed by Jeff Hankins, 29-Jun-2009.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Comp  /\  S  C_  J  /\  X  =  U. S ) 
 ->  E. s  e.  ( ~P S  i^i  Fin ) X  =  U. s )
 
Theoremcmpcov2 17112* Rewrite cmpcov 17111 for the cover  { y  e.  J  |  ph }. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Comp  /\ 
 A. x  e.  X  E. y  e.  J  ( x  e.  y  /\  ph ) )  ->  E. s  e.  ( ~P J  i^i  Fin )
 ( X  =  U. s  /\  A. y  e.  s  ph ) )
 
Theoremcmpcovf 17113* Combine cmpcov 17111 with ac6sfi 7097 to show the existence of a function that indexes the elements that are generating the open cover. (Contributed by Mario Carneiro, 14-Sep-2014.)
 |-  X  =  U. J   &    |-  (
 z  =  ( f `
  y )  ->  ( ph  <->  ps ) )   =>    |-  ( ( J  e.  Comp  /\  A. x  e.  X  E. y  e.  J  ( x  e.  y  /\  E. z  e.  A  ph ) ) 
 ->  E. s  e.  ( ~P J  i^i  Fin )
 ( X  =  U. s  /\  E. f ( f : s --> A  /\  A. y  e.  s  ps ) ) )
 
Theoremcncmp 17114 Compactness is respected by a continuous onto map. (Contributed by Jeff Hankins, 12-Jul-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
 |-  Y  =  U. K   =>    |-  (
 ( J  e.  Comp  /\  F : X -onto-> Y  /\  F  e.  ( J  Cn  K ) ) 
 ->  K  e.  Comp )
 
Theoremfincmp 17115 A finite topology is compact. (Contributed by FL, 22-Dec-2008.)
 |-  ( J  e.  ( Top  i^i  Fin )  ->  J  e.  Comp )
 
Theorem0cmp 17116 The singleton of the empty set is compact. (Contributed by FL, 2-Aug-2009.)
 |- 
 { (/) }  e.  Comp
 
Theoremcmptop 17117 A compact topology is a topology. (Contributed by Jeff Hankins, 29-Jun-2009.)
 |-  ( J  e.  Comp  ->  J  e.  Top )
 
Theoremrncmp 17118 The image of a compact set under a continuous function is compact. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( ( J  e.  Comp  /\  F  e.  ( J  Cn  K ) ) 
 ->  ( Kt  ran  F )  e. 
 Comp )
 
Theoremimacmp 17119 The image of a compact set under a continuous function is compact. (Contributed by Mario Carneiro, 18-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ( F  e.  ( J  Cn  K ) 
 /\  ( Jt  A )  e.  Comp )  ->  ( Kt  ( F " A ) )  e.  Comp )
 
Theoremdiscmp 17120 A discrete topology is compact iff the base set is finite. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  ( A  e.  Fin  <->  ~P A  e.  Comp )
 
Theoremcmpsublem 17121* Lemma for cmpsub 17122. (Contributed by Jeff Hankins, 28-Jun-2009.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( A. c  e. 
 ~P  J ( S 
 C_  U. c  ->  E. d  e.  ( ~P c  i^i 
 Fin ) S  C_  U. d )  ->  A. s  e.  ~P  ( Jt  S ) ( U. ( Jt  S )  =  U. s  ->  E. t  e.  ( ~P s  i^i  Fin ) U. ( Jt  S )  =  U. t ) ) )
 
Theoremcmpsub 17122* Two equivalent ways of describing a compact subset of a topological space. Inspired by Sue E. Goodman's Beginning Topology. (Contributed by Jeff Hankins, 22-Jun-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( ( Jt  S )  e.  Comp  <->  A. c  e.  ~P  J ( S  C_  U. c  ->  E. d  e.  ( ~P c  i^i 
 Fin ) S  C_  U. d ) ) )
 
Theoremtgcmp 17123* A topology generated by a basis is compact iff open covers drawn from the basis have finite subcovers. (See also alexsub 17734, which further specializes to subbases, assuming the ultrafilter lemma.) (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( B  e.  TopBases  /\  X  =  U. B )  ->  ( ( topGen `  B )  e.  Comp  <->  A. y  e.  ~P  B ( X  =  U. y  ->  E. z  e.  ( ~P y  i^i 
 Fin ) X  =  U. z ) ) )
 
Theoremcmpcld 17124 A closed subset of a compact space is compact. (Contributed by Jeff Hankins, 29-Jun-2009.)
 |-  ( ( J  e.  Comp  /\  S  e.  ( Clsd `  J ) )  ->  ( Jt  S )  e.  Comp )
 
Theoremuncmp 17125 The union of two compact sets is compact. (Contributed by Jeff Hankins, 30-Jan-2010.)
 |-  X  =  U. J   =>    |-  (
 ( ( J  e.  Top  /\  X  =  ( S  u.  T ) ) 
 /\  ( ( Jt  S )  e.  Comp  /\  ( Jt  T )  e.  Comp )
 )  ->  J  e.  Comp
 )
 
Theoremfiuncmp 17126* A finite union of compact sets is compact. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  ( Jt  B )  e.  Comp ) 
 ->  ( Jt  U_ x  e.  A  B )  e.  Comp )
 
Theoremsscmp 17127 A subset of a compact topology (i.e. a coarser topology) is compact. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  X  =  U. K   =>    |-  (
 ( J  e.  (TopOn `  X )  /\  K  e.  Comp  /\  J  C_  K )  ->  J  e.  Comp )
 
Theoremhauscmplem 17128* Lemma for hauscmp 17129. (Contributed by Mario Carneiro, 27-Nov-2013.)
 |-  X  =  U. J   &    |-  O  =  { y  e.  J  |  E. w  e.  J  ( A  e.  w  /\  ( ( cls `  J ) `  w )  C_  ( X  \  y ) ) }   &    |-  ( ph  ->  J  e.  Haus )   &    |-  ( ph  ->  S 
 C_  X )   &    |-  ( ph  ->  ( Jt  S )  e.  Comp )   &    |-  ( ph  ->  A  e.  ( X  \  S ) )   =>    |-  ( ph  ->  E. z  e.  J  ( A  e.  z  /\  ( ( cls `  J ) `  z )  C_  ( X  \  S ) ) )
 
Theoremhauscmp 17129 A compact subspace of a T2 space is closed. (Contributed by Jeff Hankins, 16-Jan-2010.) (Proof shortened by Mario Carneiro, 14-Dec-2013.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Haus  /\  S  C_  X  /\  ( Jt  S )  e.  Comp ) 
 ->  S  e.  ( Clsd `  J ) )
 
Theoremcmpfi 17130* If a topology is compact and a collection of closed sets has the finite intersection property, its intersection is nonempty. (Contributed by Jeff Hankins, 25-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
 |-  ( J  e.  Top  ->  ( J  e.  Comp  <->  A. x  e.  ~P  ( Clsd `  J )
 ( -.  (/)  e.  ( fi `  x )  ->  |^| x  =/=  (/) ) ) )
 
Theoremcmpfii 17131 In a compact topology, a system of closed sets with nonempty finite intersections has a nonempty intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( ( J  e.  Comp  /\  X  C_  ( Clsd `  J )  /\  -.  (/) 
 e.  ( fi `  X ) )  ->  |^| X  =/=  (/) )
 
11.1.12  Connectedness
 
Syntaxccon 17132 Extend class notation with the class of all connected topologies.
 class  Con
 
Definitiondf-con 17133 Topologies are connected when only  (/) and  U. j are both open and closed. (Contributed by FL, 17-Nov-2008.)
 |- 
 Con  =  { j  e.  Top  |  ( j  i^i  ( Clsd `  j
 ) )  =  { (/)
 ,  U. j } }
 
Theoremiscon 17134 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 17135 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 17136 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 17137 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 17138 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 17139 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 17140* 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 17141* Connectedness for a subspace. See connsub 17142. (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 17142* 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 17143 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 17144 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 17145 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 17146 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 17147 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 17148* Lemma for iuncon 17149. (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 17149* 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 17150 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 17151 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 17152* 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 17153* 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 17154* 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 17155* 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 17156* 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 17157 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 17158 Extend class definition to include the class of all first-countable topologies.
 class  1stc
 
Syntaxc2ndc 17159 Extend class definition to include the class of all second-countable topologies.
 class  2ndc
 
Definitiondf-1stc 17160* 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 17161* 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 17162* 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 17163* 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 17164 A first-countable topology is a topology. (Contributed by Jeff Hankins, 22-Aug-2009.)
 |-  ( J  e.  1stc  ->  J  e.  Top )
 
Theorem1stcclb 17165* 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 17166* 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 17167* 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 17168 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 17169 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 17170* 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 17171 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 17172 A second-countable space is first-countable. (Contributed by Jeff Hankins, 17-Jan-2010.)
 |-  ( J  e.  2ndc  ->  J  e.  1stc )
 
Theorem1stcrestlem 17173* Lemma for 1stcrest 17174. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( B  ~<_  om  ->  ran  (  x  e.  B  |->  C )  ~<_  om )
 
Theorem1stcrest 17174 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 17175 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 17176* 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 17177* 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 17178* 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 17179* 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 17180* 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 17181 A discrete space is second-countable iff it is countable. (Contributed by Mario Carneiro, 13-Apr-2015.)
 |-  ( X  ~<_  om  <->  ~P X  e.  2ndc )
 
Theorem1stcelcls 17182* 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 8057. 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 17183* 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 8057, but only via 1stcelcls 17182, 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 17184* 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 17185 Extend class notation with the "locally  A " predicate of a topological space.
 class Locally  A
 
Syntaxcnlly 17186 Extend class notation with the "N-locally  A " predicate of a topological space.
 class 𝑛Locally  A
 
Definitiondf-lly 17187* 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 17188* 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 17189* 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 17190* 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 17191 Equality theorem for the Locally  A predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( A  =  B  -> Locally  A  = Locally  B )
 
Theoremnllyeq 17192 Equality theorem for the Locally  A predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( A  =  B  -> 𝑛Locally  A  = 𝑛Locally  B )
 
Theoremllytop 17193 A locally  A space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( J  e. Locally  A  ->  J  e.  Top )
 
Theoremnllytop 17194 A locally  A space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( J  e. 𝑛Locally  A  ->  J  e.  Top )
 
Theoremllyi 17195* 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 17196* 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 17197* Eliminate the neighborhood symbol from nllyi 17196. (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 17198 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 17199 A locally  A space is n-locally  A. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |- Locally  A  C_ 𝑛Locally  A
 
Theoremllyss 17200 The "locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( A  C_  B  -> Locally  A  C_ Locally  B )
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