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Theorem List for Metamath Proof Explorer - 17101-17200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsst0 17101 A topology finer than a T0 topology is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Kol2  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  K  e.  Kol2
 )
 
Theoremsst1 17102 A topology finer than a T1 topology is T1. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Fre  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  K  e.  Fre )
 
Theoremsshaus 17103 A topology finer than a Hausdorff topology is Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Haus  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  K  e.  Haus
 )
 
Theoremregsep2 17104* In a regular space, a closed set is separated by open sets from a point not in it. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 25-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Reg  /\  ( C  e.  ( Clsd `  J )  /\  A  e.  X  /\  -.  A  e.  C ) )  ->  E. x  e.  J  E. y  e.  J  ( C  C_  x  /\  A  e.  y  /\  ( x  i^i  y
 )  =  (/) ) )
 
Theoremisreg2 17105* A topological space is regular if any closed set is separated from any point not in it by neighborhoods. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  e.  (TopOn `  X )  ->  ( J  e.  Reg  <->  A. c  e.  ( Clsd `  J ) A. x  e.  X  ( -.  x  e.  c  ->  E. o  e.  J  E. p  e.  J  ( c  C_  o  /\  x  e.  p  /\  ( o  i^i  p )  =  (/) ) ) ) )
 
Theoremdnsconst 17106 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 5643). (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 17107 The order topology is T1 for any poset. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( R  e.  PosetRel  ->  (ordTop `  R )  e.  Fre )
 
Theoremlmmo 17108 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 17109 The convergence relation is function-like in a Hausdorff space. (Contributed by Mario Carneiro, 26-Dec-2013.)
 |-  ( J  e.  Haus  ->  Fun  ( ~~> t `  J ) )
 
Theoremdishaus 17110 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 17111* Lemma for ordthaus 17112. (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 17112 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 17113 Extend class notation with the class of all compact spaces.
 class  Comp
 
Definitiondf-cmp 17114* 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 17115* 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 17116* 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 17117* Rewrite cmpcov 17116 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 17118* Combine cmpcov 17116 with ac6sfi 7101 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 17119 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 17120 A finite topology is compact. (Contributed by FL, 22-Dec-2008.)
 |-  ( J  e.  ( Top  i^i  Fin )  ->  J  e.  Comp )
 
Theorem0cmp 17121 The singleton of the empty set is compact. (Contributed by FL, 2-Aug-2009.)
 |- 
 { (/) }  e.  Comp
 
Theoremcmptop 17122 A compact topology is a topology. (Contributed by Jeff Hankins, 29-Jun-2009.)
 |-  ( J  e.  Comp  ->  J  e.  Top )
 
Theoremrncmp 17123 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 17124 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 17125 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 17126* Lemma for cmpsub 17127. (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 17127* 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 17128* A topology generated by a basis is compact iff open covers drawn from the basis have finite subcovers. (See also alexsub 17739, 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 17129 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 17130 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 17131* 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 17132 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 17133* Lemma for hauscmp 17134. (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 17134 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 17135* 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 17136 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 17137 Extend class notation with the class of all connected topologies.
 class  Con
 
Definitiondf-con 17138 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 17139 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 17140 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 17141 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 17142 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 17143 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 17144 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 17145* 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 17146* Connectedness for a subspace. See connsub 17147. (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 17147* 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 17148 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 17149 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 17150 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 17151 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 17152 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 17153* Lemma for iuncon 17154. (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 17154* 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 17155 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 17156 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 17157* 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 17158* 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 17159* 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 17160* 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 17161* 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 17162 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 17163 Extend class definition to include the class of all first-countable topologies.
 class  1stc
 
Syntaxc2ndc 17164 Extend class definition to include the class of all second-countable topologies.
 class  2ndc
 
Definitiondf-1stc 17165* 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 17166* 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 17167* 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 17168* 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 17169 A first-countable topology is a topology. (Contributed by Jeff Hankins, 22-Aug-2009.)
 |-  ( J  e.  1stc  ->  J  e.  Top )
 
Theorem1stcclb 17170* 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 17171* 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 17172* 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 17173 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 17174 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 17175* 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 17176 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 17177 A second-countable space is first-countable. (Contributed by Jeff Hankins, 17-Jan-2010.)
 |-  ( J  e.  2ndc  ->  J  e.  1stc )
 
Theorem1stcrestlem 17178* Lemma for 1stcrest 17179. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( B  ~<_  om  ->  ran  ( x  e.  B  |->  C )  ~<_  om )
 
Theorem1stcrest 17179 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 17180 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 17181* 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 17182* 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 17183* 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 17184* 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 17185* 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 17186 A discrete space is second-countable iff it is countable. (Contributed by Mario Carneiro, 13-Apr-2015.)
 |-  ( X  ~<_  om  <->  ~P X  e.  2ndc )
 
Theorem1stcelcls 17187* 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 8061. 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 17188* 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 8061, but only via 1stcelcls 17187, 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 17189* 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 17190 Extend class notation with the "locally  A " predicate of a topological space.
 class Locally  A
 
Syntaxcnlly 17191 Extend class notation with the "N-locally  A " predicate of a topological space.
 class 𝑛Locally  A
 
Definitiondf-lly 17192* 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 17193* 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 17194* 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 17195* 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 17196 Equality theorem for the Locally  A predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( A  =  B  -> Locally  A  = Locally  B )
 
Theoremnllyeq 17197 Equality theorem for the Locally  A predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( A  =  B  -> 𝑛Locally  A  = 𝑛Locally  B )
 
Theoremllytop 17198 A locally  A space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( J  e. Locally  A  ->  J  e.  Top )
 
Theoremnllytop 17199 A locally  A space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( J  e. 𝑛Locally  A  ->  J  e.  Top )
 
Theoremllyi 17200* 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 ) )
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