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Theorem List for Metamath Proof Explorer - 17001-17100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcnt0 17001 The preimage of a T0 topology under an injective map is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( ( K  e.  Kol2  /\  F : X -1-1-> Y  /\  F  e.  ( J  Cn  K ) ) 
 ->  J  e.  Kol2 )
 
Theoremist1-2 17002* An alternate characterization of T1 spaces. (Contributed by Jeff Hankins, 31-Jan-2010.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
 |-  ( J  e.  (TopOn `  X )  ->  ( J  e.  Fre  <->  A. x  e.  X  A. y  e.  X  (
 A. o  e.  J  ( x  e.  o  ->  y  e.  o ) 
 ->  x  =  y
 ) ) )
 
Theoremt1t0 17003 A T1 space is a T0 space. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |-  ( J  e.  Fre  ->  J  e.  Kol2 )
 
Theoremist1-3 17004* A space is T1 iff every point is the only point in the intersection of all open sets containing that point. (Contributed by Jeff Hankins, 31-Jan-2010.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
 |-  ( J  e.  (TopOn `  X )  ->  ( J  e.  Fre  <->  A. x  e.  X  |^|
 { o  e.  J  |  x  e.  o }  =  { x } ) )
 
Theoremcnt1 17005 The preimage of a T1 topology under an injective map is T1. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( K  e.  Fre  /\  F : X -1-1-> Y  /\  F  e.  ( J  Cn  K ) ) 
 ->  J  e.  Fre )
 
Theoremishaus2 17006* Express the predicate " J is a Hausdorff space." (Contributed by NM, 8-Mar-2007.)
 |-  ( J  e.  (TopOn `  X )  ->  ( J  e.  Haus  <->  A. x  e.  X  A. y  e.  X  ( x  =/=  y  ->  E. n  e.  J  E. m  e.  J  ( x  e.  n  /\  y  e.  m  /\  ( n  i^i  m )  =  (/) ) ) ) )
 
Theoremhaust1 17007 A Hausdorff space is a T1 space. (Contributed by FL, 11-Jun-2007.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
 |-  ( J  e.  Haus  ->  J  e.  Fre )
 
Theoremhausnei2 17008* The Hausdorff condition still holds if one considers general neighborhoods instead of open sets. (Contributed by Jeff Hankins, 5-Sep-2009.)
 |-  ( J  e.  (TopOn `  X )  ->  ( J  e.  Haus  <->  A. x  e.  X  A. y  e.  X  ( x  =/=  y  ->  E. u  e.  (
 ( nei `  J ) `  { x } ) E. v  e.  (
 ( nei `  J ) `  { y } )
 ( u  i^i  v
 )  =  (/) ) ) )
 
Theoremcnhaus 17009 The preimage of a Hausdorff topology under an injective map is Hausdorff. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( ( K  e.  Haus  /\  F : X -1-1-> Y  /\  F  e.  ( J  Cn  K ) ) 
 ->  J  e.  Haus )
 
Theoremnrmsep3 17010* In a normal space, given a closed set  B inside an open set  A, there is an open set  x such that  B  C_  x  C_  cls ( x )  C_  A. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( ( J  e.  Nrm  /\  ( A  e.  J  /\  B  e.  ( Clsd `  J )  /\  B  C_  A ) )  ->  E. x  e.  J  ( B  C_  x  /\  ( ( cls `  J ) `  x )  C_  A ) )
 
Theoremnrmsep2 17011* In a normal space, any two disjoint closed sets have the property that each one is a subset of an open set whose closure is disjoint from the other. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 24-Aug-2015.)
 |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  E. x  e.  J  ( C  C_  x  /\  ( ( ( cls `  J ) `  x )  i^i  D )  =  (/) ) )
 
Theoremnrmsep 17012* In a normal space, disjoint closed sets are separated by open sets. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  E. x  e.  J  E. y  e.  J  ( C  C_  x  /\  D  C_  y  /\  ( x  i^i  y
 )  =  (/) ) )
 
Theoremisnrm2 17013* An alternate characterization of normality. This is the important property in the proof of Urysohn's lemma. (Contributed by Jeff Hankins, 1-Feb-2010.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
 |-  ( J  e.  Nrm  <->  ( J  e.  Top  /\  A. c  e.  ( Clsd `  J ) A. d  e.  ( Clsd `  J )
 ( ( c  i^i  d )  =  (/)  ->  E. o  e.  J  ( c  C_  o  /\  ( ( ( cls `  J ) `  o
 )  i^i  d )  =  (/) ) ) ) )
 
Theoremisnrm3 17014* A topological space is normal iff any two disjoint closed sets are separated by open sets. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( J  e.  Nrm  <->  ( J  e.  Top  /\  A. c  e.  ( Clsd `  J ) A. d  e.  ( Clsd `  J )
 ( ( c  i^i  d )  =  (/)  ->  E. x  e.  J  E. y  e.  J  ( c  C_  x  /\  d  C_  y  /\  ( x  i^i  y )  =  (/) ) ) ) )
 
Theoremcnrmi 17015 A subspace of a completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( J  e. CNrm  /\  A  e.  V ) 
 ->  ( Jt  A )  e.  Nrm )
 
Theoremcnrmnrm 17016 A completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( J  e. CNrm  ->  J  e.  Nrm )
 
Theoremrestcnrm 17017 A subspace of a completely normal space is completely normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( J  e. CNrm  /\  A  e.  V ) 
 ->  ( Jt  A )  e. CNrm )
 
Theoremresthauslem 17018 Lemma for resthaus 17023 and similar theorems. If the topological property  A is preserved under injective preimages, then property  A passes to subspaces. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  e.  A  ->  J  e.  Top )   &    |-  (
 ( J  e.  A  /\  (  _I  |`  ( S  i^i  U. J ) ) : ( S  i^i  U. J ) -1-1-> ( S  i^i  U. J )  /\  (  _I  |`  ( S  i^i  U. J ) )  e.  ( ( Jt  S )  Cn  J ) )  ->  ( Jt  S )  e.  A )   =>    |-  (
 ( J  e.  A  /\  S  e.  V ) 
 ->  ( Jt  S )  e.  A )
 
Theoremlpcls 17019 The limit points of the closure of a subset are the same as the limit points of the set in a T1 space. (Contributed by Mario Carneiro, 26-Dec-2016.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Fre  /\  S  C_  X )  ->  ( ( limPt `  J ) `  ( ( cls `  J ) `  S ) )  =  (
 ( limPt `  J ) `  S ) )
 
Theoremperfcls 17020 A subset of a perfect space is perfect iff its closure is perfect (and the closure is an actual perfect set, since it is both closed and perfect in the subspace topology). (Contributed by Mario Carneiro, 26-Dec-2016.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Fre  /\  S  C_  X )  ->  ( ( Jt  S )  e. Perf 
 <->  ( Jt  ( ( cls `  J ) `  S ) )  e. Perf ) )
 
Theoremrestt0 17021 A subspace of a T0 topology is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( ( J  e.  Kol2  /\  A  e.  V ) 
 ->  ( Jt  A )  e.  Kol2 )
 
Theoremrestt1 17022 A subspace of a T1 topology is T1. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( ( J  e.  Fre  /\  A  e.  V ) 
 ->  ( Jt  A )  e.  Fre )
 
Theoremresthaus 17023 A subspace of a Hausdorff topology is Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
 |-  ( ( J  e.  Haus  /\  A  e.  V ) 
 ->  ( Jt  A )  e.  Haus )
 
Theoremt1sep2 17024* Any two points in a T1 space which have no separation are equal. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Fre  /\  A  e.  X  /\  B  e.  X )  ->  ( A. o  e.  J  ( A  e.  o  ->  B  e.  o
 )  ->  A  =  B ) )
 
Theoremt1sep 17025* Any two distinct points in a T1 space are separated by an open set. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Fre  /\  ( A  e.  X  /\  B  e.  X  /\  A  =/=  B ) ) 
 ->  E. o  e.  J  ( A  e.  o  /\  -.  B  e.  o
 ) )
 
Theoremsncld 17026 A singleton is closed in a Hausdorff space. (Contributed by NM, 5-Mar-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Haus  /\  P  e.  X ) 
 ->  { P }  e.  ( Clsd `  J )
 )
 
Theoremsshauslem 17027 Lemma for sshaus 17030 and similar theorems. If the topological property  A is preserved under injective preimages, then a topology finer than one with property  A also has property  A. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  X  =  U. J   &    |-  ( J  e.  A  ->  J  e.  Top )   &    |-  (
 ( J  e.  A  /\  (  _I  |`  X ) : X -1-1-> X  /\  (  _I  |`  X )  e.  ( K  Cn  J ) )  ->  K  e.  A )   =>    |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X )  /\  J  C_  K )  ->  K  e.  A )
 
Theoremsst0 17028 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 17029 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 17030 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 17031* 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 17032* 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 17033 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 5542). (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 17034 The order topology is T1 for any poset. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( R  e.  PosetRel  ->  (ordTop `  R )  e.  Fre )
 
Theoremlmmo 17035 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 17036 The convergence relation is function-like in a Hausdorff space. (Contributed by Mario Carneiro, 26-Dec-2013.)
 |-  ( J  e.  Haus  ->  Fun  ( ~~> t `  J ) )
 
Theoremdishaus 17037 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 17038* Lemma for ordthaus 17039. (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 17039 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 17040 Extend class notation with the class of all compact spaces.
 class  Comp
 
Definitiondf-cmp 17041* 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 17042* 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 17043* 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 17044* Rewrite cmpcov 17043 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 17045* Combine cmpcov 17043 with ac6sfi 7034 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 17046 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 17047 A finite topology is compact. (Contributed by FL, 22-Dec-2008.)
 |-  ( J  e.  ( Top  i^i  Fin )  ->  J  e.  Comp )
 
Theorem0cmp 17048 The singleton of the empty set is compact. (Contributed by FL, 2-Aug-2009.)
 |- 
 { (/) }  e.  Comp
 
Theoremcmptop 17049 A compact topology is a topology. (Contributed by Jeff Hankins, 29-Jun-2009.)
 |-  ( J  e.  Comp  ->  J  e.  Top )
 
Theoremrncmp 17050 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 17051 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 17052 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 17053* Lemma for cmpsub 17054. (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 17054* 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 17055* A topology generated by a basis is compact iff open covers drawn from the basis have finite subcovers. (See also alexsub 17666, 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 17056 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 17057 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 17058* 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 17059 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 17060* Lemma for hauscmp 17061. (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 17061 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 17062* 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 17063 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 17064 Extend class notation with the class of all connected topologies.
 class  Con
 
Definitiondf-con 17065 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 17066 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 17067 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 17068 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 17069 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 17070 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 17071 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 17072* 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 17073* Connectedness for a subspace. See connsub 17074. (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 17074* 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 17075 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 17076 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 17077 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 17078 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 17079 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 17080* Lemma for iuncon 17081. (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 17081* 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 17082 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 17083 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 17084* 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 17085* 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 17086* 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 17087* 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 17088* 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 17089 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 17090 Extend class definition to include the class of all first-countable topologies.
 class  1stc
 
Syntaxc2ndc 17091 Extend class definition to include the class of all second-countable topologies.
 class  2ndc
 
Definitiondf-1stc 17092* 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 17093* 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 17094* 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 17095* 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 17096 A first-countable topology is a topology. (Contributed by Jeff Hankins, 22-Aug-2009.)
 |-  ( J  e.  1stc  ->  J  e.  Top )
 
Theorem1stcclb 17097* 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 17098* 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 17099* 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 17100 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 )
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