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Theorem List for Metamath Proof Explorer - 17001-17100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcnrm 17001 Extend class notation with the class of all normal topologies.
 class  Nrm
 
Syntaxccnrm 17002 Extend class notation with the class of all completely normal topologies.
 class CNrm
 
Syntaxcpnrm 17003 Extend class notation with the class of all perfectly normal topologies.
 class PNrm
 
Definitiondf-t0 17004* Define T0 or Kolmogorov spaces. A T0 space satisfies a kind of "topological extensionality" principle (compare ax-ext 2239): any two points which are members of the same open sets are equal, or in contraposition, for any two distinct points there is an open set which contains one point but not the other. This differs from T1 spaces (see ist1-2 17038) in that in a T1 space you can choose which point will be in the open set and which outside; in a T0 space you only know that one of the two points is in the set. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |- 
 Kol2  =  { j  e.  Top  |  A. x  e.  U. j A. y  e.  U. j ( A. o  e.  j  ( x  e.  o  <->  y  e.  o
 )  ->  x  =  y ) }
 
Definitiondf-t1 17005* The class of all T1 spaces, also called Fréchet spaces. Morris, Topology without tears, p. 30 ex. 3. (Contributed by FL, 18-Jun-2007.)
 |- 
 Fre  =  { x  e.  Top  |  A. a  e.  U. x { a }  e.  ( Clsd `  x ) }
 
Definitiondf-haus 17006* Define the class of all Hausdorff spaces. A Hausdorff space is a topology in which distinct points have disjoint open neighborhoods. Definition of Hausdorff space in [Munkres] p. 98. (Contributed by NM, 8-Mar-2007.)
 |- 
 Haus  =  { j  e.  Top  |  A. x  e.  U. j A. y  e.  U. j ( x  =/=  y  ->  E. n  e.  j  E. m  e.  j  ( x  e.  n  /\  y  e.  m  /\  ( n  i^i  m )  =  (/) ) ) }
 
Definitiondf-reg 17007* Define regular spaces. A space is regular if a point and a closed set can be separated by neighborhoods. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |- 
 Reg  =  { j  e.  Top  |  A. x  e.  j  A. y  e.  x  E. z  e.  j  ( y  e.  z  /\  ( ( cls `  j ) `  z )  C_  x ) }
 
Definitiondf-nrm 17008* Define normal spaces. A space is normal if disjoint closed sets can be separated by neighborhoods. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |- 
 Nrm  =  { j  e.  Top  |  A. x  e.  j  A. y  e.  ( ( Clsd `  j
 )  i^i  ~P x ) E. z  e.  j  ( y  C_  z  /\  ( ( cls `  j
 ) `  z )  C_  x ) }
 
Definitiondf-cnrm 17009* Define completely normal spaces. A space is completely normal if all its subspaces are normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |- CNrm  =  { j  e.  Top  | 
 A. x  e.  ~P  U. j ( jt  x )  e.  Nrm }
 
Definitiondf-pnrm 17010* Define perfectly normal spaces. A space is perfectly normal if it is normal and every closed set is a G&delta; set, meaning that it is a countable intersection of open sets. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |- PNrm  =  { j  e.  Nrm  |  ( Clsd `  j )  C_ 
 ran  (  f  e.  ( j  ^m  NN )  |->  |^| ran  f ) }
 
Theoremist0 17011* The predicate "is a T0 space." Every pair of distinct points is topologically distinguishable. For the way this definition is usually encountered, see ist0-3 17036. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |-  X  =  U. J   =>    |-  ( J  e.  Kol2  <->  ( J  e.  Top  /\  A. x  e.  X  A. y  e.  X  (
 A. o  e.  J  ( x  e.  o  <->  y  e.  o )  ->  x  =  y )
 ) )
 
Theoremist1 17012* The predicate  J is T1. (Contributed by FL, 18-Jun-2007.)
 |-  X  =  U. J   =>    |-  ( J  e.  Fre  <->  ( J  e.  Top  /\  A. a  e.  X  { a }  e.  ( Clsd `  J )
 ) )
 
Theoremishaus 17013* Express the predicate " J is a Hausdorff space." (Contributed by NM, 8-Mar-2007.)
 |-  X  =  U. J   =>    |-  ( J  e.  Haus  <->  ( J  e.  Top  /\  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 )  =  (/) ) ) ) )
 
Theoremiscnrm 17014* The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( J  e. CNrm  <->  ( J  e.  Top  /\  A. x  e.  ~P  X ( Jt  x )  e.  Nrm ) )
 
Theoremt0sep 17015* Any two topologically indistinguishable points in a T0 space are identical. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Kol2  /\  ( A  e.  X  /\  B  e.  X ) )  ->  ( A. x  e.  J  ( A  e.  x  <->  B  e.  x )  ->  A  =  B ) )
 
Theoremt0dist 17016* Any two distinct points in a T0 space are topologically distinguishable. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Kol2  /\  ( A  e.  X  /\  B  e.  X  /\  A  =/=  B ) ) 
 ->  E. o  e.  J  -.  ( A  e.  o  <->  B  e.  o ) )
 
Theoremt1sncld 17017 In a T1 space, one-point sets are closed. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Fre  /\  A  e.  X ) 
 ->  { A }  e.  ( Clsd `  J )
 )
 
Theoremt1ficld 17018 In a T1 space, finite sets are closed. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Fre  /\  A  C_  X  /\  A  e.  Fin )  ->  A  e.  ( Clsd `  J ) )
 
Theoremhausnei 17019* Neighborhood property of a Hausdorff space. (Contributed by NM, 8-Mar-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Haus  /\  ( P  e.  X  /\  Q  e.  X  /\  P  =/=  Q ) ) 
 ->  E. n  e.  J  E. m  e.  J  ( P  e.  n  /\  Q  e.  m  /\  ( n  i^i  m )  =  (/) ) )
 
Theoremt0top 17020 A T0 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |-  ( J  e.  Kol2  ->  J  e.  Top )
 
Theoremt1top 17021 A T1 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |-  ( J  e.  Fre  ->  J  e.  Top )
 
Theoremhaustop 17022 A Hausdorff space is a topology. (Contributed by NM, 5-Mar-2007.)
 |-  ( J  e.  Haus  ->  J  e.  Top )
 
Theoremisreg 17023* The predicate "is a regular space." In a regular space, any open neighborhood has a closed subneighborhood. Note that some authors require the space to be Hausdorff (which would make it the same as T3), but we reserve the phrase "regular Hausdorff" for that as many topologists do. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  e.  Reg  <->  ( J  e.  Top  /\  A. x  e.  J  A. y  e.  x  E. z  e.  J  ( y  e.  z  /\  ( ( cls `  J ) `  z )  C_  x ) ) )
 
Theoremregtop 17024 A regular space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |-  ( J  e.  Reg  ->  J  e.  Top )
 
Theoremregsep 17025* In a regular space, every neighborhood of a point contains a closed subneighborhood. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( ( J  e.  Reg  /\  U  e.  J  /\  A  e.  U )  ->  E. x  e.  J  ( A  e.  x  /\  ( ( cls `  J ) `  x )  C_  U ) )
 
Theoremisnrm 17026* The predicate "is a normal space." Much like the case for regular spaces, normal does not imply Hausdorff or even regular. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 24-Aug-2015.)
 |-  ( J  e.  Nrm  <->  ( J  e.  Top  /\  A. x  e.  J  A. y  e.  ( ( Clsd `  J )  i^i  ~P x ) E. z  e.  J  ( y  C_  z  /\  ( ( cls `  J ) `  z )  C_  x ) ) )
 
Theoremnrmtop 17027 A normal space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |-  ( J  e.  Nrm  ->  J  e.  Top )
 
Theoremcnrmtop 17028 A completely normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( J  e. CNrm  ->  J  e.  Top )
 
Theoremiscnrm2 17029* The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( J  e.  (TopOn `  X )  ->  ( J  e. CNrm  <->  A. x  e.  ~P  X ( Jt  x )  e.  Nrm ) )
 
Theoremispnrm 17030* The property of being perfectly normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( J  e. PNrm  <->  ( J  e.  Nrm  /\  ( Clsd `  J )  C_ 
 ran  (  f  e.  ( J  ^m  NN )  |->  |^| ran  f )
 ) )
 
Theorempnrmnrm 17031 A perfectly normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( J  e. PNrm  ->  J  e.  Nrm )
 
Theorempnrmtop 17032 A perfectly normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( J  e. PNrm  ->  J  e.  Top )
 
Theorempnrmcld 17033* A closed set in a perfectly normal space is a countable intersection of open sets. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( J  e. PNrm  /\  A  e.  ( Clsd `  J ) )  ->  E. f  e.  ( J  ^m  NN ) A  =  |^| ran  f )
 
Theorempnrmopn 17034* An open set in a perfectly normal space is a countable union of closed sets. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( J  e. PNrm  /\  A  e.  J ) 
 ->  E. f  e.  (
 ( Clsd `  J )  ^m  NN ) A  =  U.
 ran  f )
 
Theoremist0-2 17035* The predicate "is a T0 space". (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( J  e.  (TopOn `  X )  ->  ( J  e.  Kol2  <->  A. x  e.  X  A. y  e.  X  (
 A. o  e.  J  ( x  e.  o  <->  y  e.  o )  ->  x  =  y )
 ) )
 
Theoremist0-3 17036* The predicate "is a T0 space," expressed in more familiar terms. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |-  ( J  e.  (TopOn `  X )  ->  ( J  e.  Kol2  <->  A. x  e.  X  A. y  e.  X  ( x  =/=  y  ->  E. o  e.  J  ( ( x  e.  o  /\  -.  y  e.  o )  \/  ( -.  x  e.  o  /\  y  e.  o
 ) ) ) ) )
 
Theoremcnt0 17037 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 17038* 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 17039 A T1 space is a T0 space. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |-  ( J  e.  Fre  ->  J  e.  Kol2 )
 
Theoremist1-3 17040* 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 17041 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 17042* 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 17043 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 17044* 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 17045 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 17046* 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 17047* 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 17048* 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 17049* 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 17050* 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 17051 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 17052 A completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( J  e. CNrm  ->  J  e.  Nrm )
 
Theoremrestcnrm 17053 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 17054 Lemma for resthaus 17059 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 17055 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 17056 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 17057 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 17058 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 17059 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 17060* 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 17061* 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 17062 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 17063 Lemma for sshaus 17066 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 17064 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 17065 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 17066 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 17067* 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 17068* 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 17069 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 5577). (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 17070 The order topology is T1 for any poset. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( R  e.  PosetRel  ->  (ordTop `  R )  e.  Fre )
 
Theoremlmmo 17071 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 17072 The convergence relation is function-like in a Hausdorff space. (Contributed by Mario Carneiro, 26-Dec-2013.)
 |-  ( J  e.  Haus  ->  Fun  ( ~~> t `  J ) )
 
Theoremdishaus 17073 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 17074* Lemma for ordthaus 17075. (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 17075 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 17076 Extend class notation with the class of all compact spaces.
 class  Comp
 
Definitiondf-cmp 17077* 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 17078* 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 17079* 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 17080* Rewrite cmpcov 17079 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 17081* Combine cmpcov 17079 with ac6sfi 7069 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 17082 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 17083 A finite topology is compact. (Contributed by FL, 22-Dec-2008.)
 |-  ( J  e.  ( Top  i^i  Fin )  ->  J  e.  Comp )
 
Theorem0cmp 17084 The singleton of the empty set is compact. (Contributed by FL, 2-Aug-2009.)
 |- 
 { (/) }  e.  Comp
 
Theoremcmptop 17085 A compact topology is a topology. (Contributed by Jeff Hankins, 29-Jun-2009.)
 |-  ( J  e.  Comp  ->  J  e.  Top )
 
Theoremrncmp 17086 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 17087 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 17088 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 17089* Lemma for cmpsub 17090. (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 17090* 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 17091* A topology generated by a basis is compact iff open covers drawn from the basis have finite subcovers. (See also alexsub 17702, 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 17092 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 17093 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 17094* 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 17095 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 17096* Lemma for hauscmp 17097. (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 17097 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 17098* 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 17099 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 17100 Extend class notation with the class of all connected topologies.
 class  Con
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