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Theorem List for Metamath Proof Explorer - 16901-17000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempnrmtop 16901 A perfectly normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( J  e. PNrm  ->  J  e.  Top )
 
Theorempnrmcld 16902* 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 16903* 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 16904* 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 16905* 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 16906 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 16907* 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 16908 A T1 space is a T0 space. (Contributed by Jeff Hankins, 1-Feb-2010.)
 |-  ( J  e.  Fre  ->  J  e.  Kol2 )
 
Theoremist1-3 16909* 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 16910 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 16911* 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 16912 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 16913* 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 16914 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 16915* 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 16916* 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 16917* 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 16918* 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 16919* 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 16920 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 16921 A completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( J  e. CNrm  ->  J  e.  Nrm )
 
Theoremrestcnrm 16922 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 16923 Lemma for resthaus 16928 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 16924 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 16925 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 16926 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 16927 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 16928 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 16929* 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 16930* 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 16931 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 16932 Lemma for sshaus 16935 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 16933 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 16934 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 16935 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 16936* 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 16937* 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 16938 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 5495). (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 16939 The order topology is T1 for any poset. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( R  e.  PosetRel  ->  (ordTop `  R )  e.  Fre )
 
Theoremlmmo 16940 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 16941 The convergence relation is function-like in a Hausdorff space. (Contributed by Mario Carneiro, 26-Dec-2013.)
 |-  ( J  e.  Haus  ->  Fun  ( ~~> t `  J ) )
 
Theoremdishaus 16942 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 16943* Lemma for ordthaus 16944. (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 16944 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 16945 Extend class notation with the class of all compact spaces.
 class  Comp
 
Definitiondf-cmp 16946* 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 16947* 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 16948* 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 16949* Rewrite cmpcov 16948 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 16950* Combine cmpcov 16948 with ac6sfi 6986 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 16951 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 16952 A finite topology is compact. (Contributed by FL, 22-Dec-2008.)
 |-  ( J  e.  ( Top  i^i  Fin )  ->  J  e.  Comp )
 
Theorem0cmp 16953 The singleton of the empty set is compact. (Contributed by FL, 2-Aug-2009.)
 |- 
 { (/) }  e.  Comp
 
Theoremcmptop 16954 A compact topology is a topology. (Contributed by Jeff Hankins, 29-Jun-2009.)
 |-  ( J  e.  Comp  ->  J  e.  Top )
 
Theoremrncmp 16955 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 16956 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 16957 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 16958* Lemma for cmpsub 16959. (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 16959* 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 16960* A topology generated by a basis is compact iff open covers drawn from the basis have finite subcovers. (See also alexsub 17571, 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 16961 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 16962 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 16963* 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 16964 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 16965* Lemma for hauscmp 16966. (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 16966 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 16967* 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 16968 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 16969 Extend class notation with the class of all connected topologies.
 class  Con
 
Definitiondf-con 16970 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 16971 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 16972 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 16973 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 16974 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 16975 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 16976 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 16977* 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 16978* Connectedness for a subspace. See connsub 16979. (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 16979* 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 16980 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 16981 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 16982 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 16983 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 16984 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 16985* Lemma for iuncon 16986. (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 16986* 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 16987 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 16988 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 16989* 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 16990* 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 16991* 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 16992* 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 16993* 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 16994 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 16995 Extend class definition to include the class of all first-countable topologies.
 class  1stc
 
Syntaxc2ndc 16996 Extend class definition to include the class of all second-countable topologies.
 class  2ndc
 
Definitiondf-1stc 16997* 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 16998* 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 16999* 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 17000* 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 ) ) ) ) )
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