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Theorem List for Metamath Proof Explorer - 20801-20900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcmphmph 20801 Compactness is a topological property-that is, for any two homeomorphic topologies, either both are compact or neither is. (Contributed by Jeff Hankins, 30-Jun-2009.) (Revised by Mario Carneiro, 23-Aug-2015.)
 |-  ( J  ~=  K  ->  ( J  e.  Comp  ->  K  e.  Comp ) )
 
Theoremconhmph 20802 Connectedness is a topological property. (Contributed by Jeff Hankins, 3-Jul-2009.)
 |-  ( J  ~=  K  ->  ( J  e.  Con  ->  K  e.  Con ) )
 
Theoremt0hmph 20803 T0 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  ~=  K  ->  ( J  e.  Kol2  ->  K  e.  Kol2 ) )
 
Theoremt1hmph 20804 T1 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  ~=  K  ->  ( J  e.  Fre  ->  K  e.  Fre ) )
 
Theoremhaushmph 20805 Hausdorff-ness is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  ~=  K  ->  ( J  e.  Haus  ->  K  e.  Haus ) )
 
Theoremreghmph 20806 Regularity is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  ~=  K  ->  ( J  e.  Reg  ->  K  e.  Reg ) )
 
Theoremnrmhmph 20807 Normality is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  ~=  K  ->  ( J  e.  Nrm  ->  K  e.  Nrm ) )
 
Theoremhmph0 20808 A topology homeomorphic to the empty set is empty. (Contributed by FL, 18-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  ( J  ~=  { (/)
 } 
 <->  J  =  { (/) } )
 
Theoremhmphdis 20809 Homeomorphisms preserve topological discretion. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  X  =  U. J   =>    |-  ( J  ~=  ~P A  ->  J  =  ~P X )
 
Theoremhmphindis 20810 Homeomorphisms preserve topological indiscretion. (Contributed by FL, 18-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  X  =  U. J   =>    |-  ( J  ~=  { (/) ,  A }  ->  J  =  { (/)
 ,  X } )
 
Theoremindishmph 20811 Equinumerous sets equipped with their indiscrete topologies are homeomorphic (which means in that particular case that a segment is homeomorphic to a circle contrary to what Wikipedia claims). (Contributed by FL, 17-Aug-2008.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
 |-  ( A  ~~  B  ->  { (/) ,  A }  ~=  { (/) ,  B }
 )
 
Theoremhmphen2 20812 Homeomorphisms preserve the cardinality of the underlying sets. (Contributed by FL, 17-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  ( J  ~=  K  ->  X  ~~  Y )
 
Theoremcmphaushmeo 20813 A continuous bijection from a compact space to a Hausdorff space is a homeomorphism. (Contributed by Mario Carneiro, 17-Feb-2015.)
 |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  ( ( J  e.  Comp  /\  K  e.  Haus  /\  F  e.  ( J  Cn  K ) )  ->  ( F  e.  ( J Homeo K )  <->  F : X -1-1-onto-> Y ) )
 
Theoremordthmeolem 20814 Lemma for ordthmeo 20815. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  X  =  dom  R   &    |-  Y  =  dom  S   =>    |-  ( ( R  e.  V  /\  S  e.  W  /\  F  Isom  R ,  S  ( X ,  Y ) )  ->  F  e.  ( (ordTop `  R )  Cn  (ordTop `  S )
 ) )
 
Theoremordthmeo 20815 An order isomorphism is a homeomorphism on the respective order topologies. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  X  =  dom  R   &    |-  Y  =  dom  S   =>    |-  ( ( R  e.  V  /\  S  e.  W  /\  F  Isom  R ,  S  ( X ,  Y ) )  ->  F  e.  ( (ordTop `  R ) Homeo (ordTop `  S )
 ) )
 
Theoremtxhmeo 20816* Lift a pair of homeomorphisms on the factors to a homeomorphism of product topologies. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  X  =  U. J   &    |-  Y  =  U. K   &    |-  ( ph  ->  F  e.  ( J Homeo L ) )   &    |-  ( ph  ->  G  e.  ( K Homeo M ) )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  <. ( F `
  x ) ,  ( G `  y
 ) >. )  e.  (
 ( J  tX  K ) Homeo ( L  tX  M ) ) )
 
Theoremtxswaphmeolem 20817* Show inverse for the "swap components" operation on a Cartesian product. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( ( y  e.  Y ,  x  e.  X  |->  <. x ,  y >. )  o.  ( x  e.  X ,  y  e.  Y  |->  <. y ,  x >. ) )  =  (  _I  |`  ( X  X.  Y ) )
 
Theoremtxswaphmeo 20818* There is a homeomorphism from  X  X.  Y to  Y  X.  X. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  ->  ( x  e.  X ,  y  e.  Y  |->  <. y ,  x >. )  e.  ( ( J  tX  K ) Homeo ( K  tX  J ) ) )
 
Theorempt1hmeo 20819* The canonical homeomorphism from a topological product on a singleton to the topology of the factor. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |-  K  =  ( Xt_ ` 
 { <. A ,  J >. } )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  {
 <. A ,  x >. } )  e.  ( J
 Homeo K ) )
 
Theoremptuncnv 20820* Exhibit the converse function of the map  G which joins two product topologies on disjoint index sets. (Contributed by Mario Carneiro, 8-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
 |-  X  =  U. K   &    |-  Y  =  U. L   &    |-  J  =  (
 Xt_ `  F )   &    |-  K  =  ( Xt_ `  ( F  |`  A ) )   &    |-  L  =  ( Xt_ `  ( F  |`  B ) )   &    |-  G  =  ( x  e.  X ,  y  e.  Y  |->  ( x  u.  y ) )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  F : C --> Top )   &    |-  ( ph  ->  C  =  ( A  u.  B ) )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   =>    |-  ( ph  ->  `' G  =  ( z  e.  U. J  |->  <. ( z  |`  A ) ,  (
 z  |`  B ) >. ) )
 
Theoremptunhmeo 20821* Define a homeomorphism from a binary product of indexed product topologies to an indexed product topology on the union of the index sets. This is the topological analogue of  ( A ^ B )  x.  ( A ^ C )  =  A ^ ( B  +  C ). (Contributed by Mario Carneiro, 8-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
 |-  X  =  U. K   &    |-  Y  =  U. L   &    |-  J  =  (
 Xt_ `  F )   &    |-  K  =  ( Xt_ `  ( F  |`  A ) )   &    |-  L  =  ( Xt_ `  ( F  |`  B ) )   &    |-  G  =  ( x  e.  X ,  y  e.  Y  |->  ( x  u.  y ) )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  F : C --> Top )   &    |-  ( ph  ->  C  =  ( A  u.  B ) )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   =>    |-  ( ph  ->  G  e.  ( ( K  tX  L ) Homeo J ) )
 
Theoremxpstopnlem1 20822* The function  F used in xpsval 15477 is a homeomorphism from the binary product topology to the indexed product topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  F  =  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } )
 )   &    |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   =>    |-  ( ph  ->  F  e.  ( ( J  tX  K ) Homeo ( Xt_ `  `' ( { J }  +c  { K } )
 ) ) )
 
Theoremxpstps 20823 A binary product of topologies is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  T  =  ( R  X.s  S )   =>    |-  ( ( R  e.  TopSp  /\  S  e.  TopSp )  ->  T  e.  TopSp )
 
Theoremxpstopnlem2 20824* Lemma for xpstopn 20825. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  J  =  (
 TopOpen `  R )   &    |-  K  =  ( TopOpen `  S )   &    |-  O  =  ( TopOpen `  T )   &    |-  X  =  ( Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  F  =  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } )
 )   =>    |-  ( ( R  e.  TopSp  /\  S  e.  TopSp )  ->  O  =  ( J  tX  K ) )
 
Theoremxpstopn 20825 The topology on a binary product of topological spaces, as we have defined it (transferring the indexed product topology on functions on  { (/) ,  1o } to  ( X  X.  Y
) by the canonical bijection), coincides with the usual topological product (generated by a base of rectangles). (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  J  =  (
 TopOpen `  R )   &    |-  K  =  ( TopOpen `  S )   &    |-  O  =  ( TopOpen `  T )   =>    |-  (
 ( R  e.  TopSp  /\  S  e.  TopSp )  ->  O  =  ( J  tX  K ) )
 
Theoremptcmpfi 20826 A topological product of finitely many compact spaces is compact. This weak version of Tychonoff's theorem does not require the axiom of choice. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  ( ( A  e.  Fin  /\  F : A --> Comp )  ->  ( Xt_ `  F )  e.  Comp )
 
Theoremxkocnv 20827* The inverse of the "currying" function  F is the uncurrying function. (Contributed by Mario Carneiro, 13-Apr-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  F  =  ( f  e.  ( ( J 
 tX  K )  Cn  L )  |->  ( x  e.  X  |->  ( y  e.  Y  |->  ( x f y ) ) ) )   &    |-  ( ph  ->  J  e. 𝑛Locally 
 Comp )   &    |-  ( ph  ->  K  e. 𝑛Locally 
 Comp )   &    |-  ( ph  ->  L  e.  Top )   =>    |-  ( ph  ->  `' F  =  ( g  e.  ( J  Cn  ( L  ^ko  K ) )  |->  ( x  e.  X ,  y  e.  Y  |->  ( ( g `  x ) `
  y ) ) ) )
 
Theoremxkohmeo 20828* The Exponential Law for topological spaces. The "currying" function  F is a homeomorphism on function spaces when  J and  K are exponentiable spaces (by xkococn 20673, it is sufficient to assume that  J ,  K are locally compact to ensure exponentiability). (Contributed by Mario Carneiro, 13-Apr-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  F  =  ( f  e.  ( ( J 
 tX  K )  Cn  L )  |->  ( x  e.  X  |->  ( y  e.  Y  |->  ( x f y ) ) ) )   &    |-  ( ph  ->  J  e. 𝑛Locally 
 Comp )   &    |-  ( ph  ->  K  e. 𝑛Locally 
 Comp )   &    |-  ( ph  ->  L  e.  Top )   =>    |-  ( ph  ->  F  e.  ( ( L 
 ^ko  ( J  tX  K )
 ) Homeo ( ( L 
 ^ko  K )  ^ko  J ) ) )
 
Theoremqtopf1 20829 If a quotient map is injective, then it is a homeomorphism. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  F : X -1-1-> Y )   =>    |-  ( ph  ->  F  e.  ( J Homeo ( J qTop 
 F ) ) )
 
Theoremqtophmeo 20830* If two functions on a base topology 
J make the same identifications in order to create quotient spaces  J qTop  F and  J qTop  G, then not only are  J qTop  F and  J qTop  G homeomorphic, but there is a unique homeomorphism that makes the diagram commute. (Contributed by Mario Carneiro, 24-Mar-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  F : X -onto-> Y )   &    |-  ( ph  ->  G : X -onto-> Y )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X )
 )  ->  ( ( F `  x )  =  ( F `  y
 ) 
 <->  ( G `  x )  =  ( G `  y ) ) )   =>    |-  ( ph  ->  E! f  e.  ( ( J qTop  F ) Homeo ( J qTop  G ) ) G  =  ( f  o.  F ) )
 
Theoremt0kq 20831* A topological space is T0 iff the quotient map is a homeomorphism onto the space's Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )   =>    |-  ( J  e.  (TopOn `  X )  ->  ( J  e.  Kol2  <->  F  e.  ( J Homeo (KQ `  J ) ) ) )
 
Theoremkqhmph 20832 A topological space is T0 iff it is homeomorphic to its Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  e.  Kol2  <->  J  ~=  (KQ `  J )
 )
 
Theoremist1-5lem 20833 Lemma for ist1-5 20835 and similar theorems. If  A is a topological property which implies T0, such as T1 or T2, the property can be "decomposed" into T0 and a non-T0 version of property  A (which is defined as stating that the Kolmogorov quotient of the space has property  A). For example, if  A is T1, then the theorem states that a space is T1 iff it is T0 and its Kolmogorov quotient is T1 (we call this property R0). (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  e.  A  ->  J  e.  Kol2 )   &    |-  ( J  ~=  (KQ `  J )  ->  ( J  e.  A  ->  (KQ `  J )  e.  A )
 )   &    |-  ( (KQ `  J )  ~=  J  ->  (
 (KQ `  J )  e.  A  ->  J  e.  A ) )   =>    |-  ( J  e.  A 
 <->  ( J  e.  Kol2  /\  (KQ `  J )  e.  A ) )
 
Theoremt1r0 20834 A T1 space is R0. That is, the Kolmogorov quotient of a T1 space is also T1 (because they are homeomorphic). (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  e.  Fre  ->  (KQ `  J )  e. 
 Fre )
 
Theoremist1-5 20835 A topological space is T1 iff it is both T0 and R0. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  e.  Fre  <->  ( J  e.  Kol2  /\  (KQ `  J )  e.  Fre ) )
 
Theoremishaus3 20836 A topological space is Hausdorff iff it is both T0 and R1 (where R1 means that any two topologically distinct points are separated by neighborhoods). (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  e.  Haus  <->  ( J  e.  Kol2  /\  (KQ `  J )  e.  Haus ) )
 
Theoremnrmreg 20837 A normal T1 space is regular Hausdorff. In other words, a T4 space is T3 . One can get away with slightly weaker assumptions; see nrmr0reg 20762. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( ( J  e.  Nrm  /\  J  e.  Fre )  ->  J  e.  Reg )
 
Theoremreghaus 20838 A regular T0 space is Hausdorff. In other words, a T3 space is T2 . A regular Hausdorff or T0 space is also known as a T3 space. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( J  e.  Reg  ->  ( J  e.  Haus  <->  J  e.  Kol2 )
 )
 
Theoremnrmhaus 20839 A T1 normal space is Hausdorff. A Hausdorff or T1 normal space is also known as a T4 space. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( J  e.  Nrm  ->  ( J  e.  Haus  <->  J  e.  Fre ) )
 
12.2  Filters and filter bases
 
12.2.1  Filter bases
 
Theoremelmptrab 20840* Membership in a one-parameter class of sets. (Contributed by Stefan O'Rear, 28-Jul-2015.)
 |-  F  =  ( x  e.  D  |->  { y  e.  B  |  ph } )   &    |-  (
 ( x  =  X  /\  y  =  Y )  ->  ( ph  <->  ps ) )   &    |-  ( x  =  X  ->  B  =  C )   &    |-  ( x  e.  D  ->  B  e.  V )   =>    |-  ( Y  e.  ( F `  X )  <-> 
 ( X  e.  D  /\  Y  e.  C  /\  ps ) )
 
Theoremelmptrab2 20841* Membership in a one-parameter class of sets, indexed by arbitrary base sets. (Contributed by Stefan O'Rear, 28-Jul-2015.)
 |-  F  =  ( x  e.  _V  |->  { y  e.  B  |  ph } )   &    |-  (
 ( x  =  X  /\  y  =  Y )  ->  ( ph  <->  ps ) )   &    |-  ( x  =  X  ->  B  =  C )   &    |-  B  e.  V   &    |-  ( Y  e.  C  ->  X  e.  W )   =>    |-  ( Y  e.  ( F `  X )  <->  ( Y  e.  C  /\  ps ) )
 
Theoremisfbas 20842* The predicate " F is a filter base." Note that some authors require filter bases to be closed under pairwise intersections, but that is not necessary under our definition. One advantage of this definition is that tails in a directed set form a filter base under our meaning. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
 |-  ( B  e.  A  ->  ( F  e.  ( fBas `  B )  <->  ( F  C_  ~P B  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P ( x  i^i  y ) )  =/=  (/) ) ) ) )
 
Theoremfbasne0 20843 There are no empty filter bases. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
 |-  ( F  e.  ( fBas `  B )  ->  F  =/=  (/) )
 
Theorem0nelfb 20844 No filter base contains the empty set. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
 |-  ( F  e.  ( fBas `  B )  ->  -.  (/)  e.  F )
 
Theoremfbsspw 20845 A filter base on a set is a subset of the power set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
 |-  ( F  e.  ( fBas `  B )  ->  F  C_  ~P B )
 
Theoremfbelss 20846 An element of the filter base is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
 |-  ( ( F  e.  ( fBas `  B )  /\  X  e.  F ) 
 ->  X  C_  B )
 
Theoremfbdmn0 20847 The domain of a filter base is nonempty. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( F  e.  ( fBas `  B )  ->  B  =/=  (/) )
 
Theoremisfbas2 20848* The predicate " F is a filter base." (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( B  e.  A  ->  ( F  e.  ( fBas `  B )  <->  ( F  C_  ~P B  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  E. z  e.  F  z  C_  ( x  i^i  y
 ) ) ) ) )
 
Theoremfbasssin 20849* A filter base contains subsets of its pairwise intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Jeff Hankins, 1-Dec-2010.)
 |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F  /\  B  e.  F )  ->  E. x  e.  F  x  C_  ( A  i^i  B ) )
 
Theoremfbssfi 20850* A filter base contains subsets of its finite intersections. (Contributed by Mario Carneiro, 26-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  ( fi
 `  F ) ) 
 ->  E. x  e.  F  x  C_  A )
 
Theoremfbssint 20851* A filter base contains subsets of its finite intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( ( F  e.  ( fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  ->  E. x  e.  F  x  C_  |^| A )
 
Theoremfbncp 20852 A filter base does not contain complements of its elements. (Contributed by Mario Carneiro, 26-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F ) 
 ->  -.  ( B  \  A )  e.  F )
 
Theoremfbun 20853* A necessary and sufficient condition for the union of two filter bases to also be a filter base. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( fBas `  X )  /\  G  e.  ( fBas `  X ) )  ->  ( ( F  u.  G )  e.  ( fBas `  X )  <->  A. x  e.  F  A. y  e.  G  E. z  e.  ( F  u.  G ) z  C_  ( x  i^i  y ) ) )
 
Theoremfbfinnfr 20854 No filter base containing a finite element is free. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( ( F  e.  ( fBas `  B )  /\  S  e.  F  /\  S  e.  Fin )  ->  |^| F  =/=  (/) )
 
Theoremopnfbas 20855* The collection of open supersets of a nonempty set in a topology is a neighborhoods of the set, one of the motivations for the filter concept. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  { x  e.  J  |  S  C_  x }  e.  ( fBas `  X ) )
 
Theoremtrfbas2 20856 Conditions for the trace of a filter base  F to be a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  ( ( F  e.  ( fBas `  Y )  /\  A  C_  Y )  ->  ( ( Ft  A )  e.  ( fBas `  A ) 
 <->  -.  (/)  e.  ( Ft  A ) ) )
 
Theoremtrfbas 20857* Conditions for the trace of a filter base  F to be a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  ( ( F  e.  ( fBas `  Y )  /\  A  C_  Y )  ->  ( ( Ft  A )  e.  ( fBas `  A ) 
 <-> 
 A. v  e.  F  ( v  i^i  A )  =/=  (/) ) )
 
12.2.2  Filters
 
Syntaxcfil 20858 Extend class notation with the set of filters on a set.
 class  Fil
 
Definitiondf-fil 20859* The set of filters on a set. Definition 1 (axioms FI, FIIa, FIIb, FIII) of [BourbakiTop1] p. I.36. Filters are used to define the concept of limit in the general case. They are a generalization of the idea of neighborhoods. Suppose you are in  RR. With neighborhoods you can express the idea of a variable that tends to a specific number but you can't express the idea of a variable that tends to infinity. Filters relax the "axioms" of neighborhoods and then succeed in expressing the idea of something that tends to infinity. Filters were invented by Cartan in 1937 and made famous by Bourbaki in his treatise. A notion similar to the notion of filter is the concept of net invented by Moore and Smith in 1922. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |- 
 Fil  =  ( z  e.  _V  |->  { f  e.  ( fBas `  z )  | 
 A. x  e.  ~P  z ( ( f  i^i  ~P x )  =/=  (/)  ->  x  e.  f ) } )
 
Theoremisfil 20860* The predicate "is a filter." (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.)
 |-  ( F  e.  ( Fil `  X )  <->  ( F  e.  ( fBas `  X )  /\  A. x  e.  ~P  X ( ( F  i^i  ~P x )  =/=  (/)  ->  x  e.  F ) ) )
 
Theoremfilfbas 20861 A filter is a filter base. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  F  e.  ( fBas `  X ) )
 
Theorem0nelfil 20862 The empty set doesn't belong to a filter. (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  -.  (/)  e.  F )
 
Theoremfileln0 20863 An element of a filter is nonempty. (Contributed by FL, 24-May-2011.) (Revised by Mario Carneiro, 28-Jul-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F ) 
 ->  A  =/=  (/) )
 
Theoremfilsspw 20864 A filter is a subset of the power set of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  F  C_  ~P X )
 
Theoremfilelss 20865 An element of a filter is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F ) 
 ->  A  C_  X )
 
Theoremfilss 20866 A filter is closed under taking supersets. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) ) 
 ->  B  e.  F )
 
Theoremfilin 20867 A filter is closed under taking intersections. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F  /\  B  e.  F )  ->  ( A  i^i  B )  e.  F )
 
Theoremfiltop 20868 The underlying set belongs to the filter. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  X  e.  F )
 
Theoremisfil2 20869* Derive the standard axioms of a filter. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( Fil `  X )  <->  ( ( F 
 C_  ~P X  /\  -.  (/) 
 e.  F  /\  X  e.  F )  /\  A. x  e.  ~P  X ( E. y  e.  F  y  C_  x  ->  x  e.  F )  /\  A. x  e.  F  A. y  e.  F  ( x  i^i  y )  e.  F ) )
 
Theoremisfildlem 20870* Lemma for isfild 20871. (Contributed by Mario Carneiro, 1-Dec-2013.)
 |-  ( ph  ->  ( x  e.  F  <->  ( x  C_  A  /\  ps ) ) )   &    |-  ( ph  ->  A  e.  _V )   =>    |-  ( ph  ->  ( B  e.  F  <->  ( B  C_  A  /\  [. B  /  x ].
 ps ) ) )
 
Theoremisfild 20871* Sufficient condition for a set of the form  { x  e.  ~P A  |  ph } to be a filter. (Contributed by Mario Carneiro, 1-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ph  ->  ( x  e.  F  <->  ( x  C_  A  /\  ps ) ) )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  [. A  /  x ].
 ps )   &    |-  ( ph  ->  -.  [. (/)  /  x ]. ps )   &    |-  ( ( ph  /\  y  C_  A  /\  z  C_  y )  ->  ( [. z  /  x ].
 ps  ->  [. y  /  x ].
 ps ) )   &    |-  (
 ( ph  /\  y  C_  A  /\  z  C_  A )  ->  ( ( [. y  /  x ]. ps  /\  [. z  /  x ].
 ps )  ->  [. (
 y  i^i  z )  /  x ]. ps )
 )   =>    |-  ( ph  ->  F  e.  ( Fil `  A ) )
 
Theoremfilfi 20872 A filter is closed under taking intersections. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  ( fi `  F )  =  F )
 
Theoremfilinn0 20873 The intersection of two elements of a filter can't be empty. (Contributed by FL, 16-Sep-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F  /\  B  e.  F )  ->  ( A  i^i  B )  =/=  (/) )
 
Theoremfilintn0 20874 A filter has the finite intersection property. Remark below definition 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 20-Sep-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  ( A  C_  F  /\  A  =/=  (/)  /\  A  e.  Fin ) )  ->  |^| A  =/=  (/) )
 
Theoremfiln0 20875 The empty set is not a filter. Remark below def. 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 30-Oct-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  F  =/=  (/) )
 
Theoreminfil 20876 The intersection of two filters is a filter. Use fiint 7857 to extend this property to the intersection of a finite set of filters. Paragraph 3 of [BourbakiTop1] p. I.36. (Contributed by FL, 17-Sep-2007.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  G  e.  ( Fil `  X ) )  ->  ( F  i^i  G )  e.  ( Fil `  X ) )
 
Theoremsnfil 20877 A singleton is a filter. Example 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 16-Sep-2007.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  { A }  e.  ( Fil `  A ) )
 
Theoremfbasweak 20878 A filter base on any set is also a filter base on any larger set. (Contributed by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( fBas `  X )  /\  F  C_  ~P Y  /\  Y  e.  V ) 
 ->  F  e.  ( fBas `  Y ) )
 
Theoremsnfbas 20879 Condition for a singleton to be a filter base. (Contributed by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  { A }  e.  ( fBas `  B ) )
 
Theoremfsubbas 20880 A condition for a set to generate a filter base. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( X  e.  V  ->  ( ( fi `  A )  e.  ( fBas `  X )  <->  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/)  e.  ( fi
 `  A ) ) ) )
 
Theoremfbasfip 20881 A filter base has the finite intersection property. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( fBas `  X )  ->  -.  (/)  e.  ( fi
 `  F ) )
 
Theoremfbunfip 20882* A helpful lemma for showing that certain sets generate filters. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( fBas `  X )  /\  G  e.  ( fBas `  Y ) )  ->  ( -.  (/)  e.  ( fi
 `  ( F  u.  G ) )  <->  A. x  e.  F  A. y  e.  G  ( x  i^i  y )  =/=  (/) ) )
 
Theoremfgval 20883* The filter generating class gives a filter for every filter base. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( fBas `  X )  ->  ( X filGen F )  =  { x  e.  ~P X  |  ( F  i^i  ~P x )  =/=  (/) } )
 
Theoremelfg 20884* A condition for elements of a generated filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( fBas `  X )  ->  ( A  e.  ( X filGen F )  <->  ( A  C_  X  /\  E. x  e.  F  x  C_  A ) ) )
 
Theoremssfg 20885 A filter base is a subset of its generated filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( fBas `  X )  ->  F  C_  ( X filGen F ) )
 
Theoremfgss 20886 A bigger base generates a bigger filter. (Contributed by NM, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( fBas `  X )  /\  G  e.  ( fBas `  X )  /\  F  C_  G )  ->  ( X filGen F )  C_  ( X filGen G ) )
 
Theoremfgss2 20887* A condition for a filter to be finer than another involving their filter bases. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( fBas `  X )  /\  G  e.  ( fBas `  X ) )  ->  ( ( X filGen F )  C_  ( X filGen G )  <->  A. x  e.  F  E. y  e.  G  y  C_  x ) )
 
Theoremfgfil 20888 A filter generates itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  ( X filGen F )  =  F )
 
Theoremelfilss 20889* An element belongs to a filter iff any element below it does. (Contributed by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X )  ->  ( A  e.  F  <->  E. t  e.  F  t 
 C_  A ) )
 
Theoremfilfinnfr 20890 No filter containing a finite element is free. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  S  e.  F  /\  S  e.  Fin )  ->  |^| F  =/=  (/) )
 
Theoremfgcl 20891 A generated filter is a filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( fBas `  X )  ->  ( X filGen F )  e.  ( Fil `  X ) )
 
Theoremfgabs 20892 Absorption law for filter generation. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( ( F  e.  ( fBas `  Y )  /\  Y  C_  X )  ->  ( X filGen ( Y
 filGen F ) )  =  ( X filGen F ) )
 
Theoremneifil 20893 The neighborhoods of a nonempty set is a filter. Example 2 of [BourbakiTop1] p. I.36. (Contributed by FL, 18-Sep-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  S  =/= 
 (/) )  ->  (
 ( nei `  J ) `  S )  e.  ( Fil `  X ) )
 
Theoremfilunibas 20894 Recover the base set from a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  U. F  =  X )
 
Theoremfilunirn 20895 Two ways to express a filter on an unspecified base. (Contributed by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  U. ran  Fil  <->  F  e.  ( Fil `  U. F ) )
 
Theoremfilcon 20896 A filter gives rise to a connected topology. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  ( F  u.  { (/) } )  e.  Con )
 
Theoremfbasrn 20897* Given a filter on a domain, produce a filter on the range. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  C  =  ran  ( x  e.  B  |->  ( F
 " x ) )   =>    |-  ( ( B  e.  ( fBas `  X )  /\  F : X --> Y  /\  Y  e.  V )  ->  C  e.  ( fBas `  Y ) )
 
Theoremfiluni 20898* The union of a nonempty set of filters with a common base and closed under pairwise union is a filter. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  C_  ( Fil `  X )  /\  F  =/=  (/)  /\  A. f  e.  F  A. g  e.  F  ( f  u.  g )  e.  F )  ->  U. F  e.  ( Fil `  X ) )
 
Theoremtrfil1 20899 Conditions for the trace of a filter  L to be a filter. (Contributed by FL, 2-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( L  e.  ( Fil `  Y )  /\  A  C_  Y )  ->  A  =  U. ( Lt  A ) )
 
Theoremtrfil2 20900* Conditions for the trace of a filter  L to be a filter. (Contributed by FL, 2-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( L  e.  ( Fil `  Y )  /\  A  C_  Y )  ->  ( ( Lt  A )  e.  ( Fil `  A ) 
 <-> 
 A. v  e.  L  ( v  i^i  A )  =/=  (/) ) )
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