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Theorem List for Metamath Proof Explorer - 17301-17400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhmphtop 17301 Reverse closure for the homeomorphic predicate. (Contributed by Mario Carneiro, 22-Aug-2015.)
 |-  ( J  ~=  K  ->  ( J  e.  Top  /\  K  e.  Top )
 )
 
Theoremhmphtop1 17302 The relation "being homeomorph to" implies the operands are topologies. (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
 |-  ( J  ~=  K  ->  J  e.  Top )
 
Theoremhmphtop2 17303 The relation "being homeomorph to" implies the operands are topologies. (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
 |-  ( J  ~=  K  ->  K  e.  Top )
 
Theoremhmphref 17304 "Is homeomorph to" is reflexive. (Contributed by FL, 25-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
 |-  ( J  e.  Top  ->  J  ~=  J )
 
Theoremhmphsym 17305 "Is homeomorph to" is symmetric. (Contributed by FL, 8-Mar-2007.) (Proof shortened by Mario Carneiro, 30-May-2014.)
 |-  ( J  ~=  K  ->  K  ~=  J )
 
Theoremhmphtr 17306 "Is homeomorph to" is transitive. (Contributed by FL, 9-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( J  ~=  K  /\  K  ~=  L )  ->  J  ~=  L )
 
Theoremhmpher 17307 "Is homeomorph to" is an equivalence relation. (Contributed by FL, 22-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
 |- 
 ~=  Er  Top
 
Theoremhmphen 17308 Homeomorphisms preserve the cardinality of the topologies. (Contributed by FL, 1-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  ( J  ~=  K  ->  J  ~~  K )
 
Theoremhmphsymb 17309 "Is homeomorph to" is symmetric. (Contributed by FL, 22-Feb-2007.)
 |-  ( J  ~=  K  <->  K 
 ~=  J )
 
Theoremhaushmphlem 17310* Lemma for haushmph 17315 and similar theorems. If the topological property  A is preserved under injective preimages, then property  A is preserved under homeomorphisms. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  e.  A  ->  J  e.  Top )   &    |-  (
 ( J  e.  A  /\  f : U. K -1-1-> U. J  /\  f  e.  ( K  Cn  J ) )  ->  K  e.  A )   =>    |-  ( J  ~=  K  ->  ( J  e.  A  ->  K  e.  A ) )
 
Theoremcmphmph 17311 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 17312 Connectedness is a topological property. (Contributed by Jeff Hankins, 3-Jul-2009.)
 |-  ( J  ~=  K  ->  ( J  e.  Con  ->  K  e.  Con ) )
 
Theoremt0hmph 17313 T0 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  ~=  K  ->  ( J  e.  Kol2  ->  K  e.  Kol2 ) )
 
Theoremt1hmph 17314 T1 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  ~=  K  ->  ( J  e.  Fre  ->  K  e.  Fre ) )
 
Theoremhaushmph 17315 Hausdorff-ness is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  ~=  K  ->  ( J  e.  Haus  ->  K  e.  Haus ) )
 
Theoremreghmph 17316 Regularity is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  ~=  K  ->  ( J  e.  Reg  ->  K  e.  Reg ) )
 
Theoremnrmhmph 17317 Normality is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  ~=  K  ->  ( J  e.  Nrm  ->  K  e.  Nrm ) )
 
Theoremhmph0 17318 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 17319 Homeomorphisms preserve topological discretion. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  X  =  U. J   =>    |-  ( J  ~=  ~P A  ->  J  =  ~P X )
 
Theoremhmphindis 17320 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 17321 Equinumerous sets equipped with their indiscrete topologies are homeomorph (which means in that particular case that a segment is homeomorph 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 17322 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 17323 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 17324 Lemma for ordthmeo 17325. (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 17325 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 17326* 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 17327* Show inverse for the "swap components" operation on a cross 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 17328* 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 17329* 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 17330* 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 17331* 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 17332* The function  F used in xpsval 13348 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 17333 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 17334* Lemma for xpstopn 17335. (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 17335 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 17336 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 17337* 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  ^ k o  K ) )  |->  ( x  e.  X ,  y  e.  Y  |->  ( ( g `  x ) `
  y ) ) ) )
 
Theoremxkohmeo 17338* 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 17186, 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 
 ^ k o  ( J  tX  K )
 )  Homeo  ( ( L 
 ^ k o  K )  ^ k o  J ) ) )
 
Theoremqtopf1 17339 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 17340* 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 homeomorhism 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 17341* 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 17342 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 17343 Lemma for ist1-5 17345 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 17344 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 17345 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 17346 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 17347 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 17272. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( ( J  e.  Nrm  /\  J  e.  Fre )  ->  J  e.  Reg )
 
Theoremreghaus 17348 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 17349 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 ) )
 
11.2  Filters and filter bases
 
11.2.1  Filter Bases
 
Syntaxcfbas 17350 Extend class definition to include the class of filter bases.
 class  fBas
 
Syntaxcfg 17351 Extend class definition to include the filter generating function.
 class  filGen
 
Definitiondf-fbas 17352* Define the class of all filter bases. Note that a filter base on one set is also a filter base for any superset, so there is not a unique base set that can be recovered. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
 |- 
 fBas  =  ( w  e.  _V  |->  { x  e.  ~P ~P w  |  ( x  =/=  (/)  /\  (/)  e/  x  /\  A. y  e.  x  A. z  e.  x  ( x  i^i  ~P (
 y  i^i  z )
 )  =/=  (/) ) }
 )
 
Definitiondf-fg 17353* Define the filter generating function. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
 |-  filGen  =  ( w  e. 
 _V ,  x  e.  ( fBas `  w )  |->  { y  e.  ~P w  |  ( x  i^i  ~P y )  =/=  (/) } )
 
Theoremelmptrab 17354* 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 17355* 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 17356* 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 17357 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 17358 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 17359 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 17360 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 17361 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 17362* 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 17363* 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 17364* 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 17365* 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 17366 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 17367* 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 17368 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 17369* 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 17370 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 17371* 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 )  =/=  (/) ) )
 
11.2.2  Filters
 
Syntaxcfil 17372 Extend class notation with the set of filters on a set.
 class  Fil
 
Definitiondf-fil 17373* 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 17374* 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 17375 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 17376 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 17377 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 17378 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 17379 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 17380 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 17381 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 17382 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 17383* 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 17384* Lemma for isfild 17385. (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 17385* 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 17386 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 17387 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 17388 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 17389 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 17390 The intersection of two filters is a filter. Use fiint 7018 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 17391 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 17392 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 17393 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 17394 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 17395 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 17396* 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 17397* 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 17398* 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 17399 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 17400 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 ) )
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