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Theorem List for Metamath Proof Explorer - 17401-17500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhmeocnv 17401 The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( F  e.  ( J  Homeo  K )  ->  `' F  e.  ( K  Homeo  J ) )
 
Theoremhmeof1o2 17402 A homeomorphism is a 1-1-onto mapping. (Contributed by Mario Carneiro, 22-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Homeo  K ) ) 
 ->  F : X -1-1-onto-> Y )
 
Theoremhmeof1o 17403 A homeomorphism is a 1-1-onto mapping. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 30-May-2014.)
 |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  ( F  e.  ( J  Homeo  K )  ->  F : X -1-1-onto-> Y )
 
Theoremhmeoima 17404 The image of an open set by a homeomorphism is an open set. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( ( F  e.  ( J  Homeo  K ) 
 /\  A  e.  J )  ->  ( F " A )  e.  K )
 
Theoremhmeoopn 17405 Homeomorphisms preserve openness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( F  e.  ( J  Homeo  K )  /\  A  C_  X )  ->  ( A  e.  J  <->  ( F " A )  e.  K ) )
 
Theoremhmeocld 17406 Homeomorphisms preserve closedness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( F  e.  ( J  Homeo  K )  /\  A  C_  X )  ->  ( A  e.  ( Clsd `  J )  <->  ( F " A )  e.  ( Clsd `  K ) ) )
 
Theoremhmeocls 17407 Homeomorphisms preserve closures. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( F  e.  ( J  Homeo  K )  /\  A  C_  X )  ->  ( ( cls `  K ) `  ( F " A ) )  =  ( F " (
 ( cls `  J ) `  A ) ) )
 
Theoremhmeontr 17408 Homeomorphisms preserve interiors. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( F  e.  ( J  Homeo  K )  /\  A  C_  X )  ->  ( ( int `  K ) `  ( F " A ) )  =  ( F " (
 ( int `  J ) `  A ) ) )
 
Theoremhmeoimaf1o 17409* The function mapping open sets to their images under a homeomorphism is a bijection of topologies. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  G  =  ( x  e.  J  |->  ( F
 " x ) )   =>    |-  ( F  e.  ( J  Homeo  K )  ->  G : J -1-1-onto-> K )
 
Theoremhmeores 17410 The restriction of a homeomorphism is a homeomorphism. (Contributed by Mario Carneiro, 14-Sep-2014.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( F  e.  ( J  Homeo  K )  /\  Y  C_  X )  ->  ( F  |`  Y )  e.  ( ( Jt  Y )  Homeo  ( Kt  ( F
 " Y ) ) ) )
 
Theoremhmeoco 17411 The composite of two homeomorphisms is a homeomorphism. (Contributed by FL, 9-Mar-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( F  e.  ( J  Homeo  K ) 
 /\  G  e.  ( K  Homeo  L ) ) 
 ->  ( G  o.  F )  e.  ( J  Homeo  L ) )
 
Theoremidhmeo 17412 The identity function is a homeomorphism. (Contributed by FL, 14-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
 |-  ( J  e.  (TopOn `  X )  ->  (  _I  |`  X )  e.  ( J  Homeo  J ) )
 
Theoremhmeocnvb 17413 The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
 |-  ( Rel  F  ->  ( `' F  e.  ( J  Homeo  K )  <->  F  e.  ( K  Homeo  J ) ) )
 
Theoremhmeoqtop 17414 A homeomorphism is a quotient map. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( F  e.  ( J  Homeo  K )  ->  K  =  ( J qTop  F ) )
 
Theoremhmph 17415 Express the predicate  J is homeomorph to  K. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( J  ~=  K  <->  ( J  Homeo  K )  =/=  (/) )
 
Theoremhmphi 17416 If there is a homeomorphism between spaces, then the spaces are homeomorphic. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( F  e.  ( J  Homeo  K )  ->  J  ~=  K )
 
Theoremhmphtop 17417 Reverse closure for the homeomorphic predicate. (Contributed by Mario Carneiro, 22-Aug-2015.)
 |-  ( J  ~=  K  ->  ( J  e.  Top  /\  K  e.  Top )
 )
 
Theoremhmphtop1 17418 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 17419 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 17420 "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 17421 "Is homeomorph to" is symmetric. (Contributed by FL, 8-Mar-2007.) (Proof shortened by Mario Carneiro, 30-May-2014.)
 |-  ( J  ~=  K  ->  K  ~=  J )
 
Theoremhmphtr 17422 "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 17423 "Is homeomorph to" is an equivalence relation. (Contributed by FL, 22-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
 |- 
 ~=  Er  Top
 
Theoremhmphen 17424 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 17425 "Is homeomorph to" is symmetric. (Contributed by FL, 22-Feb-2007.)
 |-  ( J  ~=  K  <->  K 
 ~=  J )
 
Theoremhaushmphlem 17426* Lemma for haushmph 17431 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 17427 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 17428 Connectedness is a topological property. (Contributed by Jeff Hankins, 3-Jul-2009.)
 |-  ( J  ~=  K  ->  ( J  e.  Con  ->  K  e.  Con ) )
 
Theoremt0hmph 17429 T0 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  ~=  K  ->  ( J  e.  Kol2  ->  K  e.  Kol2 ) )
 
Theoremt1hmph 17430 T1 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  ~=  K  ->  ( J  e.  Fre  ->  K  e.  Fre ) )
 
Theoremhaushmph 17431 Hausdorff-ness is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  ~=  K  ->  ( J  e.  Haus  ->  K  e.  Haus ) )
 
Theoremreghmph 17432 Regularity is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  ~=  K  ->  ( J  e.  Reg  ->  K  e.  Reg ) )
 
Theoremnrmhmph 17433 Normality is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  ~=  K  ->  ( J  e.  Nrm  ->  K  e.  Nrm ) )
 
Theoremhmph0 17434 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 17435 Homeomorphisms preserve topological discretion. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  X  =  U. J   =>    |-  ( J  ~=  ~P A  ->  J  =  ~P X )
 
Theoremhmphindis 17436 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 17437 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 17438 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 17439 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 17440 Lemma for ordthmeo 17441. (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 17441 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 17442* 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 17443* 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 17444* 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 17445* 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 17446* 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 17447* 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 17448* The function  F used in xpsval 13422 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 17449 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 17450* Lemma for xpstopn 17451. (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 17451 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 17452 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 17453* 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 17454* 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 17302, 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 17455 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 17456* 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 17457* 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 17458 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 17459 Lemma for ist1-5 17461 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 17460 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 17461 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 17462 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 17463 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 17388. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( ( J  e.  Nrm  /\  J  e.  Fre )  ->  J  e.  Reg )
 
Theoremreghaus 17464 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 17465 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 17466 Extend class definition to include the class of filter bases.
 class  fBas
 
Syntaxcfg 17467 Extend class definition to include the filter generating function.
 class  filGen
 
Definitiondf-fbas 17468* 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 17469* 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 17470* 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 17471* 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 17472* 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 17473 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 17474 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 17475 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 17476 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 17477 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 17478* 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 17479* 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 17480* 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 17481* 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 17482 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 17483* 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 17484 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 17485* 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 17486 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 17487* 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 17488 Extend class notation with the set of filters on a set.
 class  Fil
 
Definitiondf-fil 17489* 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 17490* 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 17491 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 17492 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 17493 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 17494 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 17495 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 17496 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 17497 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 17498 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 17499* 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 17500* Lemma for isfild 17501. (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 ) ) )
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