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Theorem List for Intuitionistic Logic Explorer - 12101-12200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtopgrpplusgd 12101 The additive operation of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
 |-  W  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. }   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  .+  e.  W )   &    |-  ( ph  ->  J  e.  X )   =>    |-  ( ph  ->  .+  =  ( +g  `  W )
 )
 
Theoremtopgrptsetd 12102 The topology of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
 |-  W  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. }   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  .+  e.  W )   &    |-  ( ph  ->  J  e.  X )   =>    |-  ( ph  ->  J  =  (TopSet `  W )
 )
 
Theoremplendx 12103 Index value of the df-ple 12030 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.)
 |-  ( le `  ndx )  = ; 1 0
 
Theorempleid 12104 Utility theorem: self-referencing, index-independent form of df-ple 12030. (Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.)
 |- 
 le  = Slot  ( le ` 
 ndx )
 
Theorempleslid 12105 Slot property of  le. (Contributed by Jim Kingdon, 9-Feb-2023.)
 |-  ( le  = Slot  ( le `  ndx )  /\  ( le `  ndx )  e.  NN )
 
Theoremdsndx 12106 Index value of the df-ds 12032 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( dist `  ndx )  = ; 1
 2
 
Theoremdsid 12107 Utility theorem: index-independent form of df-ds 12032. (Contributed by Mario Carneiro, 23-Dec-2013.)
 |- 
 dist  = Slot  ( dist `  ndx )
 
Theoremdsslid 12108 Slot property of  dist. (Contributed by Jim Kingdon, 6-May-2023.)
 |-  ( dist  = Slot  ( dist ` 
 ndx )  /\  ( dist `  ndx )  e. 
 NN )
 
6.1.3  Definition of the structure product
 
Syntaxcrest 12109 Extend class notation with the function returning a subspace topology.
 classt
 
Syntaxctopn 12110 Extend class notation with the topology extractor function.
 class  TopOpen
 
Definitiondf-rest 12111* Function returning the subspace topology induced by the topology  y and the set  x. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.)
 |-t  =  ( j  e.  _V ,  x  e.  _V  |->  ran  ( y  e.  j  |->  ( y  i^i  x ) ) )
 
Definitiondf-topn 12112 Define the topology extractor function. This differs from df-tset 12029 when a structure has been restricted using df-ress 11956; in this case the TopSet component will still have a topology over the larger set, and this function fixes this by restricting the topology as well. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  TopOpen  =  ( w  e. 
 _V  |->  ( (TopSet `  w )t  ( Base `  w )
 ) )
 
Theoremrestfn 12113 The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.)
 |-t  Fn  ( _V  X.  _V )
 
Theoremtopnfn 12114 The topology extractor function is a function on the universe. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  TopOpen 
 Fn  _V
 
Theoremrestval 12115* The subspace topology induced by the topology  J on the set  A. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.)
 |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( Jt  A )  =  ran  ( x  e.  J  |->  ( x  i^i  A ) ) )
 
Theoremelrest 12116* The predicate "is an open set of a subspace topology". (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  ( ( J  e.  V  /\  B  e.  W )  ->  ( A  e.  ( Jt  B )  <->  E. x  e.  J  A  =  ( x  i^i  B ) ) )
 
Theoremelrestr 12117 Sufficient condition for being an open set in a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  ( ( J  e.  V  /\  S  e.  W  /\  A  e.  J ) 
 ->  ( A  i^i  S )  e.  ( Jt  S ) )
 
Theoremrestid2 12118 The subspace topology over a subset of the base set is the original topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ( Jt  A )  =  J )
 
Theoremrestsspw 12119 The subspace topology is a collection of subsets of the restriction set. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( Jt  A )  C_  ~P A
 
Theoremrestid 12120 The subspace topology of the base set is the original topology. (Contributed by Jeff Hankins, 9-Jul-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( J  e.  V  ->  ( Jt  X )  =  J )
 
Theoremtopnvalg 12121 Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by Jim Kingdon, 11-Feb-2023.)
 |-  B  =  ( Base `  W )   &    |-  J  =  (TopSet `  W )   =>    |-  ( W  e.  V  ->  ( Jt  B )  =  (
 TopOpen `  W ) )
 
Theoremtopnidg 12122 Value of the topology extractor function when the topology is defined over the same set as the base. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  B  =  ( Base `  W )   &    |-  J  =  (TopSet `  W )   =>    |-  ( ( W  e.  V  /\  J  C_  ~P B )  ->  J  =  (
 TopOpen `  W ) )
 
Theoremtopnpropgd 12123 The topology extractor function depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Jim Kingdon, 13-Feb-2023.)
 |-  ( ph  ->  ( Base `  K )  =  ( Base `  L )
 )   &    |-  ( ph  ->  (TopSet `  K )  =  (TopSet `  L ) )   &    |-  ( ph  ->  K  e.  V )   &    |-  ( ph  ->  L  e.  W )   =>    |-  ( ph  ->  ( TopOpen `  K )  =  (
 TopOpen `  L ) )
 
Syntaxctg 12124 Extend class notation with a function that converts a basis to its corresponding topology.
 class  topGen
 
Syntaxcpt 12125 Extend class notation with a function whose value is a product topology.
 class  Xt_
 
Syntaxc0g 12126 Extend class notation with group identity element.
 class  0g
 
Syntaxcgsu 12127 Extend class notation to include finitely supported group sums.
 class  gsumg
 
Definitiondf-0g 12128* Define group identity element. Remark: this definition is required here because the symbol  0g is already used in df-gsum 12129. The related theorems will be provided later. (Contributed by NM, 20-Aug-2011.)
 |- 
 0g  =  ( g  e.  _V  |->  ( iota
 e ( e  e.  ( Base `  g )  /\  A. x  e.  ( Base `  g ) ( ( e ( +g  `  g ) x )  =  x  /\  ( x ( +g  `  g
 ) e )  =  x ) ) ) )
 
Definitiondf-gsum 12129* Define the group sum for the structure  G of a finite sequence of elements whose values are defined by the expression  B and whose set of indices is  A. It may be viewed as a product (if 
G is a multiplication), a sum (if 
G is an addition) or any other operation. The variable  k is normally a free variable in  B (i.e.,  B can be thought of as  B ( k )). The definition is meaningful in different contexts, depending on the size of the index set  A and each demanding different properties of  G.

1. If  A  =  (/) and  G has an identity element, then the sum equals this identity.

2. If  A  =  ( M ... N ) and 
G is any magma, then the sum is the sum of the elements, evaluated left-to-right, i.e.  ( B ( 1 )  +  B
( 2 ) )  +  B ( 3 ) etc.

3. If  A is a finite set (or is nonzero for finitely many indices) and  G is a commutative monoid, then the sum adds up these elements in some order, which is then uniquely defined.

4. If  A is an infinite set and  G is a Hausdorff topological group, then there is a meaningful sum, but  gsumg cannot handle this case. (Contributed by FL, 5-Sep-2010.) (Revised by FL, 17-Oct-2011.) (Revised by Mario Carneiro, 7-Dec-2014.)

 |- 
 gsumg  =  ( w  e.  _V ,  f  e.  _V  |->  [_
 { x  e.  ( Base `  w )  | 
 A. y  e.  ( Base `  w ) ( ( x ( +g  `  w ) y )  =  y  /\  (
 y ( +g  `  w ) x )  =  y ) }  /  o ]_ if ( ran  f  C_  o ,  ( 0g
 `  w ) ,  if ( dom  f  e.  ran  ... ,  ( iota
 x E. m E. n  e.  ( ZZ>= `  m ) ( dom  f  =  ( m
 ... n )  /\  x  =  (  seq m ( ( +g  `  w ) ,  f
 ) `  n )
 ) ) ,  ( iota x E. g [. ( `' f " ( _V  \  o ) )  /  y ]. ( g : ( 1 ... ( `  y ) ) -1-1-onto-> y  /\  x  =  (  seq 1 ( ( +g  `  w ) ,  (
 f  o.  g ) ) `  ( `  y
 ) ) ) ) ) ) )
 
Definitiondf-topgen 12130* Define a function that converts a basis to its corresponding topology. Equivalent to the definition of a topology generated by a basis in [Munkres] p. 78. (Contributed by NM, 16-Jul-2006.)
 |-  topGen  =  ( x  e. 
 _V  |->  { y  |  y 
 C_  U. ( x  i^i  ~P y ) } )
 
Definitiondf-pt 12131* Define the product topology on a collection of topologies. For convenience, it is defined on arbitrary collections of sets, expressed as a function from some index set to the subbases of each factor space. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |- 
 Xt_  =  ( f  e.  _V  |->  ( topGen `  { x  |  E. g ( ( g  Fn  dom  f  /\  A. y  e.  dom  f ( g `  y )  e.  (
 f `  y )  /\  E. z  e.  Fin  A. y  e.  ( dom  f  \  z ) ( g `  y
 )  =  U. (
 f `  y )
 )  /\  x  =  X_ y  e.  dom  f
 ( g `  y
 ) ) } )
 )
 
Syntaxcprds 12132 The function constructing structure products.
 class  X_s
 
Syntaxcpws 12133 The function constructing structure powers.
 class  ^s
 
Definitiondf-prds 12134* Define a structure product. This can be a product of groups, rings, modules, or ordered topological fields; any unused components will have garbage in them but this is usually not relevant for the purpose of inheriting the structures present in the factors. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.)
 |-  X_s  =  ( s  e.  _V ,  r  e.  _V  |->  [_ X_ x  e.  dom  r ( Base `  (
 r `  x )
 )  /  v ]_ [_ ( f  e.  v ,  g  e.  v  |->  X_ x  e.  dom  r ( ( f `
  x ) ( Hom  `  ( r `  x ) ) ( g `  x ) ) )  /  h ]_ ( ( { <. (
 Base `  ndx ) ,  v >. ,  <. ( +g  ` 
 ndx ) ,  (
 f  e.  v ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( ( f `  x ) ( +g  `  ( r `  x ) ) ( g `
  x ) ) ) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( ( f `
  x ) ( .r `  ( r `
  x ) ) ( g `  x ) ) ) )
 >. }  u.  { <. (Scalar `  ndx ) ,  s >. ,  <. ( .s `  ndx ) ,  ( f  e.  ( Base `  s
 ) ,  g  e.  v  |->  ( x  e. 
 dom  r  |->  ( f ( .s `  (
 r `  x )
 ) ( g `  x ) ) ) ) >. ,  <. ( .i
 `  ndx ) ,  (
 f  e.  v ,  g  e.  v  |->  ( s  gsumg  ( x  e.  dom  r  |->  ( ( f `
  x ) ( .i `  ( r `
  x ) ) ( g `  x ) ) ) ) ) >. } )  u.  ( { <. (TopSet `  ndx ) ,  ( Xt_ `  ( TopOpen  o.  r )
 ) >. ,  <. ( le ` 
 ndx ) ,  { <. f ,  g >.  |  ( { f ,  g }  C_  v  /\  A. x  e.  dom  r ( f `  x ) ( le `  ( r `  x ) ) ( g `
  x ) ) } >. ,  <. ( dist ` 
 ndx ) ,  (
 f  e.  v ,  g  e.  v  |->  sup ( ( ran  ( x  e.  dom  r  |->  ( ( f `  x ) ( dist `  (
 r `  x )
 ) ( g `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) >. }  u.  { <. ( Hom  `  ndx ) ,  h >. , 
 <. (comp `  ndx ) ,  ( a  e.  (
 v  X.  v ) ,  c  e.  v  |->  ( d  e.  (
 c h ( 2nd `  a ) ) ,  e  e.  ( h `
  a )  |->  ( x  e.  dom  r  |->  ( ( d `  x ) ( <. ( ( 1st `  a
 ) `  x ) ,  ( ( 2nd `  a
 ) `  x ) >. (comp `  ( r `  x ) ) ( c `  x ) ) ( e `  x ) ) ) ) ) >. } )
 ) )
 
Theoremreldmprds 12135 The structure product is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.)
 |- 
 Rel  dom  X_s
 
Definitiondf-pws 12136* Define a structure power, which is just a structure product where all the factors are the same. (Contributed by Mario Carneiro, 11-Jan-2015.)
 |- 
 ^s  =  ( r  e. 
 _V ,  i  e. 
 _V  |->  ( (Scalar `  r
 ) X_s ( i  X.  {
 r } ) ) )
 
6.2  The complex numbers as an algebraic extensible structure
 
6.2.1  Definition and basic properties
 
Syntaxcpsmet 12137 Extend class notation with the class of all pseudometric spaces.
 class PsMet
 
Syntaxcxmet 12138 Extend class notation with the class of all extended metric spaces.
 class  *Met
 
Syntaxcmet 12139 Extend class notation with the class of all metrics.
 class  Met
 
Syntaxcbl 12140 Extend class notation with the metric space ball function.
 class  ball
 
Syntaxcfbas 12141 Extend class definition to include the class of filter bases.
 class  fBas
 
Syntaxcfg 12142 Extend class definition to include the filter generating function.
 class  filGen
 
Syntaxcmopn 12143 Extend class notation with a function mapping each metric space to the family of its open sets.
 class  MetOpen
 
Syntaxcmetu 12144 Extend class notation with the function mapping metrics to the uniform structure generated by that metric.
 class metUnif
 
Definitiondf-psmet 12145* Define the set of all pseudometrics on a given base set. In a pseudo metric, two distinct points may have a distance zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |- PsMet  =  ( x  e.  _V  |->  { d  e.  ( RR*  ^m  ( x  X.  x ) )  |  A. y  e.  x  ( (
 y d y )  =  0  /\  A. z  e.  x  A. w  e.  x  (
 y d z ) 
 <_  ( ( w d y ) +e
 ( w d z ) ) ) }
 )
 
Definitiondf-xmet 12146* Define the set of all extended metrics on a given base set. The definition is similar to df-met 12147, but we also allow the metric to take on the value +oo. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |- 
 *Met  =  ( x  e.  _V  |->  { d  e.  ( RR*  ^m  ( x  X.  x ) )  |  A. y  e.  x  A. z  e.  x  ( ( ( y d z )  =  0  <->  y  =  z
 )  /\  A. w  e.  x  ( y d z )  <_  (
 ( w d y ) +e ( w d z ) ) ) } )
 
Definitiondf-met 12147* Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric. However, we will often also call the metric itself a "metric space".) Equivalent to Definition 14-1.1 of [Gleason] p. 223. (Contributed by NM, 25-Aug-2006.)
 |- 
 Met  =  ( x  e.  _V  |->  { d  e.  ( RR  ^m  ( x  X.  x ) )  | 
 A. y  e.  x  A. z  e.  x  ( ( ( y d z )  =  0  <-> 
 y  =  z ) 
 /\  A. w  e.  x  ( y d z )  <_  ( ( w d y )  +  ( w d z ) ) ) } )
 
Definitiondf-bl 12148* Define the metric space ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Thierry Arnoux, 11-Feb-2018.)
 |- 
 ball  =  ( d  e.  _V  |->  ( x  e. 
 dom  dom  d ,  z  e.  RR*  |->  { y  e.  dom  dom  d  |  ( x d y )  < 
 z } ) )
 
Definitiondf-mopn 12149 Define a function whose value is the family of open sets of a metric space. (Contributed by NM, 1-Sep-2006.)
 |-  MetOpen  =  ( d  e. 
 U. ran  *Met  |->  ( topGen `  ran  ( ball `  d ) ) )
 
Definitiondf-fbas 12150* 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 12151* 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 )  =/=  (/) } )
 
Definitiondf-metu 12152* Define the function mapping metrics to the uniform structure generated by that metric. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
 |- metUnif  =  ( d  e.  U. ran PsMet 
 |->  ( ( dom  dom  d  X.  dom  dom  d )
 filGen ran  ( a  e.  RR+  |->  ( `' d " ( 0 [,) a
 ) ) ) ) )
 
PART 7  BASIC TOPOLOGY
 
7.1  Topology
 
7.1.1  Topological spaces

A topology on a set is a set of subsets of that set, called open sets, which satisfy certain conditions. One condition is that the whole set be an open set. Therefore, a set is recoverable from a topology on it (as its union), and it may sometimes be more convenient to consider topologies without reference to the underlying set.

 
7.1.1.1  Topologies
 
Syntaxctop 12153 Syntax for the class of topologies.
 class  Top
 
Definitiondf-top 12154* Define the class of topologies. It is a proper class. See istopg 12155 and istopfin 12156 for the corresponding characterizations, using respectively binary intersections like in this definition and nonempty finite intersections.

The final form of the definition is due to Bourbaki (Def. 1 of [BourbakiTop1] p. I.1), while the idea of defining a topology in terms of its open sets is due to Aleksandrov. For the convoluted history of the definitions of these notions, see

Gregory H. Moore, The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica 35 (2008) 220--241.

(Contributed by NM, 3-Mar-2006.) (Revised by BJ, 20-Oct-2018.)

 |- 
 Top  =  { x  |  ( A. y  e. 
 ~P  x U. y  e.  x  /\  A. y  e.  x  A. z  e.  x  ( y  i^i  z )  e.  x ) }
 
Theoremistopg 12155* Express the predicate " J is a topology". See istopfin 12156 for another characterization using nonempty finite intersections instead of binary intersections.

Note: In the literature, a topology is often represented by a calligraphic letter T, which resembles the letter J. This confusion may have led to J being used by some authors (e.g., K. D. Joshi, Introduction to General Topology (1983), p. 114) and it is convenient for us since we later use  T to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

 |-  ( J  e.  A  ->  ( J  e.  Top  <->  ( A. x ( x  C_  J  ->  U. x  e.  J )  /\  A. x  e.  J  A. y  e.  J  ( x  i^i  y )  e.  J ) ) )
 
Theoremistopfin 12156* Express the predicate " J is a topology" using nonempty finite intersections instead of binary intersections as in istopg 12155. It is not clear we can prove the converse without adding additional conditions. (Contributed by NM, 19-Jul-2006.) (Revised by Jim Kingdon, 14-Jan-2023.)
 |-  ( J  e.  Top  ->  ( A. x ( x 
 C_  J  ->  U. x  e.  J )  /\  A. x ( ( x 
 C_  J  /\  x  =/= 
 (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) ) )
 
Theoremuniopn 12157 The union of a subset of a topology (that is, the union of any family of open sets of a topology) is an open set. (Contributed by Stefan Allan, 27-Feb-2006.)
 |-  ( ( J  e.  Top  /\  A  C_  J )  ->  U. A  e.  J )
 
Theoremiunopn 12158* The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)
 |-  ( ( J  e.  Top  /\  A. x  e.  A  B  e.  J )  -> 
 U_ x  e.  A  B  e.  J )
 
Theoreminopn 12159 The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
 |-  ( ( J  e.  Top  /\  A  e.  J  /\  B  e.  J )  ->  ( A  i^i  B )  e.  J )
 
Theoremfiinopn 12160 The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)
 |-  ( J  e.  Top  ->  ( ( A  C_  J  /\  A  =/=  (/)  /\  A  e.  Fin )  ->  |^| A  e.  J ) )
 
Theoremunopn 12161 The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( J  e.  Top  /\  A  e.  J  /\  B  e.  J )  ->  ( A  u.  B )  e.  J )
 
Theorem0opn 12162 The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.)
 |-  ( J  e.  Top  ->  (/) 
 e.  J )
 
Theorem0ntop 12163 The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)
 |- 
 -.  (/)  e.  Top
 
Theoremtopopn 12164 The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  X  e.  J )
 
Theoremeltopss 12165 A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  A  e.  J ) 
 ->  A  C_  X )
 
7.1.1.2  Topologies on sets
 
Syntaxctopon 12166 Syntax for the function of topologies on sets.
 class TopOn
 
Definitiondf-topon 12167* Define the function that associates with a set the set of topologies on it. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |- TopOn  =  ( b  e.  _V  |->  { j  e.  Top  |  b  =  U. j }
 )
 
Theoremfuntopon 12168 The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.)
 |- 
 Fun TopOn
 
Theoremistopon 12169 Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  ( J  e.  (TopOn `  B )  <->  ( J  e.  Top  /\  B  =  U. J ) )
 
Theoremtopontop 12170 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( J  e.  (TopOn `  B )  ->  J  e.  Top )
 
Theoremtoponuni 12171 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( J  e.  (TopOn `  B )  ->  B  =  U. J )
 
Theoremtopontopi 12172 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  J  e.  (TopOn `  B )   =>    |-  J  e.  Top
 
Theoremtoponunii 12173 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  J  e.  (TopOn `  B )   =>    |-  B  =  U. J
 
Theoremtoptopon 12174 Alternative definition of  Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
 
Theoremtoptopon2 12175 A topology is the same thing as a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)
 |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
 
Theoremtopontopon 12176 A topology on a set is a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)
 |-  ( J  e.  (TopOn `  X )  ->  J  e.  (TopOn `  U. J ) )
 
Theoremtoponrestid 12177 Given a topology on a set, restricting it to that same set has no effect. (Contributed by Jim Kingdon, 6-Jul-2022.)
 |-  A  e.  (TopOn `  B )   =>    |-  A  =  ( At  B )
 
Theoremtoponsspwpwg 12178 The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.) (Revised by Jim Kingdon, 16-Jan-2023.)
 |-  ( A  e.  V  ->  (TopOn `  A )  C_ 
 ~P ~P A )
 
Theoremdmtopon 12179 The domain of TopOn is  _V. (Contributed by BJ, 29-Apr-2021.)
 |- 
 dom TopOn  =  _V
 
Theoremfntopon 12180 The class TopOn is a function with domain  _V. (Contributed by BJ, 29-Apr-2021.)
 |- TopOn  Fn  _V
 
Theoremtoponmax 12181 The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( J  e.  (TopOn `  B )  ->  B  e.  J )
 
Theoremtoponss 12182 A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  J )  ->  A  C_  X )
 
Theoremtoponcom 12183 If  K is a topology on the base set of topology  J, then  J is a topology on the base of  K. (Contributed by Mario Carneiro, 22-Aug-2015.)
 |-  ( ( J  e.  Top  /\  K  e.  (TopOn `  U. J ) )  ->  J  e.  (TopOn `  U. K ) )
 
Theoremtoponcomb 12184 Biconditional form of toponcom 12183. (Contributed by BJ, 5-Dec-2021.)
 |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J  e.  (TopOn ` 
 U. K )  <->  K  e.  (TopOn ` 
 U. J ) ) )
 
Theoremtopgele 12185 The topologies over the same set have the greatest element (the discrete topology) and the least element (the indiscrete topology). (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 16-Sep-2015.)
 |-  ( J  e.  (TopOn `  X )  ->  ( { (/) ,  X }  C_  J  /\  J  C_  ~P X ) )
 
7.1.1.3  Topological spaces
 
Syntaxctps 12186 Syntax for the class of topological spaces.
 class  TopSp
 
Definitiondf-topsp 12187 Define the class of topological spaces (as extensible structures). (Contributed by Stefan O'Rear, 13-Aug-2015.)
 |- 
 TopSp  =  { f  |  ( TopOpen `  f )  e.  (TopOn `  ( Base `  f ) ) }
 
Theoremistps 12188 Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  A  =  ( Base `  K )   &    |-  J  =  (
 TopOpen `  K )   =>    |-  ( K  e.  TopSp  <->  J  e.  (TopOn `  A ) )
 
Theoremistps2 12189 Express the predicate "is a topological space." (Contributed by NM, 20-Oct-2012.)
 |-  A  =  ( Base `  K )   &    |-  J  =  (
 TopOpen `  K )   =>    |-  ( K  e.  TopSp  <->  ( J  e.  Top  /\  A  =  U. J ) )
 
Theoremtpsuni 12190 The base set of a topological space. (Contributed by FL, 27-Jun-2014.)
 |-  A  =  ( Base `  K )   &    |-  J  =  (
 TopOpen `  K )   =>    |-  ( K  e.  TopSp  ->  A  =  U. J )
 
Theoremtpstop 12191 The topology extractor on a topological space is a topology. (Contributed by FL, 27-Jun-2014.)
 |-  J  =  ( TopOpen `  K )   =>    |-  ( K  e.  TopSp  ->  J  e.  Top )
 
Theoremtpspropd 12192 A topological space depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  ( ph  ->  ( Base `  K )  =  ( Base `  L )
 )   &    |-  ( ph  ->  ( TopOpen `  K )  =  (
 TopOpen `  L ) )   =>    |-  ( ph  ->  ( K  e.  TopSp 
 <->  L  e.  TopSp ) )
 
Theoremtopontopn 12193 Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  A  =  ( Base `  K )   &    |-  J  =  (TopSet `  K )   =>    |-  ( J  e.  (TopOn `  A )  ->  J  =  ( TopOpen `  K )
 )
 
Theoremtsettps 12194 If the topology component is already correctly truncated, then it forms a topological space (with the topology extractor function coming out the same as the component). (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  A  =  ( Base `  K )   &    |-  J  =  (TopSet `  K )   =>    |-  ( J  e.  (TopOn `  A )  ->  K  e.  TopSp )
 
Theoremistpsi 12195 Properties that determine a topological space. (Contributed by NM, 20-Oct-2012.)
 |-  ( Base `  K )  =  A   &    |-  ( TopOpen `  K )  =  J   &    |-  A  =  U. J   &    |-  J  e.  Top   =>    |-  K  e.  TopSp
 
Theoremeltpsg 12196 Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  K  =  { <. (
 Base `  ndx ) ,  A >. ,  <. (TopSet `  ndx ) ,  J >. }   =>    |-  ( J  e.  (TopOn `  A )  ->  K  e.  TopSp )
 
Theoremeltpsi 12197 Properties that determine a topological space from a construction (using no explicit indices). (Contributed by NM, 20-Oct-2012.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  K  =  { <. (
 Base `  ndx ) ,  A >. ,  <. (TopSet `  ndx ) ,  J >. }   &    |-  A  =  U. J   &    |-  J  e.  Top   =>    |-  K  e.  TopSp
 
7.1.2  Topological bases
 
Syntaxctb 12198 Syntax for the class of topological bases.
 class  TopBases
 
Definitiondf-bases 12199* Define the class of topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 12201). Note that "bases" is the plural of "basis". (Contributed by NM, 17-Jul-2006.)
 |-  TopBases 
 =  { x  |  A. y  e.  x  A. z  e.  x  ( y  i^i  z ) 
 C_  U. ( x  i^i  ~P ( y  i^i  z
 ) ) }
 
Theoremisbasisg 12200* Express the predicate "the set 
B is a basis for a topology". (Contributed by NM, 17-Jul-2006.)
 |-  ( B  e.  C  ->  ( B  e.  TopBases  <->  A. x  e.  B  A. y  e.  B  ( x  i^i  y ) 
 C_  U. ( B  i^i  ~P ( x  i^i  y
 ) ) ) )
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