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Theorem List for Metamath Proof Explorer - 17001-17100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremindistpsALT 17001 The indiscrete topology on a set  A expressed as a topological space. Here we show how to derive the structural version indistps 16999 from the direct component assignment version indistps2 17000. (Contributed by NM, 24-Oct-2012.) (Proof modification is discouraged.)
 |-  A  e.  _V   &    |-  K  =  { <. ( Base `  ndx ) ,  A >. , 
 <. (TopSet `  ndx ) ,  { (/) ,  A } >. }   =>    |-  K  e.  TopSp
 
Theoremindistps2ALT 17002 The indiscrete topology on a set  A expressed as a topological space, using direct component assignments. Here we show how to derive the direct component assignment version indistps2 17000 from the structural version indistps 16999. (Contributed by NM, 24-Oct-2012.) (Proof modification is discouraged.)
 |-  ( Base `  K )  =  A   &    |-  ( TopOpen `  K )  =  { (/) ,  A }   =>    |-  K  e.  TopSp
 
Theoremdistps 17003 The discrete topology on a set  A expressed as a topological space. (Contributed by FL, 20-Aug-2006.)
 |-  A  e.  _V   &    |-  K  =  { <. ( Base `  ndx ) ,  A >. , 
 <. (TopSet `  ndx ) ,  ~P A >. }   =>    |-  K  e.  TopSp
 
11.1.4  Closure and interior
 
Syntaxccld 17004 Extend class notation with the set of closed sets of a topology.
 class  Clsd
 
Syntaxcnt 17005 Extend class notation with interior of a subset of a topology base set.
 class  int
 
Syntaxccl 17006 Extend class notation with closure of a subset of a topology base set.
 class  cls
 
Definitiondf-cld 17007* Define a function on topologies whose value is the set of closed sets of the topology. (Contributed by NM, 2-Oct-2006.)
 |- 
 Clsd  =  ( j  e.  Top  |->  { x  e.  ~P U. j  |  ( U. j  \  x )  e.  j } )
 
Definitiondf-ntr 17008* Define a function on topologies whose value is the interior function on the subsets of the base set. See ntrval 17024. (Contributed by NM, 10-Sep-2006.)
 |- 
 int  =  ( j  e.  Top  |->  ( x  e. 
 ~P U. j  |->  U. (
 j  i^i  ~P x ) ) )
 
Definitiondf-cls 17009* Define a function on topologies whose value is the closure function on the subsets of the base set. See clsval 17025. (Contributed by NM, 3-Oct-2006.)
 |- 
 cls  =  ( j  e.  Top  |->  ( x  e. 
 ~P U. j  |->  |^| { y  e.  ( Clsd `  j )  |  x  C_  y }
 ) )
 
Theoremfncld 17010 The closed-set generator is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |- 
 Clsd  Fn  Top
 
Theoremcldval 17011* The set of closed sets of a topology. (Note that the set of open sets is just the topology itself, so we don't have a separate definition.) (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  ( Clsd `  J )  =  { x  e.  ~P X  |  ( X  \  x )  e.  J } )
 
Theoremntrfval 17012* The interior function on the subsets of a topology's base set. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  ( int `  J )  =  ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) ) )
 
Theoremclsfval 17013* The closure function on the subsets of a topology's base set. (Contributed by NM, 3-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  ( cls `  J )  =  ( x  e.  ~P X  |->  |^| { y  e.  ( Clsd `  J )  |  x  C_  y }
 ) )
 
Theoremcldrcl 17014 Reverse closure of the closed set operation. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( C  e.  ( Clsd `  J )  ->  J  e.  Top )
 
Theoremiscld 17015 The predicate " S is a closed set." (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  ( S  e.  ( Clsd `  J )  <->  ( S  C_  X  /\  ( X  \  S )  e.  J ) ) )
 
Theoremiscld2 17016 A subset of the underlying set of a topology is closed iff its complement is open. (Contributed by NM, 4-Oct-2006.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( S  e.  ( Clsd `  J )  <->  ( X  \  S )  e.  J ) )
 
Theoremcldss 17017 A closed set is a subset of the underlying set of a topology. (Contributed by NM, 5-Oct-2006.) (Revised by Stefan O'Rear, 22-Feb-2015.)
 |-  X  =  U. J   =>    |-  ( S  e.  ( Clsd `  J )  ->  S  C_  X )
 
Theoremcldss2 17018 The set of closed sets is contained in the powerset of the base. (Contributed by Mario Carneiro, 6-Jan-2014.)
 |-  X  =  U. J   =>    |-  ( Clsd `  J )  C_  ~P X
 
Theoremcldopn 17019 The complement of a closed set is open. (Contributed by NM, 5-Oct-2006.) (Revised by Stefan O'Rear, 22-Feb-2015.)
 |-  X  =  U. J   =>    |-  ( S  e.  ( Clsd `  J )  ->  ( X  \  S )  e.  J )
 
Theoremisopn2 17020 A subset of the underlying set of a topology is open iff its complement is closed. (Contributed by NM, 4-Oct-2006.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( S  e.  J  <->  ( X  \  S )  e.  ( Clsd `  J ) ) )
 
Theoremopncld 17021 The complement of an open set is closed. (Contributed by NM, 6-Oct-2006.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  e.  J ) 
 ->  ( X  \  S )  e.  ( Clsd `  J ) )
 
Theoremdifopn 17022 The difference of a closed set with an open set is open. (Contributed by Mario Carneiro, 6-Jan-2014.)
 |-  X  =  U. J   =>    |-  (
 ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  ->  ( A  \  B )  e.  J )
 
Theoremtopcld 17023 The underlying set of a topology is closed. Part of Theorem 6.1(1) of [Munkres] p. 93. (Contributed by NM, 3-Oct-2006.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  X  e.  ( Clsd `  J )
 )
 
Theoremntrval 17024 The interior of a subset of a topology's base set is the union of all the open sets it includes. Definition of interior of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( ( int `  J ) `  S )  = 
 U. ( J  i^i  ~P S ) )
 
Theoremclsval 17025* The closure of a subset of a topology's base set is the intersection of all the closed sets that include it. Definition of closure of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( ( cls `  J ) `  S )  = 
 |^| { x  e.  ( Clsd `  J )  |  S  C_  x }
 )
 
Theorem0cld 17026 The empty set is closed. Part of Theorem 6.1(1) of [Munkres] p. 93. (Contributed by NM, 4-Oct-2006.)
 |-  ( J  e.  Top  ->  (/) 
 e.  ( Clsd `  J ) )
 
Theoremiincld 17027* The indexed intersection of a collection  B ( x ) of closed sets is closed. Theorem 6.1(2) of [Munkres] p. 93. (Contributed by NM, 5-Oct-2006.) (Revised by Mario Carneiro, 3-Sep-2015.)
 |-  ( ( A  =/=  (/)  /\  A. x  e.  A  B  e.  ( Clsd `  J ) )  ->  |^|_
 x  e.  A  B  e.  ( Clsd `  J )
 )
 
Theoremintcld 17028 The intersection of a set of closed sets is closed. (Contributed by NM, 5-Oct-2006.)
 |-  ( ( A  =/=  (/)  /\  A  C_  ( Clsd `  J ) )  ->  |^| A  e.  ( Clsd `  J ) )
 
Theoremuncld 17029 The union of two closed sets is closed. Equivalent to Theorem 6.1(3) of [Munkres] p. 93. (Contributed by NM, 5-Oct-2006.)
 |-  ( ( A  e.  ( Clsd `  J )  /\  B  e.  ( Clsd `  J ) )  ->  ( A  u.  B )  e.  ( Clsd `  J ) )
 
Theoremcldcls 17030 A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.)
 |-  ( S  e.  ( Clsd `  J )  ->  ( ( cls `  J ) `  S )  =  S )
 
Theoremincld 17031 The intersection of two closed sets is closed. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
 |-  ( ( A  e.  ( Clsd `  J )  /\  B  e.  ( Clsd `  J ) )  ->  ( A  i^i  B )  e.  ( Clsd `  J ) )
 
Theoremriincld 17032* An indexed relative intersection of closed sets is closed. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\ 
 A. x  e.  A  B  e.  ( Clsd `  J ) )  ->  ( X  i^i  |^|_ x  e.  A  B )  e.  ( Clsd `  J )
 )
 
Theoremiuncld 17033* A finite indexed union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  B  e.  ( Clsd `  J )
 )  ->  U_ x  e.  A  B  e.  ( Clsd `  J ) )
 
Theoremunicld 17034 A finite union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  A  e.  Fin  /\  A  C_  ( Clsd `  J ) )  ->  U. A  e.  ( Clsd `  J )
 )
 
Theoremclscld 17035 The closure of a subset of a topology's underlying set is closed. (Contributed by NM, 4-Oct-2006.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( ( cls `  J ) `  S )  e.  ( Clsd `  J )
 )
 
Theoremclsf 17036 The closure function is a function from subsets of the base to closed sets. (Contributed by Mario Carneiro, 11-Apr-2015.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  ( cls `  J ) : ~P X --> ( Clsd `  J ) )
 
Theoremntropn 17037 The interior of a subset of a topology's underlying set is open. (Contributed by NM, 11-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( ( int `  J ) `  S )  e.  J )
 
Theoremclsval2 17038 Express closure in terms of interior. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( ( cls `  J ) `  S )  =  ( X  \  (
 ( int `  J ) `  ( X  \  S ) ) ) )
 
Theoremntrval2 17039 Interior expressed in terms of closure. (Contributed by NM, 1-Oct-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( ( int `  J ) `  S )  =  ( X  \  (
 ( cls `  J ) `  ( X  \  S ) ) ) )
 
Theoremntrdif 17040 An interior of a complement is the complement of the closure. This set is also known as the exterior of  A. (Contributed by Jeff Hankins, 31-Aug-2009.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  A  C_  X )  ->  ( ( int `  J ) `  ( X  \  A ) )  =  ( X  \  (
 ( cls `  J ) `  A ) ) )
 
Theoremclsdif 17041 A closure of a complement is the complement of the interior. (Contributed by Jeff Hankins, 31-Aug-2009.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  A  C_  X )  ->  ( ( cls `  J ) `  ( X  \  A ) )  =  ( X  \  (
 ( int `  J ) `  A ) ) )
 
Theoremclsss 17042 Subset relationship for closure. (Contributed by NM, 10-Feb-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  ( ( cls `  J ) `  T )  C_  ( ( cls `  J ) `  S ) )
 
Theoremntrss 17043 Subset relationship for interior. (Contributed by NM, 3-Oct-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  ( ( int `  J ) `  T )  C_  ( ( int `  J ) `  S ) )
 
Theoremsscls 17044 A subset of a topology's underlying set is included in its closure. (Contributed by NM, 22-Feb-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  S  C_  ( ( cls `  J ) `  S ) )
 
Theoremntrss2 17045 A subset includes its interior. (Contributed by NM, 3-Oct-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( ( int `  J ) `  S )  C_  S )
 
Theoremssntr 17046 An open subset of a set is a subset of the set's interior. (Contributed by Jeff Hankins, 31-Aug-2009.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  (
 ( ( J  e.  Top  /\  S  C_  X )  /\  ( O  e.  J  /\  O  C_  S )
 )  ->  O  C_  (
 ( int `  J ) `  S ) )
 
Theoremclsss3 17047 The closure of a subset of a topological space is included in the space. (Contributed by NM, 26-Feb-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( ( cls `  J ) `  S )  C_  X )
 
Theoremntrss3 17048 The interior of a subset of a topological space is included in the space. (Contributed by NM, 1-Oct-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( ( int `  J ) `  S )  C_  X )
 
Theoremntrin 17049 A pairwise intersection of interiors is the interior of the intersection. This does not always hold for arbitrary intersections. (Contributed by Jeff Hankins, 31-Aug-2009.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  ( ( int `  J ) `  ( A  i^i  B ) )  =  ( ( ( int `  J ) `  A )  i^i  ( ( int `  J ) `  B ) ) )
 
Theoremcmclsopn 17050 The complement of a closure is open. (Contributed by NM, 11-Sep-2006.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( X  \  (
 ( cls `  J ) `  S ) )  e.  J )
 
Theoremcmntrcld 17051 The complement of an interior is closed. (Contributed by NM, 1-Oct-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( X  \  (
 ( int `  J ) `  S ) )  e.  ( Clsd `  J )
 )
 
Theoremiscld3 17052 A subset is closed iff it equals its own closure. (Contributed by NM, 2-Oct-2006.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( S  e.  ( Clsd `  J )  <->  ( ( cls `  J ) `  S )  =  S )
 )
 
Theoremiscld4 17053 A subset is closed iff it contains its own closure. (Contributed by NM, 31-Jan-2008.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( S  e.  ( Clsd `  J )  <->  ( ( cls `  J ) `  S )  C_  S ) )
 
Theoremisopn3 17054 A subset is open iff it equals its own interior. (Contributed by NM, 9-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( S  e.  J  <->  ( ( int `  J ) `  S )  =  S ) )
 
Theoremclsidm 17055 The closure operation is idempotent. (Contributed by NM, 2-Oct-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( ( cls `  J ) `  ( ( cls `  J ) `  S ) )  =  (
 ( cls `  J ) `  S ) )
 
Theoremntridm 17056 The interior operation is idempotent. (Contributed by NM, 2-Oct-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( ( int `  J ) `  ( ( int `  J ) `  S ) )  =  (
 ( int `  J ) `  S ) )
 
Theoremclstop 17057 The closure of a topology's underlying set is entire set. (Contributed by NM, 5-Oct-2007.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  (
 ( cls `  J ) `  X )  =  X )
 
Theoremntrtop 17058 The interior of a topology's underlying set is entire set. (Contributed by NM, 12-Sep-2006.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  (
 ( int `  J ) `  X )  =  X )
 
Theorem0ntr 17059 A subset with an empty interior cannot cover a whole (nonempty) topology. (Contributed by NM, 12-Sep-2006.)
 |-  X  =  U. J   =>    |-  (
 ( ( J  e.  Top  /\  X  =/=  (/) )  /\  ( S  C_  X  /\  ( ( int `  J ) `  S )  =  (/) ) )  ->  ( X  \  S )  =/=  (/) )
 
Theoremclsss2 17060 If a subset is included in a closed set, so is the subset's closure. (Contributed by NM, 22-Feb-2007.)
 |-  X  =  U. J   =>    |-  (
 ( C  e.  ( Clsd `  J )  /\  S  C_  C )  ->  ( ( cls `  J ) `  S )  C_  C )
 
Theoremelcls 17061* Membership in a closure. Theorem 6.5(a) of [Munkres] p. 95. (Contributed by NM, 22-Feb-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X  /\  P  e.  X )  ->  ( P  e.  (
 ( cls `  J ) `  S )  <->  A. x  e.  J  ( P  e.  x  ->  ( x  i^i  S )  =/=  (/) ) ) )
 
Theoremelcls2 17062* Membership in a closure. (Contributed by NM, 5-Mar-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( P  e.  (
 ( cls `  J ) `  S )  <->  ( P  e.  X  /\  A. x  e.  J  ( P  e.  x  ->  ( x  i^i  S )  =/=  (/) ) ) ) )
 
Theoremclsndisj 17063 Any open set containing a point that belongs to the closure of a subset intersects the subset. One direction of Theorem 6.5(a) of [Munkres] p. 95. (Contributed by NM, 26-Feb-2007.)
 |-  X  =  U. J   =>    |-  (
 ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  ( ( cls `  J ) `  S ) )  /\  ( U  e.  J  /\  P  e.  U ) )  ->  ( U  i^i  S )  =/=  (/) )
 
Theoremntrcls0 17064 A subset whose closure has an empty interior also has an empty interior. (Contributed by NM, 4-Oct-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X  /\  ( ( int `  J ) `  ( ( cls `  J ) `  S ) )  =  (/) )  ->  ( ( int `  J ) `  S )  =  (/) )
 
Theoremntreq0 17065* Two ways to say that a subset has an empty interior. (Contributed by NM, 3-Oct-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( ( ( int `  J ) `  S )  =  (/)  <->  A. x  e.  J  ( x  C_  S  ->  x  =  (/) ) ) )
 
Theoremcldmre 17066 The closed sets of a topology comprise a Moore system on the points of the topology. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  ( Clsd `  J )  e.  (Moore `  X )
 )
 
Theoremmrccls 17067 Moore closure generalizes closure in a topology. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  ( Clsd `  J )
 )   =>    |-  ( J  e.  Top  ->  ( cls `  J )  =  F )
 
Theoremcls0 17068 The closure of the empty set. (Contributed by NM, 2-Oct-2007.)
 |-  ( J  e.  Top  ->  ( ( cls `  J ) `  (/) )  =  (/) )
 
Theoremntr0 17069 The interior of the empty set. (Contributed by NM, 2-Oct-2007.)
 |-  ( J  e.  Top  ->  ( ( int `  J ) `  (/) )  =  (/) )
 
Theoremisopn3i 17070 An open subset equals its own interior. (Contributed by Mario Carneiro, 30-Dec-2016.)
 |-  ( ( J  e.  Top  /\  S  e.  J ) 
 ->  ( ( int `  J ) `  S )  =  S )
 
Theoremelcls3 17071* Membership in a closure in terms of the members of a basis. Theorem 6.5(b) of [Munkres] p. 95. (Contributed by NM, 26-Feb-2007.) (Revised by Mario Carneiro, 3-Sep-2015.)
 |-  ( ph  ->  J  =  ( topGen `  B )
 )   &    |-  ( ph  ->  X  =  U. J )   &    |-  ( ph  ->  B  e.  TopBases )   &    |-  ( ph  ->  S  C_  X )   &    |-  ( ph  ->  P  e.  X )   =>    |-  ( ph  ->  ( P  e.  ( ( cls `  J ) `  S )  <->  A. x  e.  B  ( P  e.  x  ->  ( x  i^i  S )  =/=  (/) ) ) )
 
Theoremopncldf1 17072* A bijection useful for converting statements about open sets to statements about closed sets and vice versa. (Contributed by Jeff Hankins, 27-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
 |-  X  =  U. J   &    |-  F  =  ( u  e.  J  |->  ( X  \  u ) )   =>    |-  ( J  e.  Top  ->  ( F : J -1-1-onto-> ( Clsd `  J )  /\  `' F  =  ( x  e.  ( Clsd `  J )  |->  ( X 
 \  x ) ) ) )
 
Theoremopncldf2 17073* The values of the open-closed bijection. (Contributed by Jeff Hankins, 27-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
 |-  X  =  U. J   &    |-  F  =  ( u  e.  J  |->  ( X  \  u ) )   =>    |-  ( ( J  e.  Top  /\  A  e.  J ) 
 ->  ( F `  A )  =  ( X  \  A ) )
 
Theoremopncldf3 17074* The values of the converse/inverse of the open-closed bijection. (Contributed by Jeff Hankins, 27-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
 |-  X  =  U. J   &    |-  F  =  ( u  e.  J  |->  ( X  \  u ) )   =>    |-  ( B  e.  ( Clsd `  J )  ->  ( `' F `  B )  =  ( X  \  B ) )
 
Theoremisclo 17075* A set  A is clopen iff for every point  x in the space there is a neighborhood  y such that all the points in  y are in  A iff  x is. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  A  C_  X )  ->  ( A  e.  ( J  i^i  ( Clsd `  J ) )  <->  A. x  e.  X  E. y  e.  J  ( x  e.  y  /\  A. z  e.  y  ( x  e.  A  <->  z  e.  A ) ) ) )
 
Theoremisclo2 17076* A set  A is clopen iff for every point  x in the space there is a neighborhood  y of  x which is either disjoint from  A or contained in  A. (Contributed by Mario Carneiro, 7-Jul-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  A  C_  X )  ->  ( A  e.  ( J  i^i  ( Clsd `  J ) )  <->  A. x  e.  X  E. y  e.  J  ( x  e.  y  /\  A. z  e.  y  ( z  e.  A  ->  y  C_  A )
 ) ) )
 
Theoremdiscld 17077 The open sets of a discrete topology are closed and its closed sets are open. (Contributed by FL, 7-Jun-2007.) (Revised by Mario Carneiro, 7-Apr-2015.)
 |-  ( A  e.  V  ->  ( Clsd `  ~P A )  =  ~P A )
 
Theoremsn0cld 17078 The closed sets of the topology 
{ (/) }. (Contributed by FL, 5-Jan-2009.)
 |-  ( Clsd `  { (/) } )  =  { (/) }
 
Theoremindiscld 17079 The closed sets of an indiscrete topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  ( Clsd `  { (/) ,  A } )  =  { (/)
 ,  A }
 
Theoremmretopd 17080* A Moore collection which is closed under finite unions called topological; such a collection is the closed sets of a canonically associated topology. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( ph  ->  M  e.  (Moore `  B )
 )   &    |-  ( ph  ->  (/)  e.  M )   &    |-  ( ( ph  /\  x  e.  M  /\  y  e.  M )  ->  ( x  u.  y )  e.  M )   &    |-  J  =  {
 z  e.  ~P B  |  ( B  \  z
 )  e.  M }   =>    |-  ( ph  ->  ( J  e.  (TopOn `  B )  /\  M  =  ( Clsd `  J ) ) )
 
Theoremtoponmre 17081 The topologies over a given base set form a Moore collection: the intersection of any family of them is a topology, including the empty (relative) intersection which gives the discrete topology distop 16984. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 5-May-2015.)
 |-  ( B  e.  V  ->  (TopOn `  B )  e.  (Moore `  ~P B ) )
 
Theoremcldmreon 17082 The closed sets of a topology over a set are a Moore collection over the same set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  ( J  e.  (TopOn `  B )  ->  ( Clsd `  J )  e.  (Moore `  B )
 )
 
Theoremiscldtop 17083* A family is the closed sets of a topology iff it is a Moore collection and closed under finite union. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( K  e.  ( Clsd " (TopOn `  B ) )  <->  ( K  e.  (Moore `  B )  /\  (/) 
 e.  K  /\  A. x  e.  K  A. y  e.  K  ( x  u.  y )  e.  K ) )
 
Theoremmreclatdemo 17084 The closed subspaces of a topology-bearing module form a complete lattice. Demonstration for mreclat 14541. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  ( W  e.  ( TopSp  i^i  LMod )  ->  (toInc `  ( ( LSubSp `  W )  i^i  ( Clsd `  ( TopOpen `  W ) ) ) )  e.  CLat )
 
11.1.5  Neighborhoods
 
Syntaxcnei 17085 Extend class notation with neighborhood relation for topologies.
 class  nei
 
Definitiondf-nei 17086* Define a function on topologies whose value is a map from a subset to its neighborhoods. (Contributed by NM, 11-Feb-2007.)
 |- 
 nei  =  ( j  e.  Top  |->  ( x  e. 
 ~P U. j  |->  { y  e.  ~P U. j  | 
 E. g  e.  j  ( x  C_  g  /\  g  C_  y ) }
 ) )
 
Theoremneifval 17087* The neighborhood function on the subsets of a topology's base set. (Contributed by NM, 11-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  ( nei `  J )  =  ( x  e.  ~P X  |->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v
 ) } ) )
 
Theoremneif 17088 The neighborhood function is a function of the subsets of a topology's base set. (Contributed by NM, 12-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  ( nei `  J )  Fn 
 ~P X )
 
Theoremneiss2 17089 A set with a neighborhood is a subset of the topology's base set. (This theorem depends on a function's value being empty outside of its domain, but it will make later theorems simpler to state.) (Contributed by NM, 12-Feb-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  X )
 
Theoremneival 17090* The set of neighborhoods of a subset of the base set of a topology. (Contributed by NM, 11-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( ( nei `  J ) `  S )  =  { v  e.  ~P X  |  E. g  e.  J  ( S  C_  g  /\  g  C_  v
 ) } )
 
Theoremisnei 17091* The predicate " N is a neighborhood of  S." (Contributed by FL, 25-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( N  e.  (
 ( nei `  J ) `  S )  <->  ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) ) )
 
Theoremneiint 17092 An intuitive definition of a neighborhood in terms of interior. (Contributed by Szymon Jaroszewicz, 18-Dec-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  ->  ( N  e.  (
 ( nei `  J ) `  S )  <->  S  C_  ( ( int `  J ) `  N ) ) )
 
Theoremisneip 17093* The predicate " N is a neighborhood of point  P." (Contributed by NM, 26-Feb-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  P  e.  X ) 
 ->  ( N  e.  (
 ( nei `  J ) `  { P } )  <->  ( N  C_  X  /\  E. g  e.  J  ( P  e.  g  /\  g  C_  N ) ) ) )
 
Theoremneii1 17094 A neighborhood is included in the topology's base set. (Contributed by NM, 12-Feb-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  N  C_  X )
 
Theoremneisspw 17095 The neighborhoods of any set are subsets of the base set. (Contributed by Stefan O'Rear, 6-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  (
 ( nei `  J ) `  S )  C_  ~P X )
 
Theoremneii2 17096* Property of a neighborhood. (Contributed by NM, 12-Feb-2007.)
 |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
 
Theoremneiss 17097 Any neighborhood of a set  S is also a neighborhood of any subset  R  C_  S. Theorem of [BourbakiTop1] p. I.2. (Contributed by FL, 25-Sep-2006.)
 |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  N  e.  ( ( nei `  J ) `  R ) )
 
Theoremssnei 17098 A set is included in its neighborhoods. Proposition Viii of [BourbakiTop1] p. I.3 . (Contributed by FL, 16-Nov-2006.)
 |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  N )
 
Theoremelnei 17099 A point belongs to any of its neighborhoods. Proposition Viii of [BourbakiTop1] p. I.3. (Contributed by FL, 28-Sep-2006.)
 |-  ( ( J  e.  Top  /\  P  e.  A  /\  N  e.  ( ( nei `  J ) `  { P } ) ) 
 ->  P  e.  N )
 
Theorem0nnei 17100 The empty set is not a neighborhood of a nonempty set. (Contributed by FL, 18-Sep-2007.)
 |-  ( ( J  e.  Top  /\  S  =/=  (/) )  ->  -.  (/)  e.  ( ( nei `  J ) `  S ) )
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