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Theorem List for Metamath Proof Explorer - 16601-16700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisopn2 16601 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 16602 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 16603 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 16604 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 16605 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 16606* 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 16607 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 16608* 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 16609 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 16610 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 16611 A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.)
 |-  ( S  e.  ( Clsd `  J )  ->  ( ( cls `  J ) `  S )  =  S )
 
Theoremincld 16612 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 16613* A 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 16614* 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 16615 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 16616 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 16617 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 16618 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 16619 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 16620 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 16621 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 16622 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 16623 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 16624 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 16625 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 16626 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 16627 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 16628 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 16629 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 16630 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 16631 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 16632 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 16633 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 16634 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 16635 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 16636 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 16637 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 16638 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 16639 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 16640 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 16641 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 16642* 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 16643* 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 16644 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 16645 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 16646* 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 16647 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 16648 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 16649 The closure of the empty set. (Contributed by NM, 2-Oct-2007.)
 |-  ( J  e.  Top  ->  ( ( cls `  J ) `  (/) )  =  (/) )
 
Theoremntr0 16650 The interior of the empty set. (Contributed by NM, 2-Oct-2007.)
 |-  ( J  e.  Top  ->  ( ( int `  J ) `  (/) )  =  (/) )
 
Theoremisopn3i 16651 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 16652* 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 16653* 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 16654* 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 16655* 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 16656* 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 16657* 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 16658 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 16659 The closed sets of the topology 
{ (/) }. (Contributed by FL, 5-Jan-2009.)
 |-  ( Clsd `  { (/) } )  =  { (/) }
 
Theoremindiscld 16660 The closed sets of an indiscrete topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  ( Clsd `  { (/) ,  A } )  =  { (/)
 ,  A }
 
Theoremmretopd 16661* 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 16662 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 16565. (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 16663 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 16664* 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 16665 The closed subspaces of a topology-bearing module form a complete lattice. Demonstration for mreclat 14125. (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 16666 Extend class notation with neighborhood relation for topologies.
 class  nei
 
Definitiondf-nei 16667* 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 16668* 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 16669 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 16670 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 16671* 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 16672* 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 16673 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 16674* 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 16675 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 16676 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 16677* 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 16678 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 16679 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 16680 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 16681 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 ) )
 
Theoremneips 16682* A neighborhood of a set is a neighborhood of every point in the set. Proposition of [BourbakiTop1] p. I.2. (Contributed by FL, 16-Nov-2006.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  ( N  e.  ( ( nei `  J ) `  S )  <->  A. p  e.  S  N  e.  ( ( nei `  J ) `  { p } ) ) )
 
Theoremopnneissb 16683 An open set is a neighborhood of any of its subsets. (Contributed by FL, 2-Oct-2006.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  N  e.  J  /\  S  C_  X )  ->  ( S  C_  N  <->  N  e.  (
 ( nei `  J ) `  S ) ) )
 
Theoremopnssneib 16684 Any superset of an open set is a neighborhood of it. (Contributed by NM, 14-Feb-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  e.  J  /\  N  C_  X )  ->  ( S  C_  N  <->  N  e.  (
 ( nei `  J ) `  S ) ) )
 
Theoremssnei2 16685 Any subset of  X containing a neigborhood of a set is a neighborhood of this set. Proposition Vi of [BourbakiTop1] p. I.3. (Contributed by FL, 2-Oct-2006.)
 |-  X  =  U. J   =>    |-  (
 ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  /\  ( N  C_  M  /\  M  C_  X ) ) 
 ->  M  e.  ( ( nei `  J ) `  S ) )
 
Theoremneindisj 16686 Any neighborhood of an element in the closure of a subset intersects the subset. Part of proof of Theorem 6.6 of [Munkres] p. 97. (Contributed by NM, 26-Feb-2007.)
 |-  X  =  U. J   =>    |-  (
 ( ( J  e.  Top  /\  S  C_  X )  /\  ( P  e.  (
 ( cls `  J ) `  S )  /\  N  e.  ( ( nei `  J ) `  { P }
 ) ) )  ->  ( N  i^i  S )  =/=  (/) )
 
Theoremopnneiss 16687 An open set is a neighborhood of any of its subsets. (Contributed by NM, 13-Feb-2007.)
 |-  ( ( J  e.  Top  /\  N  e.  J  /\  S  C_  N )  ->  N  e.  ( ( nei `  J ) `  S ) )
 
Theoremopnneip 16688 An open set is a neighborhood of any of its members. (Contributed by NM, 8-Mar-2007.)
 |-  ( ( J  e.  Top  /\  N  e.  J  /\  P  e.  N )  ->  N  e.  ( ( nei `  J ) `  { P } )
 )
 
Theoremopnnei 16689* A set is open iff it is a neighborhood of all its points. ( Contributed by Jeff Hankins, 15-Sep-2009.) (Contributed by NM, 16-Sep-2009.)
 |-  ( J  e.  Top  ->  ( S  e.  J  <->  A. x  e.  S  S  e.  ( ( nei `  J ) `  { x }
 ) ) )
 
Theoremtpnei 16690 The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 16687. (Contributed by FL, 2-Oct-2006.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  ( S  C_  X  <->  X  e.  (
 ( nei `  J ) `  S ) ) )
 
Theoremneiuni 16691 The union of the neighborhoods of a set equals the topology's underlying set. (Contributed by FL, 18-Sep-2007.) (Revised by Mario Carneiro, 9-Apr-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  X  =  U. (
 ( nei `  J ) `  S ) )
 
Theoremneindisj2 16692* A point  P belongs to the closure of a set  S iff every neighborhood of  P meets  S. (Contributed by FL, 15-Sep-2013.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X  /\  P  e.  X )  ->  ( P  e.  (
 ( cls `  J ) `  S )  <->  A. n  e.  (
 ( nei `  J ) `  { P } )
 ( n  i^i  S )  =/=  (/) ) )
 
Theoremtopssnei 16693 A finer topology has more neighborhoods. (Contributed by Mario Carneiro, 9-Apr-2015.)
 |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y ) 
 /\  J  C_  K )  ->  ( ( nei `  J ) `  S )  C_  ( ( nei `  K ) `  S ) )
 
Theoreminnei 16694 The intersection of two neighborhoods of a set is also a neighborhood of the set. Proposition Vii of [BourbakiTop1] p. I.3 . (Contributed by FL, 28-Sep-2006.)
 |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  M  e.  ( ( nei `  J ) `  S ) ) 
 ->  ( N  i^i  M )  e.  ( ( nei `  J ) `  S ) )
 
Theoremopnneiid 16695 Only an open set is a neighborhood of itself. (Contributed by FL, 2-Oct-2006.)
 |-  ( J  e.  Top  ->  ( N  e.  (
 ( nei `  J ) `  N )  <->  N  e.  J ) )
 
Theoremneissex 16696* For any neighborhood  N of  S, there is a neighborhood  x of  S such that  N is a neighborhood of all subsets of  x. Proposition Viv of [BourbakiTop1] p. I.3 . (Contributed by FL, 2-Oct-2006.)
 |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  E. x  e.  (
 ( nei `  J ) `  S ) A. y
 ( y  C_  x  ->  N  e.  ( ( nei `  J ) `  y ) ) )
 
Theorem0nei 16697 The empty set is a neighborhood of itself. (Contributed by FL, 10-Dec-2006.)
 |-  ( J  e.  Top  ->  (/) 
 e.  ( ( nei `  J ) `  (/) ) )
 
11.1.6  Limit points and perfect sets
 
Syntaxclp 16698 Extend class notation with the limit point function for topologies.
 class  limPt
 
Syntaxcperf 16699 Extend class notation with the class of all perfect spaces.
 class Perf
 
Definitiondf-lp 16700* Define a function on topologies whose value is the set of limit points of the subsets of the base set. See lpval 16703. (Contributed by NM, 10-Feb-2007.)
 |- 
 limPt  =  ( j  e.  Top  |->  ( x  e. 
 ~P U. j  |->  { y  |  y  e.  (
 ( cls `  j ) `  ( x  \  {
 y } ) ) } ) )
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