HomeHome Metamath Proof Explorer
Theorem List (p. 202 of 410)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-26627)
  Hilbert Space Explorer  Hilbert Space Explorer
(26628-28150)
  Users' Mathboxes  Users' Mathboxes
(28151-40909)
 

Theorem List for Metamath Proof Explorer - 20101-20200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcldopn 20101 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 20102 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 20103 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 20104 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 20105 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 20106 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 20107* 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 20108 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 20109* 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 20110 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 20111 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 20112 A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.)
 |-  ( S  e.  ( Clsd `  J )  ->  ( ( cls `  J ) `  S )  =  S )
 
Theoremincld 20113 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 20114* 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 20115* 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 20116 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 20117 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 20118 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 20119 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 20120 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 20121 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 20122 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 20123 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 20124 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 20125 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 20126 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 20127 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 20128 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 20129 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 20130 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 20131 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 20132 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 20133 The complement of an interior is closed. (Contributed by NM, 1-Oct-2007.) (Proof shortened by OpenAI, 3-Jul-2020.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( X  \  (
 ( int `  J ) `  S ) )  e.  ( Clsd `  J )
 )
 
TheoremcmntrcldOLD 20134 The complement of an interior is closed. (Contributed by NM, 1-Oct-2007.) Obsolete version of cmntrcld 20133 as of 3-Jul-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( X  \  (
 ( int `  J ) `  S ) )  e.  ( Clsd `  J )
 )
 
Theoremiscld3 20135 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 20136 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 20137 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 20138 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 20139 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 20140 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 20141 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 20142 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 20143 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 20144* 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 20145* 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 20146 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 20147 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 20148* 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 20149 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 20150 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 20151 The closure of the empty set. (Contributed by NM, 2-Oct-2007.)
 |-  ( J  e.  Top  ->  ( ( cls `  J ) `  (/) )  =  (/) )
 
Theoremntr0 20152 The interior of the empty set. (Contributed by NM, 2-Oct-2007.)
 |-  ( J  e.  Top  ->  ( ( int `  J ) `  (/) )  =  (/) )
 
Theoremisopn3i 20153 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 20154* 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 20155* 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 20156* 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 20157* 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 20158* 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 20159* 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 20160 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 20161 The closed sets of the topology 
{ (/) }. (Contributed by FL, 5-Jan-2009.)
 |-  ( Clsd `  { (/) } )  =  { (/) }
 
Theoremindiscld 20162 The closed sets of an indiscrete topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  ( Clsd `  { (/) ,  A } )  =  { (/)
 ,  A }
 
Theoremmretopd 20163* 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 20164 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 20066. (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 20165 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 20166* 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 ) )
 
TheoremmreclatdemoBAD 20167 The closed subspaces of a topology-bearing module form a complete lattice. Demonstration for mreclatBAD 16488. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 6282 update): This proof uses the old df-clat 16409 and references the required instance of mreclatBAD 16488 as a hypothesis. When mreclatBAD 16488 is corrected to become mreclat, delete this theorem and uncomment the mreclatdemo below.
 |-  ( ( ( LSubSp `  W )  i^i  ( Clsd `  ( TopOpen `  W )
 ) )  e.  (Moore ` 
 U. ( TopOpen `  W ) )  ->  (toInc `  ( ( LSubSp `  W )  i^i  ( Clsd `  ( TopOpen `  W ) ) ) )  e.  CLat )   =>    |-  ( W  e.  ( TopSp  i^i  LMod )  ->  (toInc `  (
 ( LSubSp `  W )  i^i  ( Clsd `  ( TopOpen `  W ) ) ) )  e.  CLat )
 
12.1.5  Neighborhoods
 
Syntaxcnei 20168 Extend class notation with neighborhood relation for topologies.
 class  nei
 
Definitiondf-nei 20169* 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 20170* 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 20171 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 20172 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 20173* 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 20174* 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 20175 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 20176* 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 20177 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 20178 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 20179* 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 20180 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 20181 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 20182 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 20183 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 20184* 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 20185 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 20186 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 20187 Any subset of  X containing a neighborhood 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 20188 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 20189 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 20190 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 20191* A set is open iff it is a neighborhood of all of its points. (Contributed by Jeff Hankins, 15-Sep-2009.)
 |-  ( J  e.  Top  ->  ( S  e.  J  <->  A. x  e.  S  S  e.  ( ( nei `  J ) `  { x }
 ) ) )
 
Theoremtpnei 20192 The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 20189. (Contributed by FL, 2-Oct-2006.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  ( S  C_  X  <->  X  e.  (
 ( nei `  J ) `  S ) ) )
 
Theoremneiuni 20193 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 20194* 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 20195 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 20196 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 20197 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 20198* 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 20199 The empty set is a neighborhood of itself. (Contributed by FL, 10-Dec-2006.)
 |-  ( J  e.  Top  ->  (/) 
 e.  ( ( nei `  J ) `  (/) ) )
 
Theoremneipeltop 20200* Lemma for neiptopreu 20204. (Contributed by Thierry Arnoux, 6-Jan-2018.)
 |-  J  =  { a  e.  ~P X  |  A. p  e.  a  a  e.  ( N `  p ) }   =>    |-  ( C  e.  J  <->  ( C  C_  X  /\  A. p  e.  C  C  e.  ( N `  p ) ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-40909
  Copyright terms: Public domain < Previous  Next >