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Theorem List for Metamath Proof Explorer - 16801-16900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremneival 16801* 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 16802* 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 16803 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 16804* 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 16805 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 16806 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 16807* 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 16808 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 16809 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 16810 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 16811 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 16812* 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 16813 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 16814 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 16815 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 16816 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 16817 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 16818 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 16819* 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 16820 The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 16817. (Contributed by FL, 2-Oct-2006.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  ( S  C_  X  <->  X  e.  (
 ( nei `  J ) `  S ) ) )
 
Theoremneiuni 16821 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 16822* 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 16823 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 16824 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 16825 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 16826* 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 16827 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 16828 Extend class notation with the limit point function for topologies.
 class  limPt
 
Syntaxcperf 16829 Extend class notation with the class of all perfect spaces.
 class Perf
 
Definitiondf-lp 16830* Define a function on topologies whose value is the set of limit points of the subsets of the base set. See lpval 16833. (Contributed by NM, 10-Feb-2007.)
 |- 
 limPt  =  ( j  e.  Top  |->  ( x  e. 
 ~P U. j  |->  { y  |  y  e.  (
 ( cls `  j ) `  ( x  \  {
 y } ) ) } ) )
 
Definitiondf-perf 16831 Define the class of all perfect spaces. A perfect space is one for which every point in the set is a limit point of the whole space. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |- Perf  =  { j  e.  Top  |  ( ( limPt `  j
 ) `  U. j )  =  U. j }
 
Theoremlpfval 16832* The limit point function on the subsets of a topology's base set. (Contributed by NM, 10-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  ( limPt `  J )  =  ( x  e.  ~P X  |->  { y  |  y  e.  ( ( cls `  J ) `  ( x  \  { y }
 ) ) } )
 )
 
Theoremlpval 16833* The set of limit points of a subset of the base set of a topology. Alternate definition of limit point in [Munkres] p. 97. (Contributed by NM, 10-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( ( limPt `  J ) `  S )  =  { x  |  x  e.  ( ( cls `  J ) `  ( S  \  { x } ) ) } )
 
Theoremislp 16834 The predicate " P is a limit point of  S." (Contributed by NM, 10-Feb-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( P  e.  (
 ( limPt `  J ) `  S )  <->  P  e.  (
 ( cls `  J ) `  ( S  \  { P } ) ) ) )
 
Theoremlpsscls 16835 The limits points of a subset are included in the subset's closure. (Contributed by NM, 26-Feb-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( ( limPt `  J ) `  S )  C_  ( ( cls `  J ) `  S ) )
 
Theoremlpss 16836 The limits points of a subset are included in the base set. (Contributed by NM, 9-Nov-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( ( limPt `  J ) `  S )  C_  X )
 
Theoremlpdifsn 16837  P is a limit point of  S iff it is a limit point of  S  \  { P }. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( P  e.  (
 ( limPt `  J ) `  S )  <->  P  e.  (
 ( limPt `  J ) `  ( S  \  { P } ) ) ) )
 
Theoremlpss3 16838 Subset relationship for limit points. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  ( ( limPt `  J ) `  T )  C_  ( ( limPt `  J ) `  S ) )
 
Theoremislp2 16839* The predicate " P is a limit point of  S," in terms of neighborhoods. Definition of limit point in [Munkres] p. 97. Although Munkres uses open neighborhoods, it also works for our more general neighborhoods. (Contributed by NM, 26-Feb-2007.) (Proof shortened by Mario Carneiro, 25-Dec-2016.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X  /\  P  e.  X )  ->  ( P  e.  (
 ( limPt `  J ) `  S )  <->  A. n  e.  (
 ( nei `  J ) `  { P } )
 ( n  i^i  ( S  \  { P }
 ) )  =/=  (/) ) )
 
Theoremmaxlp 16840 A point is a limit point of the whole space iff the singleton of the point is not open. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  ( P  e.  ( ( limPt `  J ) `  X )  <->  ( P  e.  X  /\  -.  { P }  e.  J )
 ) )
 
Theoremclslp 16841 The closure of a subset of a topological space is the subset together with its limit points. Theorem 6.6 of [Munkres] p. 97. (Contributed by NM, 26-Feb-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( ( cls `  J ) `  S )  =  ( S  u.  (
 ( limPt `  J ) `  S ) ) )
 
Theoremislpi 16842 A point belonging to a set's closure but not the set itself is a limit point. (Contributed by NM, 8-Nov-2007.)
 |-  X  =  U. J   =>    |-  (
 ( ( J  e.  Top  /\  S  C_  X )  /\  ( P  e.  (
 ( cls `  J ) `  S )  /\  -.  P  e.  S )
 )  ->  P  e.  ( ( limPt `  J ) `  S ) )
 
Theoremcldlp 16843 A subset of a topological space is closed iff it contains all its limit points. Corollary 6.7 of [Munkres] p. 97. (Contributed by NM, 26-Feb-2007.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( S  e.  ( Clsd `  J )  <->  ( ( limPt `  J ) `  S )  C_  S ) )
 
Theoremisperf 16844 Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  X  =  U. J   =>    |-  ( J  e. Perf  <->  ( J  e.  Top  /\  ( ( limPt `  J ) `  X )  =  X ) )
 
Theoremisperf2 16845 Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  X  =  U. J   =>    |-  ( J  e. Perf  <->  ( J  e.  Top  /\  X  C_  ( ( limPt `  J ) `  X ) ) )
 
Theoremisperf3 16846* A perfect space is a topology which has no open singletons. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  X  =  U. J   =>    |-  ( J  e. Perf  <->  ( J  e.  Top  /\  A. x  e.  X  -.  { x }  e.  J ) )
 
Theoremperflp 16847 The limit points of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  X  =  U. J   =>    |-  ( J  e. Perf  ->  ( (
 limPt `  J ) `  X )  =  X )
 
Theoremperfi 16848 Property of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  X  =  U. J   =>    |-  (
 ( J  e. Perf  /\  P  e.  X )  ->  -.  { P }  e.  J )
 
Theoremperftop 16849 A perfect space is a topology. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  ( J  e. Perf  ->  J  e.  Top )
 
11.1.7  Subspace topologies
 
Theoremrestrcl 16850 Reverse closure for the subspace topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 1-May-2015.)
 |-  ( ( Jt  A )  e.  Top  ->  ( J  e.  _V  /\  A  e.  _V ) )
 
Theoremrestbas 16851 A subspace topology basis is a basis.  Y is normally a subset of the base set of  J. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  ( B  e.  TopBases  ->  ( Bt  A )  e.  TopBases )
 
Theoremtgrest 16852 A subspace can be generated by restricted sets from a basis for the original topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Proof shortened by Mario Carneiro, 30-Aug-2015.)
 |-  ( ( B  e.  V  /\  A  e.  W )  ->  ( topGen `  ( Bt  A ) )  =  ( ( topGen `  B )t  A ) )
 
Theoremresttop 16853 A subspace topology is a topology. Definition of subspace topology in [Munkres] p. 89.  A is normally a subset of the base set of  J. (Contributed by FL, 15-Apr-2007.) (Revised by Mario Carneiro, 1-May-2015.)
 |-  ( ( J  e.  Top  /\  A  e.  V ) 
 ->  ( Jt  A )  e.  Top )
 
Theoremresttopon 16854 A subspace topology is a topology on the base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( Jt  A )  e.  (TopOn `  A ) )
 
Theoremrestuni 16855 The underlying set of a subspace topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  A  C_  X )  ->  A  =  U. ( Jt  A ) )
 
Theoremstoig 16856 The topological space built with a subspace topology. (Contributed by FL, 5-Jan-2009.) (Proof shortened by Mario Carneiro, 1-May-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  A  C_  X )  ->  { <. ( Base `  ndx ) ,  A >. , 
 <. (TopSet `  ndx ) ,  ( Jt  A ) >. }  e.  TopSp
 )
 
Theoremrestco 16857 Composition of subspaces. (Contributed by Mario Carneiro, 15-Dec-2013.) (Revised by Mario Carneiro, 1-May-2015.)
 |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X ) 
 ->  ( ( Jt  A )t  B )  =  ( Jt  ( A  i^i  B ) ) )
 
Theoremrestabs 16858 Equivalence of being a subspace of a subspace and being a subspace of the original. (Contributed by Jeff Hankins, 11-Jul-2009.) (Proof shortened by Mario Carneiro, 1-May-2015.)
 |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W ) 
 ->  ( ( Jt  T )t  S )  =  ( Jt  S ) )
 
Theoremrestin 16859 When the subspace region is not a subset of the base of the topology, the resulting set is the same as the subspace restricted to the base. (Contributed by Mario Carneiro, 15-Dec-2013.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  V  /\  A  e.  W ) 
 ->  ( Jt  A )  =  ( Jt  ( A  i^i  X ) ) )
 
Theoremrestuni2 16860 The underlying set of a subspace topology. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  A  e.  V ) 
 ->  ( A  i^i  X )  =  U. ( Jt  A ) )
 
Theoremresttopon2 16861 The underlying set of a subspace topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  V )  ->  ( Jt  A )  e.  (TopOn `  ( A  i^i  X ) ) )
 
Theoremrest0 16862 The subspace topology induced by the topology  J on the empty set. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 1-May-2015.)
 |-  ( J  e.  Top  ->  ( Jt  (/) )  =  { (/)
 } )
 
Theoremrestsn 16863 The only subspace topology induced by the topology  { (/)
}. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  ( A  e.  V  ->  ( { (/) }t  A )  =  { (/) } )
 
Theoremrestsn2 16864 The subspace topology induced by a singleton. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  ( Jt  { A } )  =  ~P { A }
 )
 
Theoremrestcld 16865* A closed set of a subspace topology is a closed set of the original topology intersected with the subset. (Contributed by FL, 11-Jul-2009.) (Proof shortened by Mario Carneiro, 15-Dec-2013.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X )  ->  ( A  e.  ( Clsd `  ( Jt  S ) )  <->  E. x  e.  ( Clsd `  J ) A  =  ( x  i^i  S ) ) )
 
Theoremrestcldi 16866 A closed set is closed in the subspace topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  X  =  U. J   =>    |-  (
 ( A  C_  X  /\  B  e.  ( Clsd `  J )  /\  B  C_  A )  ->  B  e.  ( Clsd `  ( Jt  A ) ) )
 
Theoremrestcldr 16867 A set which is closed in the subspace topology induced by a closed set is closed in the original topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( A  e.  ( Clsd `  J )  /\  B  e.  ( Clsd `  ( Jt  A ) ) ) 
 ->  B  e.  ( Clsd `  J ) )
 
Theoremrestopnb 16868 If  B is an open subset of the subspace base set  A, then any subset of  B is open iff it is open in  A. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B ) )  ->  ( C  e.  J  <->  C  e.  ( Jt  A ) ) )
 
Theoremssrest 16869 If  K is a finer topology than  J, then the subspace topologies induced by  A maintain this relationship. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 1-May-2015.)
 |-  ( ( K  e.  V  /\  J  C_  K )  ->  ( Jt  A ) 
 C_  ( Kt  A ) )
 
Theoremrestopn2 16870 The if  A is open, then  B is open in  A iff it is an open subset of  A. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( ( J  e.  Top  /\  A  e.  J ) 
 ->  ( B  e.  ( Jt  A )  <->  ( B  e.  J  /\  B  C_  A ) ) )
 
Theoremrestdis 16871 A subspace of a discrete topology is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  ( ( A  e.  V  /\  B  C_  A )  ->  ( ~P At  B )  =  ~P B )
 
Theoremrestfpw 16872 The restriction of the set of finite subsets of  A is the set of finite subsets of  B. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |-  ( ( A  e.  V  /\  B  C_  A )  ->  ( ( ~P A  i^i  Fin )t  B )  =  ( ~P B  i^i  Fin ) )
 
Theoremrestcls 16873 A closure in a subspace topology. (Contributed by Jeff Hankins, 22-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  X  =  U. J   &    |-  K  =  ( Jt  Y )   =>    |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  ( ( cls `  K ) `  S )  =  ( ( ( cls `  J ) `  S )  i^i  Y ) )
 
Theoremrestntr 16874 An interior in a subspace topology. Willard in General Topology says that there is no analog of restcls 16873 for interiors. In some sense, that is true. (Contributed by Jeff Hankins, 23-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  X  =  U. J   &    |-  K  =  ( Jt  Y )   =>    |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  ( ( int `  K ) `  S )  =  ( ( ( int `  J ) `  ( S  u.  ( X  \  Y ) ) )  i^i  Y ) )
 
Theoremrestlp 16875 The limit points of a subset restrict naturally in a subspace. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  X  =  U. J   &    |-  K  =  ( Jt  Y )   =>    |-  ( ( J  e.  Top  /\  Y  C_  X  /\  S  C_  Y )  ->  ( ( limPt `  K ) `  S )  =  ( ( ( limPt `  J ) `  S )  i^i  Y ) )
 
Theoremrestperf 16876 Perfection of a subspace. Note that the term "perfect set" is reserved for closed sets which are perfect in the subspace topology. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  X  =  U. J   &    |-  K  =  ( Jt  Y )   =>    |-  ( ( J  e.  Top  /\  Y  C_  X )  ->  ( K  e. Perf  <->  Y  C_  ( (
 limPt `  J ) `  Y ) ) )
 
Theoremperfopn 16877 An open subset of a perfect space is perfect. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  X  =  U. J   &    |-  K  =  ( Jt  Y )   =>    |-  ( ( J  e. Perf  /\  Y  e.  J ) 
 ->  K  e. Perf )
 
Theoremresstopn 16878 The topology of a restricted structure. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  H  =  ( Ks  A )   &    |-  J  =  (
 TopOpen `  K )   =>    |-  ( Jt  A )  =  ( TopOpen `  H )
 
Theoremresstps 16879 A restricted topological space is a topological space. Note that this theorem would not be true if  TopSp was defined directly in terms of the TopSet slot instead of the  TopOpen derived function. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( ( K  e.  TopSp  /\  A  e.  V ) 
 ->  ( Ks  A )  e.  TopSp )
 
11.1.8  Order topology
 
Theoremordtbaslem 16880* Lemma for ordtbas 16884. In a total order, unbounded-above intervals are closed under intersection. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  X  =  dom  R   &    |-  A  =  ran  (  x  e.  X  |->  { y  e.  X  |  -.  y R x } )   =>    |-  ( R  e.  TosetRel  ->  ( fi `  A )  =  A )
 
Theoremordtval 16881* Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  X  =  dom  R   &    |-  A  =  ran  (  x  e.  X  |->  { y  e.  X  |  -.  y R x } )   &    |-  B  =  ran  (  x  e.  X  |->  { y  e.  X  |  -.  x R y }
 )   =>    |-  ( R  e.  V  ->  (ordTop `  R )  =  ( topGen `  ( fi `  ( { X }  u.  ( A  u.  B ) ) ) ) )
 
Theoremordtuni 16882* Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  X  =  dom  R   &    |-  A  =  ran  (  x  e.  X  |->  { y  e.  X  |  -.  y R x } )   &    |-  B  =  ran  (  x  e.  X  |->  { y  e.  X  |  -.  x R y }
 )   =>    |-  ( R  e.  V  ->  X  =  U. ( { X }  u.  ( A  u.  B ) ) )
 
Theoremordtbas2 16883* Lemma for ordtbas 16884. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  X  =  dom  R   &    |-  A  =  ran  (  x  e.  X  |->  { y  e.  X  |  -.  y R x } )   &    |-  B  =  ran  (  x  e.  X  |->  { y  e.  X  |  -.  x R y }
 )   &    |-  C  =  ran  (  a  e.  X ,  b  e.  X  |->  { y  e.  X  |  ( -.  y R a  /\  -.  b R y ) } )   =>    |-  ( R  e.  TosetRel  ->  ( fi `  ( A  u.  B ) )  =  ( ( A  u.  B )  u.  C ) )
 
Theoremordtbas 16884* In a total order, the finite intersections of the open rays generates the set of open intervals, but no more - these four collections form a subbasis for the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  X  =  dom  R   &    |-  A  =  ran  (  x  e.  X  |->  { y  e.  X  |  -.  y R x } )   &    |-  B  =  ran  (  x  e.  X  |->  { y  e.  X  |  -.  x R y }
 )   &    |-  C  =  ran  (  a  e.  X ,  b  e.  X  |->  { y  e.  X  |  ( -.  y R a  /\  -.  b R y ) } )   =>    |-  ( R  e.  TosetRel  ->  ( fi `  ( { X }  u.  ( A  u.  B ) ) )  =  ( ( { X }  u.  ( A  u.  B ) )  u.  C ) )
 
Theoremordttopon 16885 Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  X  =  dom  R   =>    |-  ( R  e.  V  ->  (ordTop `  R )  e.  (TopOn `  X ) )
 
Theoremordtopn1 16886* An upward ray  ( P ,  +oo ) is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  V  /\  P  e.  X ) 
 ->  { x  e.  X  |  -.  x R P }  e.  (ordTop `  R ) )
 
Theoremordtopn2 16887* A downward ray  (  -oo ,  P ) is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  V  /\  P  e.  X ) 
 ->  { x  e.  X  |  -.  P R x }  e.  (ordTop `  R ) )
 
Theoremordtopn3 16888* An open interval  ( A ,  B ) is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  { x  e.  X  |  ( -.  x R A  /\  -.  B R x ) }  e.  (ordTop `  R ) )
 
Theoremordtcld1 16889* A downward ray  (  -oo ,  P ] is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  V  /\  P  e.  X ) 
 ->  { x  e.  X  |  x R P }  e.  ( Clsd `  (ordTop `  R ) ) )
 
Theoremordtcld2 16890* An upward ray  [ P ,  +oo ) is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  V  /\  P  e.  X ) 
 ->  { x  e.  X  |  P R x }  e.  ( Clsd `  (ordTop `  R ) ) )
 
Theoremordtcld3 16891* An closed interval  [ A ,  B ] is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  { x  e.  X  |  ( A R x 
 /\  x R B ) }  e.  ( Clsd `  (ordTop `  R ) ) )
 
Theoremordttop 16892 The order topology is a topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( R  e.  V  ->  (ordTop `  R )  e.  Top )
 
Theoremordtcnv 16893 The order dual generates the same topology as the original order. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( R  e.  PosetRel  ->  (ordTop `  `' R )  =  (ordTop `  R ) )
 
Theoremordtrest 16894 The subspace topology of an order topology is in general finer than the topology generated by the restricted order, but we do have inclusion in one direction. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( ( R  e.  PosetRel  /\  A  e.  V ) 
 ->  (ordTop `  ( R  i^i  ( A  X.  A ) ) )  C_  ( (ordTop `  R )t  A ) )
 
Theoremordtrest2lem 16895* Lemma for ordtrest2 16896. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  X  =  dom  R   &    |-  ( ph  ->  R  e.  TosetRel  )   &    |-  ( ph  ->  A  C_  X )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A )
 )  ->  { z  e.  X  |  ( x R z  /\  z R y ) }  C_  A )   =>    |-  ( ph  ->  A. v  e.  ran  (  z  e.  X  |->  { w  e.  X  |  -.  w R z } ) ( v  i^i  A )  e.  (ordTop `  ( R  i^i  ( A  X.  A ) ) ) )
 
Theoremordtrest2 16896* An interval-closed set  A in a total order has the same subspace topology as the restricted order topology. (An interval-closed set is the same thing as an open or half-open or closed interval in  RR, but in other sets like  QQ there are interval-closed sets like  ( pi ,  +oo )  i^i  QQ that are not intervals.) (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  X  =  dom  R   &    |-  ( ph  ->  R  e.  TosetRel  )   &    |-  ( ph  ->  A  C_  X )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A )
 )  ->  { z  e.  X  |  ( x R z  /\  z R y ) }  C_  A )   =>    |-  ( ph  ->  (ordTop `  ( R  i^i  ( A  X.  A ) ) )  =  ( (ordTop `  R )t  A ) )
 
Theoremletopon 16897 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  (ordTop `  <_  )  e.  (TopOn `  RR* )
 
Theoremletop 16898 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  (ordTop `  <_  )  e. 
 Top
 
Theoremletopuni 16899 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  RR*  =  U. (ordTop `  <_  )
 
Theoremxrstopn 16900 The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  (ordTop `  <_  )  =  ( TopOpen `  RR* s )
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