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Theorem List for Metamath Proof Explorer - 17001-17100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremistpsi 17001 Properties that determine a topological space. (Contributed by NM, 20-Oct-2012.)
 |-  ( Base `  K )  =  A   &    |-  ( TopOpen `  K )  =  J   &    |-  A  =  U. J   &    |-  J  e.  Top   =>    |-  K  e.  TopSp
 
Theoremeltpsg 17002 Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  K  =  { <. (
 Base `  ndx ) ,  A >. ,  <. (TopSet `  ndx ) ,  J >. }   =>    |-  ( J  e.  (TopOn `  A )  ->  K  e.  TopSp )
 
Theoremeltpsi 17003 Properties that determine a topological space from a construction (using no explicit indices). (Contributed by NM, 20-Oct-2012.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  K  =  { <. (
 Base `  ndx ) ,  A >. ,  <. (TopSet `  ndx ) ,  J >. }   &    |-  A  =  U. J   &    |-  J  e.  Top   =>    |-  K  e.  TopSp
 
11.1.2  TopBases for topologies
 
Theoremisbasisg 17004* Express the predicate " B is a basis for a topology." (Contributed by NM, 17-Jul-2006.)
 |-  ( B  e.  C  ->  ( B  e.  TopBases  <->  A. x  e.  B  A. y  e.  B  ( x  i^i  y ) 
 C_  U. ( B  i^i  ~P ( x  i^i  y
 ) ) ) )
 
Theoremisbasis2g 17005* Express the predicate " B is a basis for a topology." (Contributed by NM, 17-Jul-2006.)
 |-  ( B  e.  C  ->  ( B  e.  TopBases  <->  A. x  e.  B  A. y  e.  B  A. z  e.  ( x  i^i  y ) E. w  e.  B  ( z  e.  w  /\  w  C_  ( x  i^i  y ) ) ) )
 
Theoremisbasis3g 17006* Express the predicate " B is a basis for a topology." Definition of basis in [Munkres] p. 78. (Contributed by NM, 17-Jul-2006.)
 |-  ( B  e.  C  ->  ( B  e.  TopBases  <->  ( A. x  e.  B  x  C_  U. B  /\  A. x  e.  U. B E. y  e.  B  x  e.  y  /\  A. x  e.  B  A. y  e.  B  A. z  e.  ( x  i^i  y
 ) E. w  e.  B  ( z  e.  w  /\  w  C_  ( x  i^i  y ) ) ) ) )
 
Theorembasis1 17007 Property of a basis. (Contributed by NM, 16-Jul-2006.)
 |-  ( ( B  e.  TopBases  /\  C  e.  B  /\  D  e.  B )  ->  ( C  i^i  D )  C_  U. ( B  i^i  ~P ( C  i^i  D ) ) )
 
Theorembasis2 17008* Property of a basis. (Contributed by NM, 17-Jul-2006.)
 |-  ( ( ( B  e.  TopBases  /\  C  e.  B )  /\  ( D  e.  B  /\  A  e.  ( C  i^i  D ) ) )  ->  E. x  e.  B  ( A  e.  x  /\  x  C_  ( C  i^i  D ) ) )
 
Theoremfiinbas 17009* If a set is closed under finite intersection, then it is a basis for a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( B  e.  C  /\  A. x  e.  B  A. y  e.  B  ( x  i^i  y )  e.  B )  ->  B  e.  TopBases )
 
Theorembasdif0 17010 A basis is not affected by the addition or removal of the empty set. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( ( B  \  { (/) } )  e.  TopBases  <->  B  e.  TopBases )
 
Theorembaspartn 17011* A disjoint system of sets is a basis for a topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( ( P  e.  V  /\  A. x  e.  P  A. y  e.  P  ( x  =  y  \/  ( x  i^i  y )  =  (/) ) )  ->  P  e. 
 TopBases )
 
Theoremtgval 17012* The topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
 |-  ( B  e.  V  ->  ( topGen `  B )  =  { x  |  x  C_ 
 U. ( B  i^i  ~P x ) } )
 
Theoremtgval2 17013* Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 17026) that  ( topGen `  B ) is indeed a topology (on  U. B; see unitg 17024). (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
 |-  ( B  e.  V  ->  ( topGen `  B )  =  { x  |  ( x  C_  U. B  /\  A. y  e.  x  E. z  e.  B  (
 y  e.  z  /\  z  C_  x ) ) } )
 
Theoremeltg 17014 Membership in a topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
 |-  ( B  e.  V  ->  ( A  e.  ( topGen `
  B )  <->  A  C_  U. ( B  i^i  ~P A ) ) )
 
Theoremeltg2 17015* Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
 |-  ( B  e.  V  ->  ( A  e.  ( topGen `
  B )  <->  ( A  C_  U. B  /\  A. x  e.  A  E. y  e.  B  ( x  e.  y  /\  y  C_  A ) ) ) )
 
Theoremeltg2b 17016* Membership in a topology generated by a basis. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 10-Jan-2015.)
 |-  ( B  e.  V  ->  ( A  e.  ( topGen `
  B )  <->  A. x  e.  A  E. y  e.  B  ( x  e.  y  /\  y  C_  A ) ) )
 
Theoremeltg4i 17017 An open set in a topology generated by a basis is the union of all basic open sets contained in it. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( A  e.  ( topGen `
  B )  ->  A  =  U. ( B  i^i  ~P A ) )
 
Theoremeltg3i 17018 The union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 30-Aug-2015.)
 |-  ( ( B  e.  V  /\  A  C_  B )  ->  U. A  e.  ( topGen `
  B ) )
 
Theoremeltg3 17019* Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Mario Carneiro, 30-Aug-2015.)
 |-  ( B  e.  V  ->  ( A  e.  ( topGen `
  B )  <->  E. x ( x 
 C_  B  /\  A  =  U. x ) ) )
 
Theoremtgval3 17020* Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80. (Contributed by NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.)
 |-  ( B  e.  V  ->  ( topGen `  B )  =  { x  |  E. y ( y  C_  B  /\  x  =  U. y ) } )
 
Theoremtg1 17021 Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
 |-  ( A  e.  ( topGen `
  B )  ->  A  C_  U. B )
 
Theoremtg2 17022* Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
 |-  ( ( A  e.  ( topGen `  B )  /\  C  e.  A ) 
 ->  E. x  e.  B  ( C  e.  x  /\  x  C_  A ) )
 
Theorembastg 17023 A member of a basis is a subset of the topology it generates. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
 |-  ( B  e.  V  ->  B  C_  ( topGen `  B ) )
 
Theoremunitg 17024 The topology generated by a basis 
B is a topology on 
U. B. Importantly, this theorem means that we don't have to specify separately the base set for the topological space generated by a basis. In other words, any member of the class  TopBases completely specifies the basis it corresponds to. (Contributed by NM, 16-Jul-2006.)
 |-  ( B  e.  V  ->  U. ( topGen `  B )  =  U. B )
 
Theoremtgss 17025 Subset relation for generated topologies. (Contributed by NM, 7-May-2007.)
 |-  ( ( C  e.  V  /\  B  C_  C )  ->  ( topGen `  B )  C_  ( topGen `  C ) )
 
Theoremtgcl 17026 Show that a basis generates a topology. Remark in [Munkres] p. 79. (Contributed by NM, 17-Jul-2006.)
 |-  ( B  e.  TopBases  ->  (
 topGen `  B )  e. 
 Top )
 
Theoremtgclb 17027 The property tgcl 17026 can be reversed: if the topology generated by  B is actually a topology, then 
B must be a topological basis. This yields an alternative definition of  TopBases. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  ( B  e.  TopBases  <->  ( topGen `  B )  e.  Top )
 
Theoremtgtopon 17028 A basis generates a topology on 
U. B. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( B  e.  TopBases  ->  (
 topGen `  B )  e.  (TopOn `  U. B ) )
 
Theoremtopbas 17029 A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
 |-  ( J  e.  Top  ->  J  e.  TopBases )
 
Theoremtgtop 17030 A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
 |-  ( J  e.  Top  ->  ( topGen `  J )  =  J )
 
Theoremeltop 17031 Membership in a topology, expressed without quantifiers. (Contributed by NM, 19-Jul-2006.)
 |-  ( J  e.  Top  ->  ( A  e.  J  <->  A 
 C_  U. ( J  i^i  ~P A ) ) )
 
Theoremeltop2 17032* Membership in a topology. (Contributed by NM, 19-Jul-2006.)
 |-  ( J  e.  Top  ->  ( A  e.  J  <->  A. x  e.  A  E. y  e.  J  ( x  e.  y  /\  y  C_  A ) ) )
 
Theoremeltop3 17033* Membership in a topology. (Contributed by NM, 19-Jul-2006.)
 |-  ( J  e.  Top  ->  ( A  e.  J  <->  E. x ( x  C_  J  /\  A  =  U. x ) ) )
 
Theoremfibas 17034 A collection of finite intersections is a basis. The initial set is a subbasis for the topology. (Contributed by Jeff Hankins, 25-Aug-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
 |-  ( fi `  A )  e.  TopBases
 
Theoremtgdom 17035 A space has no more open sets than subsets of a basis. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 9-Apr-2015.)
 |-  ( B  e.  V  ->  ( topGen `  B )  ~<_  ~P B )
 
Theoremtgiun 17036* The indexed union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  ( ( B  e.  V  /\  A. x  e.  A  C  e.  B )  ->  U_ x  e.  A  C  e.  ( topGen `  B ) )
 
Theoremtgidm 17037 The topology generator function is idempotent. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.)
 |-  ( B  e.  V  ->  ( topGen `  ( topGen `  B ) )  =  ( topGen `
  B ) )
 
Theorembastop 17038 Two ways to express that a basis is a topology. (Contributed by NM, 18-Jul-2006.)
 |-  ( B  e.  TopBases  ->  ( B  e.  Top  <->  ( topGen `  B )  =  B )
 )
 
Theoremtgtop11 17039 The topology generation function is one-to-one when applied to completed topologies. (Contributed by NM, 18-Jul-2006.)
 |-  ( ( J  e.  Top  /\  K  e.  Top  /\  ( topGen `  J )  =  ( topGen `  K )
 )  ->  J  =  K )
 
Theorem0top 17040 The singleton of the empty set is the only topology possible for an empty underlying set. (Contributed by NM, 9-Sep-2006.)
 |-  ( J  e.  Top  ->  ( U. J  =  (/)  <->  J  =  { (/) } ) )
 
Theoremen1top 17041  { (/) } is the only topology with one element. (Contributed by FL, 18-Aug-2008.)
 |-  ( J  e.  Top  ->  ( J  ~~  1o  <->  J  =  { (/)
 } ) )
 
Theoremen2top 17042 If a topology has two elements, it is the indiscrete topology. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  ( J  e.  (TopOn `  X )  ->  ( J  ~~  2o  <->  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) ) )
 
Theoremtgss3 17043 A criterion for determining whether one topology is finer than another. Lemma 2.2 of [Munkres] p. 80 using abbreviations. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
 |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( ( topGen `  B )  C_  ( topGen `  C )  <->  B  C_  ( topGen `  C ) ) )
 
Theoremtgss2 17044* A criterion for determining whether one topology is finer than another, based on a comparison of their bases. Lemma 2.2 of [Munkres] p. 80. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
 |-  ( ( B  e.  V  /\  U. B  =  U. C )  ->  (
 ( topGen `  B )  C_  ( topGen `  C )  <->  A. x  e.  U. B A. y  e.  B  ( x  e.  y  ->  E. z  e.  C  ( x  e.  z  /\  z  C_  y ) ) ) )
 
Theorembasgen 17045 Given a topology  J, show that a subset  B satisfying the third antecedent is a basis for it. Lemma 2.3 of [Munkres] p. 81 using abbreviations. (Contributed by NM, 22-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.)
 |-  ( ( J  e.  Top  /\  B  C_  J  /\  J  C_  ( topGen `  B ) )  ->  ( topGen `  B )  =  J )
 
Theorembasgen2 17046* Given a topology  J, show that a subset  B satisfying the third antecedent is a basis for it. Lemma 2.3 of [Munkres] p. 81. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
 |-  ( ( J  e.  Top  /\  B  C_  J  /\  A. x  e.  J  A. y  e.  x  E. z  e.  B  (
 y  e.  z  /\  z  C_  x ) ) 
 ->  ( topGen `  B )  =  J )
 
Theorem2basgen 17047 Conditions that determine the equality of two generated topologies. (Contributed by NM, 8-May-2007.) (Revised by Mario Carneiro, 2-Sep-2015.)
 |-  ( ( B  C_  C  /\  C  C_  ( topGen `
  B ) ) 
 ->  ( topGen `  B )  =  ( topGen `  C )
 )
 
Theoremtgfiss 17048 If a subbase is included into a topology, so is the generated topology. (Contributed by FL, 20-Apr-2012.) (Proof shortened by Mario Carneiro, 10-Jan-2015.)
 |-  ( ( J  e.  Top  /\  A  C_  J )  ->  ( topGen `  ( fi `  A ) )  C_  J )
 
Theoremtgdif0 17049 A generated topology is not affected by the addition or removal of the empty set from the base. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( topGen `  ( B  \  { (/) } ) )  =  ( topGen `  B )
 
Theorembastop1 17050* A subset of a topology is a basis for the topology iff every member of the topology is a union of members of the basis. We use the idiom " ( topGen `  B
)  =  J " to express " B is a basis for topology  J," since we do not have a separate notation for this. Definition 15.35 of [Schechter] p. 428. (Contributed by NM, 2-Feb-2008.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
 |-  ( ( J  e.  Top  /\  B  C_  J )  ->  ( ( topGen `  B )  =  J  <->  A. x  e.  J  E. y ( y  C_  B  /\  x  =  U. y ) ) )
 
Theorembastop2 17051* A version of bastop1 17050 that doesn't have  B  C_  J in the antecedent. (Contributed by NM, 3-Feb-2008.)
 |-  ( J  e.  Top  ->  ( ( topGen `  B )  =  J  <->  ( B  C_  J  /\  A. x  e.  J  E. y ( y  C_  B  /\  x  =  U. y ) ) ) )
 
11.1.3  Examples of topologies
 
Theoremdistop 17052 The discrete topology on a set  A. Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 17-Jul-2006.) (Revised by Mario Carneiro, 19-Mar-2015.)
 |-  ( A  e.  V  ->  ~P A  e.  Top )
 
Theoremdistopon 17053 The discrete topology on a set  A, with base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( A  e.  V  ->  ~P A  e.  (TopOn `  A ) )
 
Theoremsn0topon 17054 The singleton of the empty set is a topology on the empty set. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |- 
 { (/) }  e.  (TopOn `  (/) )
 
Theoremsn0top 17055 The singleton of the empty set is a topology. (Contributed by Stefan Allan, 3-Mar-2006.) (Proof shortened by Mario Carneiro, 13-Aug-2015.)
 |- 
 { (/) }  e.  Top
 
Theoremindislem 17056 A lemma to eliminate some sethood hypotheses when dealing with the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |- 
 { (/) ,  (  _I  `  A ) }  =  { (/) ,  A }
 
Theoremindistopon 17057 The indiscrete topology on a set  A. Part of Example 2 in [Munkres] p. 77. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( A  e.  V  ->  { (/) ,  A }  e.  (TopOn `  A )
 )
 
Theoremindistop 17058 The indiscrete topology on a set  A. Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 16-Jul-2006.) (Revised by Stefan Allan, 6-Nov-2008.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |- 
 { (/) ,  A }  e.  Top
 
Theoremindisuni 17059 The base set of the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  (  _I  `  A )  =  U. { (/) ,  A }
 
Theoremfctop 17060* The finite complement topology on a set  A. Example 3 in [Munkres] p. 77. (Contributed by FL, 15-Aug-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  ( A  e.  V  ->  { x  e.  ~P A  |  ( ( A  \  x )  e. 
 Fin  \/  x  =  (/) ) }  e.  (TopOn `  A ) )
 
Theoremfctop2 17061* The finite complement topology on a set  A. Example 3 in [Munkres] p. 77. (This version of fctop 17060 requires the Axiom of Infinity.) (Contributed by FL, 20-Aug-2006.)
 |-  ( A  e.  V  ->  { x  e.  ~P A  |  ( ( A  \  x )  ~<  om 
 \/  x  =  (/) ) }  e.  (TopOn `  A ) )
 
Theoremcctop 17062* The countable complement topology on a set  A. Example 4 in [Munkres] p. 77. (Contributed by FL, 23-Aug-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  ( A  e.  V  ->  { x  e.  ~P A  |  ( ( A  \  x )  ~<_  om 
 \/  x  =  (/) ) }  e.  (TopOn `  A ) )
 
Theoremppttop 17063* The particular point topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( ( A  e.  V  /\  P  e.  A )  ->  { x  e. 
 ~P A  |  ( P  e.  x  \/  x  =  (/) ) }  e.  (TopOn `  A )
 )
 
Theorempptbas 17064* The particular point topology is generated by a basis consisting of pairs  { x ,  P } for each  x  e.  A. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( ( A  e.  V  /\  P  e.  A )  ->  { x  e. 
 ~P A  |  ( P  e.  x  \/  x  =  (/) ) }  =  ( topGen `  ran  ( x  e.  A  |->  { x ,  P } ) ) )
 
Theoremepttop 17065* The excluded point topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( ( A  e.  V  /\  P  e.  A )  ->  { x  e. 
 ~P A  |  ( P  e.  x  ->  x  =  A ) }  e.  (TopOn `  A ) )
 
Theoremindistpsx 17066 The indiscrete topology on a set  A expressed as a topological space, using explicit structure component references. Compare with indistps 17067 and indistps2 17068. The advantage of this version is that the actual function for the structure is evident, and df-ndx 13464 is not needed, nor any other special definition outside of basic set theory. The disadvantage is that if the indices of the component definitions df-base 13466 and df-tset 13540 are changed in the future, this theorem will also have to be changed. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use; use indistps 17067 instead. (New usage is discouraged.) (Contributed by FL, 19-Jul-2006.)
 |-  A  e.  _V   &    |-  K  =  { <. 1 ,  A >. ,  <. 9 ,  { (/)
 ,  A } >. }   =>    |-  K  e.  TopSp
 
Theoremindistps 17067 The indiscrete topology on a set  A expressed as a topological space, using implicit structure indices. The advantage of this version over indistpsx 17066 is that it is independent of the indices of the component definitions df-base 13466 and df-tset 13540, and if they are changed in the future, this theorem will not be affected. The advantage over indistps2 17068 is that it is easy to eliminate the hypotheses with eqid 2435 and vtoclg 3003 to result in a closed theorem. Theorems indistpsALT 17069 and indistps2ALT 17070 show that the two forms can be derived from each other. (Contributed by FL, 19-Jul-2006.)
 |-  A  e.  _V   &    |-  K  =  { <. ( Base `  ndx ) ,  A >. , 
 <. (TopSet `  ndx ) ,  { (/) ,  A } >. }   =>    |-  K  e.  TopSp
 
Theoremindistps2 17068 The indiscrete topology on a set  A expressed as a topological space, using direct component assignments. Compare with indistps 17067. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 17069 and indistps2ALT 17070 show that the two forms can be derived from each other. (Contributed by NM, 24-Oct-2012.)
 |-  ( Base `  K )  =  A   &    |-  ( TopOpen `  K )  =  { (/) ,  A }   =>    |-  K  e.  TopSp
 
TheoremindistpsALT 17069 The indiscrete topology on a set  A expressed as a topological space. Here we show how to derive the structural version indistps 17067 from the direct component assignment version indistps2 17068. (Contributed by NM, 24-Oct-2012.) (Proof modification is discouraged.)
 |-  A  e.  _V   &    |-  K  =  { <. ( Base `  ndx ) ,  A >. , 
 <. (TopSet `  ndx ) ,  { (/) ,  A } >. }   =>    |-  K  e.  TopSp
 
Theoremindistps2ALT 17070 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 17068 from the structural version indistps 17067. (Contributed by NM, 24-Oct-2012.) (Proof modification is discouraged.)
 |-  ( Base `  K )  =  A   &    |-  ( TopOpen `  K )  =  { (/) ,  A }   =>    |-  K  e.  TopSp
 
Theoremdistps 17071 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 17072 Extend class notation with the set of closed sets of a topology.
 class  Clsd
 
Syntaxcnt 17073 Extend class notation with interior of a subset of a topology base set.
 class  int
 
Syntaxccl 17074 Extend class notation with closure of a subset of a topology base set.
 class  cls
 
Definitiondf-cld 17075* 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 17076* Define a function on topologies whose value is the interior function on the subsets of the base set. See ntrval 17092. (Contributed by NM, 10-Sep-2006.)
 |- 
 int  =  ( j  e.  Top  |->  ( x  e. 
 ~P U. j  |->  U. (
 j  i^i  ~P x ) ) )
 
Definitiondf-cls 17077* Define a function on topologies whose value is the closure function on the subsets of the base set. See clsval 17093. (Contributed by NM, 3-Oct-2006.)
 |- 
 cls  =  ( j  e.  Top  |->  ( x  e. 
 ~P U. j  |->  |^| { y  e.  ( Clsd `  j )  |  x  C_  y }
 ) )
 
Theoremfncld 17078 The closed-set generator is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |- 
 Clsd  Fn  Top
 
Theoremcldval 17079* 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 17080* 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 17081* 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 17082 Reverse closure of the closed set operation. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( C  e.  ( Clsd `  J )  ->  J  e.  Top )
 
Theoremiscld 17083 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 17084 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 17085 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 17086 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 17087 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 17088 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 17089 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 17090 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 17091 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 17092 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 17093* 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 17094 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 17095* 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 17096 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 17097 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 17098 A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.)
 |-  ( S  e.  ( Clsd `  J )  ->  ( ( cls `  J ) `  S )  =  S )
 
Theoremincld 17099 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 17100* 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 )
 )
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