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Theorem List for Metamath Proof Explorer - 16701-16800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeltg3i 16701 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 16702* 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 16703* 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 16704 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 16705* 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 16706 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 16707 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 16708 Subset relation for generated topologies. (Contributed by NM, 7-May-2007.)
 |-  ( ( C  e.  V  /\  B  C_  C )  ->  ( topGen `  B )  C_  ( topGen `  C ) )
 
Theoremtgcl 16709 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 16710 The property tgcl 16709 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 16711 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 16712 A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
 |-  ( J  e.  Top  ->  J  e.  TopBases )
 
Theoremtgtop 16713 A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
 |-  ( J  e.  Top  ->  ( topGen `  J )  =  J )
 
Theoremeltop 16714 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 16715* 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 16716* 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 16717 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 16718 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 16719* 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 16720 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 16721 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 16722 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 16723 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 16724  { (/) } is the only topology with one element. (Contributed by FL, 18-Aug-2008.)
 |-  ( J  e.  Top  ->  ( J  ~~  1o  <->  J  =  { (/)
 } ) )
 
Theoremen2top 16725 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 16726 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 16727* 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 16728 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 16729* 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 16730 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 16731 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 16732 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 16733* 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 16734* A version of bastop1 16733 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 16735 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 16736 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 16737 The singleton of the empty set is a topology on the empty set. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |- 
 { (/) }  e.  (TopOn `  (/) )
 
Theoremsn0top 16738 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 16739 A lemma to eliminate some sethood hypotheses when dealing with the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |- 
 { (/) ,  (  _I  `  A ) }  =  { (/) ,  A }
 
Theoremindistopon 16740 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 16741 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 16742 The base set of the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  (  _I  `  A )  =  U. { (/) ,  A }
 
Theoremfctop 16743* 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 16744* The finite complement topology on a set  A. Example 3 in [Munkres] p. 77. (This version of fctop 16743 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 16745* 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 16746* 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 16747* 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 16748* 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 16749 The indiscrete topology on a set  A expressed as a topological space, using explicit structure component references. Compare with indistps 16750 and indistps2 16751. The advantage of this version is that the actual function for the structure is evident, and df-ndx 13153 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 13155 and df-tset 13229 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 16750 instead. (New usage is discouraged.) (Contributed by FL, 19-Jul-2006.)
 |-  A  e.  _V   &    |-  K  =  { <. 1 ,  A >. ,  <. 9 ,  { (/)
 ,  A } >. }   =>    |-  K  e.  TopSp
 
Theoremindistps 16750 The indiscrete topology on a set  A expressed as a topological space, using implicit structure indices. The advantage of this version over indistpsx 16749 is that it is independent of the indices of the component definitions df-base 13155 and df-tset 13229, and if they are changed in the future, this theorem will not be affected. The advantage over indistps2 16751 is that it is easy to eliminate the hypotheses with eqid 2285 and vtoclg 2845 to result in a closed theorem. Theorems indistpsALT 16752 and indistps2ALT 16753 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 16751 The indiscrete topology on a set  A expressed as a topological space, using direct component assignments. Compare with indistps 16750. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 16752 and indistps2ALT 16753 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 16752 The indiscrete topology on a set  A expressed as a topological space. Here we show how to derive the structural version indistps 16750 from the direct component assignment version indistps2 16751. (Contributed by NM, 24-Oct-2012.) (Proof modification is discouraged.)
 |-  A  e.  _V   &    |-  K  =  { <. ( Base `  ndx ) ,  A >. , 
 <. (TopSet `  ndx ) ,  { (/) ,  A } >. }   =>    |-  K  e.  TopSp
 
Theoremindistps2ALT 16753 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 16751 from the structural version indistps 16750. (Contributed by NM, 24-Oct-2012.) (Proof modification is discouraged.)
 |-  ( Base `  K )  =  A   &    |-  ( TopOpen `  K )  =  { (/) ,  A }   =>    |-  K  e.  TopSp
 
Theoremdistps 16754 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 16755 Extend class notation with the set of closed sets of a topology.
 class  Clsd
 
Syntaxcnt 16756 Extend class notation with interior of a subset of a topology base set.
 class  int
 
Syntaxccl 16757 Extend class notation with closure of a subset of a topology base set.
 class  cls
 
Definitiondf-cld 16758* 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 16759* Define a function on topologies whose value is the interior function on the subsets of the base set. See ntrval 16775. (Contributed by NM, 10-Sep-2006.)
 |- 
 int  =  ( j  e.  Top  |->  ( x  e. 
 ~P U. j  |->  U. (
 j  i^i  ~P x ) ) )
 
Definitiondf-cls 16760* Define a function on topologies whose value is the closure function on the subsets of the base set. See clsval 16776. (Contributed by NM, 3-Oct-2006.)
 |- 
 cls  =  ( j  e.  Top  |->  ( x  e. 
 ~P U. j  |->  |^| { y  e.  ( Clsd `  j )  |  x  C_  y }
 ) )
 
Theoremfncld 16761 The closed-set generator is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |- 
 Clsd  Fn  Top
 
Theoremcldval 16762* 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 16763* 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 16764* 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 16765 Reverse closure of the closed set operation. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( C  e.  ( Clsd `  J )  ->  J  e.  Top )
 
Theoremiscld 16766 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 16767 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 16768 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 16769 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 16770 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 16771 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 16772 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 16773 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 16774 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 16775 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 16776* 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 16777 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 16778* 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 16779 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 16780 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 16781 A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.)
 |-  ( S  e.  ( Clsd `  J )  ->  ( ( cls `  J ) `  S )  =  S )
 
Theoremincld 16782 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 16783* 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 16784* 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 16785 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 16786 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 16787 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 16788 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 16789 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 16790 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 16791 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 16792 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 16793 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 16794 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 16795 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 16796 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 16797 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 16798 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 16799 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 16800 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 ) ) )
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