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Theorem List for Metamath Proof Explorer - 16501-16600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtopontopi 16501 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  J  e.  (TopOn `  B )   =>    |-  J  e.  Top
 
Theoremtoponunii 16502 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  J  e.  (TopOn `  B )   =>    |-  B  =  U. J
 
Theoremtoptopon 16503 Alternative definition of  Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
 
Theoremtopgele 16504 The topologies over the same set have a greatest element (the discrete topology) and a least element (the indiscrete topology). (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 16-Sep-2015.)
 |-  ( J  e.  (TopOn `  X )  ->  ( { (/) ,  X }  C_  J  /\  J  C_  ~P X ) )
 
Theoremtopsn 16505 The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 3721). (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
 |-  ( J  e.  (TopOn ` 
 { A } )  ->  J  =  ~P { A } )
 
Theoremistps 16506 Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  A  =  ( Base `  K )   &    |-  J  =  (
 TopOpen `  K )   =>    |-  ( K  e.  TopSp  <->  J  e.  (TopOn `  A ) )
 
Theoremistps2 16507 Express the predicate "is a topological space." (Contributed by NM, 20-Oct-2012.)
 |-  A  =  ( Base `  K )   &    |-  J  =  (
 TopOpen `  K )   =>    |-  ( K  e.  TopSp  <->  ( J  e.  Top  /\  A  =  U. J ) )
 
Theoremtpsuni 16508 The base set of a topological space. (Contributed by FL, 27-Jun-2014.)
 |-  A  =  ( Base `  K )   &    |-  J  =  (
 TopOpen `  K )   =>    |-  ( K  e.  TopSp  ->  A  =  U. J )
 
Theoremtpstop 16509 The topology extractor on a topological space is a topology. (Contributed by FL, 27-Jun-2014.)
 |-  J  =  ( TopOpen `  K )   =>    |-  ( K  e.  TopSp  ->  J  e.  Top )
 
Theoremtpspropd 16510 A topological space depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  ( ph  ->  ( Base `  K )  =  ( Base `  L )
 )   &    |-  ( ph  ->  ( TopOpen `  K )  =  (
 TopOpen `  L ) )   =>    |-  ( ph  ->  ( K  e.  TopSp 
 <->  L  e.  TopSp ) )
 
Theoremtpsprop2d 16511 A topological space depends only on the base and topology components. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( ph  ->  ( Base `  K )  =  ( Base `  L )
 )   &    |-  ( ph  ->  (TopSet `  K )  =  (TopSet `  L ) )   =>    |-  ( ph  ->  ( K  e.  TopSp  <->  L  e.  TopSp ) )
 
Theoremtopontopn 16512 Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  A  =  ( Base `  K )   &    |-  J  =  (TopSet `  K )   =>    |-  ( J  e.  (TopOn `  A )  ->  J  =  ( TopOpen `  K )
 )
 
Theoremtsettps 16513 If the topology component is already correctly truncated, then it forms a topological space (with the topology extractor function coming out the same as the component). (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  A  =  ( Base `  K )   &    |-  J  =  (TopSet `  K )   =>    |-  ( J  e.  (TopOn `  A )  ->  K  e.  TopSp )
 
Theoremistpsi 16514 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 16515 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 16516 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 16517* 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 16518* 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 16519* 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 16520 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 16521* 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 16522* 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 16523 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 16524* 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 16525* 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 16526* Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 16539) that  ( topGen `  B ) is indeed a topology (on  U. B; see unitg 16537). (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 16527 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 16528* 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 16529* 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 16530 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 16531 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 16532* 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 16533* 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 16534 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 16535* 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 16536 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 16537 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 16538 Subset relation for generated topologies. (Contributed by NM, 7-May-2007.)
 |-  ( ( C  e.  V  /\  B  C_  C )  ->  ( topGen `  B )  C_  ( topGen `  C ) )
 
Theoremtgcl 16539 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 16540 The property tgcl 16539 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 16541 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 16542 A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
 |-  ( J  e.  Top  ->  J  e.  TopBases )
 
Theoremtgtop 16543 A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
 |-  ( J  e.  Top  ->  ( topGen `  J )  =  J )
 
Theoremeltop 16544 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 16545* 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 16546* 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 16547 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 16548 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 16549* 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 16550 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 16551 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 16552 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 16553 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 16554  { (/) } is the only topology with one element. (Contributed by FL, 18-Aug-2008.)
 |-  ( J  e.  Top  ->  ( J  ~~  1o  <->  J  =  { (/)
 } ) )
 
Theoremen2top 16555 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 16556 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 16557* 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 16558 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 16559* 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 16560 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 16561 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 16562 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 16563* 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 16564* A version of bastop1 16563 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 16565 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 16566 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 16567 The singleton of the empty set is a topology on the empty set. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |- 
 { (/) }  e.  (TopOn `  (/) )
 
Theoremsn0top 16568 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 16569 A lemma to eliminate some sethood hypotheses when dealing with the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |- 
 { (/) ,  (  _I  `  A ) }  =  { (/) ,  A }
 
Theoremindistopon 16570 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 16571 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 16572 Th base set of the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  (  _I  `  A )  =  U. { (/) ,  A }
 
Theoremfctop 16573* 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 16574* The finite complement topology on a set  A. Example 3 in [Munkres] p. 77. (This version of fctop 16573 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 16575* 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 16576* 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 16577* 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 16578* 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 16579 The indiscrete topology on a set  A expressed as a topological space, using explicit structure component references. Compare with indistps 16580 and indistps2 16581. The advantage of this version is that the actual function for the structure is evident, and df-ndx 13025 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 13027 and df-tset 13101 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 16580 instead. (Contributed by FL, 19-Jul-2006.)
 |-  A  e.  _V   &    |-  K  =  { <. 1 ,  A >. ,  <. 9 ,  { (/)
 ,  A } >. }   =>    |-  K  e.  TopSp
 
Theoremindistps 16580 The indiscrete topology on a set  A expressed as a topological space, using implicit structure indices. The advantage of this version over indistpsx 16579 is that it is independent of the indices of the component definitions df-base 13027 and df-tset 13101, and if they are changed in the future, this theorem will not be affected. The advantage over indistps2 16581 is that it is easy to eliminate the hypotheses with eqid 2253 and vtoclg 2781 to result in a closed theorem. Theorems indistpsALT 16582 and indistps2ALT 16583 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 16581 The indiscrete topology on a set  A expressed as a topological space, using direct component assignments. Compare with indistps 16580. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 16582 and indistps2ALT 16583 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 16582 The indiscrete topology on a set  A expressed as a topological space. Here we show how to derive the structural version indistps 16580 from the direct component assignment version indistps2 16581. (Contributed by NM, 24-Oct-2012.) (Proof modification is discouraged.)
 |-  A  e.  _V   &    |-  K  =  { <. ( Base `  ndx ) ,  A >. , 
 <. (TopSet `  ndx ) ,  { (/) ,  A } >. }   =>    |-  K  e.  TopSp
 
Theoremindistps2ALT 16583 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 16581 from the structural version indistps 16580. (Contributed by NM, 24-Oct-2012.) (Proof modification is discouraged.)
 |-  ( Base `  K )  =  A   &    |-  ( TopOpen `  K )  =  { (/) ,  A }   =>    |-  K  e.  TopSp
 
Theoremdistps 16584 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 16585 Extend class notation with the set of closed sets of a topology.
 class  Clsd
 
Syntaxcnt 16586 Extend class notation with interior of a subset of a topology base set.
 class  int
 
Syntaxccl 16587 Extend class notation with closure of a subset of a topology base set.
 class  cls
 
Definitiondf-cld 16588* 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 16589* Define a function on topologies whose value is the interior function on the subsets of the base set. See ntrval 16605. (Contributed by NM, 10-Sep-2006.)
 |- 
 int  =  ( j  e.  Top  |->  ( x  e. 
 ~P U. j  |->  U. (
 j  i^i  ~P x ) ) )
 
Definitiondf-cls 16590* Define a function on topologies whose value is the closure function on the subsets of the base set. See clsval 16606. (Contributed by NM, 3-Oct-2006.)
 |- 
 cls  =  ( j  e.  Top  |->  ( x  e. 
 ~P U. j  |->  |^| { y  e.  ( Clsd `  j )  |  x  C_  y }
 ) )
 
Theoremfncld 16591 The closed-set generator is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |- 
 Clsd  Fn  Top
 
Theoremcldval 16592* 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 16593* 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 16594* 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 16595 Reverse closure of the closed set operation. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( C  e.  ( Clsd `  J )  ->  J  e.  Top )
 
Theoremiscld 16596 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 16597 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 16598 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 16599 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 16600 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 )
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