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Theorem List for Metamath Proof Explorer - 16601-16700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeltopss 16601 A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  A  e.  J ) 
 ->  A  C_  X )
 
Theoremriinopn 16602* A finite indexed relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  B  e.  J )  ->  ( X  i^i  |^|_ x  e.  A  B )  e.  J )
 
Theoremrintopn 16603 A finite relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  A  C_  J  /\  A  e.  Fin )  ->  ( X  i^i  |^| A )  e.  J )
 
TheoremeltopspOLD 16604 Construct a topological space from a topology and vice-versa. We say that  A is a topology on  U. A. (This could be proved more efficiently from istpsOLD 16606, but the proof here does not require the Axiom of Regularity.) (Contributed by NM, 8-Mar-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( <. U. J ,  J >.  e.  TopSp OLD  <->  J  e.  Top )
 
TheoremtpsexOLD 16605 Existence implied by membership in a topological space. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( <. A ,  J >.  e.  TopSp OLD  ->  ( A  e.  _V  /\  J  e.  _V ) )
 
TheoremistpsOLD 16606 Express the predicate "is a topological space." (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( <. A ,  J >.  e.  TopSp OLD  <->  ( J  e.  Top  /\  A  =  U. J ) )
 
Theoremistps2OLD 16607 Express the predicate "is a topological space." (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( <. A ,  J >.  e.  TopSp OLD  <->  ( ( J  e.  Top  /\  J  C_  ~P A )  /\  ( (/) 
 e.  J  /\  A  e.  J ) ) )
 
Theoremistps3OLD 16608* A standard textbook definition of a topological space. (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( <. A ,  J >.  e.  TopSp OLD  <->  ( ( J 
 C_  ~P A  /\  (/)  e.  J  /\  A  e.  J ) 
 /\  ( A. x ( x  C_  J  ->  U. x  e.  J ) 
 /\  A. x  e.  J  A. y  e.  J  ( x  i^i  y )  e.  J ) ) )
 
Theoremistps4OLD 16609* A standard textbook definition of a topological space. (Contributed by NM, 19-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( <. A ,  J >.  e.  TopSp OLD  <->  ( ( J 
 C_  ~P A  /\  (/)  e.  J  /\  A  e.  J ) 
 /\  ( A. x ( x  C_  J  ->  U. x  e.  J ) 
 /\  A. x ( ( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) ) ) )
 
Theoremistps5OLD 16610* A standard textbook definition of a topological space  <. A ,  J >.: a topology on  A is a collection  J of subsets of  A such that  (/) and  A are in  J and that is closed under union and finite intersection. Definition of topological space in [Munkres] p. 76. (Contributed by NM, 19-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( <. A ,  J >.  e.  TopSp OLD  <->  ( ( A. x  e.  J  x  C_  A  /\  (/)  e.  J  /\  A  e.  J ) 
 /\  ( A. x ( x  C_  J  ->  U. x  e.  J ) 
 /\  A. x ( ( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) ) ) )
 
Theoremistopon 16611 Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  ( J  e.  (TopOn `  B )  <->  ( J  e.  Top  /\  B  =  U. J ) )
 
Theoremtopontop 16612 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( J  e.  (TopOn `  B )  ->  J  e.  Top )
 
Theoremtoponuni 16613 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 )
 
Theoremtoponmax 16614 The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( J  e.  (TopOn `  B )  ->  B  e.  J )
 
Theoremtoponss 16615 A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  J )  ->  A  C_  X )
 
Theoremtoponcom 16616 If  K is a topology on the base set of topology  J, then  J is a topology on the base of  K. (Contributed by Mario Carneiro, 22-Aug-2015.)
 |-  ( ( J  e.  Top  /\  K  e.  (TopOn `  U. J ) )  ->  J  e.  (TopOn `  U. K ) )
 
Theoremtopontopi 16617 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 16618 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 16619 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 16620 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 16621 The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 3781). (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
 |-  ( J  e.  (TopOn ` 
 { A } )  ->  J  =  ~P { A } )
 
Theoremistps 16622 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 16623 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 16624 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 16625 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 16626 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 16627 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 16628 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 16629 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 16630 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 16631 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 16632 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 16633* 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 16634* 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 16635* 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 16636 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 16637* 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 16638* 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 16639 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 16640* 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 16641* 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 16642* Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 16655) that  ( topGen `  B ) is indeed a topology (on  U. B; see unitg 16653). (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 16643 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 16644* 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 16645* 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 16646 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 16647 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 16648* 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 16649* 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 16650 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 16651* 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 16652 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 16653 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 16654 Subset relation for generated topologies. (Contributed by NM, 7-May-2007.)
 |-  ( ( C  e.  V  /\  B  C_  C )  ->  ( topGen `  B )  C_  ( topGen `  C ) )
 
Theoremtgcl 16655 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 16656 The property tgcl 16655 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 16657 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 16658 A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
 |-  ( J  e.  Top  ->  J  e.  TopBases )
 
Theoremtgtop 16659 A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
 |-  ( J  e.  Top  ->  ( topGen `  J )  =  J )
 
Theoremeltop 16660 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 16661* 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 16662* 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 16663 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 16664 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 16665* 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 16666 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 16667 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 16668 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 16669 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 16670  { (/) } is the only topology with one element. (Contributed by FL, 18-Aug-2008.)
 |-  ( J  e.  Top  ->  ( J  ~~  1o  <->  J  =  { (/)
 } ) )
 
Theoremen2top 16671 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 16672 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 16673* 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 16674 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 16675* 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 16676 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 16677 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 16678 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 16679* 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 16680* A version of bastop1 16679 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 16681 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 16682 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 16683 The singleton of the empty set is a topology on the empty set. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |- 
 { (/) }  e.  (TopOn `  (/) )
 
Theoremsn0top 16684 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 16685 A lemma to eliminate some sethood hypotheses when dealing with the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |- 
 { (/) ,  (  _I  `  A ) }  =  { (/) ,  A }
 
Theoremindistopon 16686 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 16687 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 16688 Th base set of the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  (  _I  `  A )  =  U. { (/) ,  A }
 
Theoremfctop 16689* 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 16690* The finite complement topology on a set  A. Example 3 in [Munkres] p. 77. (This version of fctop 16689 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 16691* 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 16692* 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 16693* 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 16694* 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 16695 The indiscrete topology on a set  A expressed as a topological space, using explicit structure component references. Compare with indistps 16696 and indistps2 16697. The advantage of this version is that the actual function for the structure is evident, and df-ndx 13099 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 13101 and df-tset 13175 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 16696 instead. (Contributed by FL, 19-Jul-2006.)
 |-  A  e.  _V   &    |-  K  =  { <. 1 ,  A >. ,  <. 9 ,  { (/)
 ,  A } >. }   =>    |-  K  e.  TopSp
 
Theoremindistps 16696 The indiscrete topology on a set  A expressed as a topological space, using implicit structure indices. The advantage of this version over indistpsx 16695 is that it is independent of the indices of the component definitions df-base 13101 and df-tset 13175, and if they are changed in the future, this theorem will not be affected. The advantage over indistps2 16697 is that it is easy to eliminate the hypotheses with eqid 2256 and vtoclg 2811 to result in a closed theorem. Theorems indistpsALT 16698 and indistps2ALT 16699 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 16697 The indiscrete topology on a set  A expressed as a topological space, using direct component assignments. Compare with indistps 16696. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 16698 and indistps2ALT 16699 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 16698 The indiscrete topology on a set  A expressed as a topological space. Here we show how to derive the structural version indistps 16696 from the direct component assignment version indistps2 16697. (Contributed by NM, 24-Oct-2012.) (Proof modification is discouraged.)
 |-  A  e.  _V   &    |-  K  =  { <. ( Base `  ndx ) ,  A >. , 
 <. (TopSet `  ndx ) ,  { (/) ,  A } >. }   =>    |-  K  e.  TopSp
 
Theoremindistps2ALT 16699 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 16697 from the structural version indistps 16696. (Contributed by NM, 24-Oct-2012.) (Proof modification is discouraged.)
 |-  ( Base `  K )  =  A   &    |-  ( TopOpen `  K )  =  { (/) ,  A }   =>    |-  K  e.  TopSp
 
Theoremdistps 16700 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
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