P. 208 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 20701-20800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	istop2g 20701*	Express the predicate " J is a topology," using nonempty finite intersections instead of binary intersections as in istopg 20700. (Contributed by NM, 19-Jul-2006.)
|- ( J e. A -> ( J e. Top <-> ( A. x ( x C_ J -> U. x e. J ) /\ A. x ( ( x C_ J /\ x =/= (/) /\ x e. Fin ) -> |^| x e. J ) ) ) )
Theorem	uniopn 20702	The union of a subset of a topology (that is, the union of any family of open sets of a topology) is an open set. (Contributed by Stefan Allan, 27-Feb-2006.)
|- ( ( J e. Top /\ A C_ J ) -> U. A e. J )
Theorem	iunopn 20703*	The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)
|- ( ( J e. Top /\ A. x e. A B e. J ) -> U_ x e. A B e. J )
Theorem	inopn 20704	The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
|- ( ( J e. Top /\ A e. J /\ B e. J ) -> ( A i^i B ) e. J )
Theorem	fitop 20705	A topology is closed under finite intersections. (Contributed by Jeff Hankins, 7-Oct-2009.)
|- ( J e. Top -> ( fi ` J ) = J )
Theorem	fiinopn 20706	The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)
|- ( J e. Top -> ( ( A C_ J /\ A =/= (/) /\ A e. Fin ) -> |^| A e. J ) )
Theorem	iinopn 20707*	The intersection of a nonempty finite family of open sets is open. (Contributed by Mario Carneiro, 14-Sep-2014.)
|- ( ( J e. Top /\ ( A e. Fin /\ A =/= (/) /\ A. x e. A B e. J ) ) -> |^|_ x e. A B e. J )
Theorem	unopn 20708	The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( J e. Top /\ A e. J /\ B e. J ) -> ( A u. B ) e. J )
Theorem	0opn 20709	The empty set is an open subset of a topology. (Contributed by Stefan Allan, 27-Feb-2006.)
|- ( J e. Top -> (/) e. J )
Theorem	0ntop 20710	The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)
|- -. (/) e. Top
Theorem	topopn 20711	The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
|- X = U. J   =>    |- ( J e. Top -> X e. J )
Theorem	eltopss 20712	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 )
Theorem	riinopn 20713*	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 )
Theorem	rintopn 20714	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 )
12.1.1.2  Topologies on sets
Syntax	ctopon 20715	Syntax for the function of topologies on sets.
class TopOn
Definition	df-topon 20716*	Define the function that associates with a set the set of topologies on it. (Contributed by Stefan O'Rear, 31-Jan-2015.)
|- TopOn = ( b e. _V |-> { j e. Top | b = U. j } )
Theorem	istopon 20717	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 ) )
Theorem	topontop 20718	A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- ( J e. (TopOn ` B ) -> J e. Top )
Theorem	toponuni 20719	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 )
Theorem	topontopi 20720	A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- J e. (TopOn ` B )   =>    |- J e. Top
Theorem	toponunii 20721	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
Theorem	toptopon 20722	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 ) )
Theorem	toptopon2 20723	A topology is the same thing as a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)
|- ( J e. Top <-> J e. (TopOn ` U. J ) )
Theorem	topontopon 20724	A topology on a set is a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)
|- ( J e. (TopOn ` X ) -> J e. (TopOn ` U. J ) )
Theorem	funtopon 20725	The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.)
|- Fun TopOn
Theorem	toponsspwpw 20726	The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.)
|- (TopOn ` A ) C_ ~P ~P A
Theorem	dmtopon 20727	The domain of TopOn is _V. (Contributed by BJ, 29-Apr-2021.)
|- dom TopOn = _V
Theorem	fntopon 20728	The class TopOn is a function with domain _V. Analogue for topologies of fnmre 16251 for Moore collections. (Contributed by BJ, 29-Apr-2021.)
|- TopOn Fn _V
Theorem	toprntopon 20729	A topology is the same thing as a topology on a set (variable-free version). (Contributed by BJ, 27-Apr-2021.)
|- Top = U. ran TopOn
Theorem	toponmax 20730	The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- ( J e. (TopOn ` B ) -> B e. J )
Theorem	toponss 20731	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 )
Theorem	toponcom 20732	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 ) )
Theorem	toponcomb 20733	Biconditional form of toponcom 20732. (Contributed by BJ, 5-Dec-2021.)
|- ( ( J e. Top /\ K e. Top ) -> ( J e. (TopOn ` U. K ) <-> K e. (TopOn ` U. J ) ) )
Theorem	topgele 20734	The topologies over the same set have the greatest element (the discrete topology) and the 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 ) )
Theorem	topsn 20735	The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 4428). (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
|- ( J e. (TopOn ` { A } ) -> J = ~P { A } )
12.1.1.3  Topological spaces
Syntax	ctps 20736	Syntax for the class of topological spaces.
class TopSp
Definition	df-topsp 20737	Define the class of topological spaces (as extensible structures). (Contributed by Stefan O'Rear, 13-Aug-2015.)
|- TopSp = { f | ( TopOpen ` f ) e. (TopOn ` ( Base ` f ) ) }
Theorem	istps 20738	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 ) )
Theorem	istps2 20739	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 ) )
Theorem	tpsuni 20740	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 )
Theorem	tpstop 20741	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 )
Theorem	tpspropd 20742	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 ) )
Theorem	tpsprop2d 20743	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 ) )
Theorem	topontopn 20744	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 ) )
Theorem	tsettps 20745	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 )
Theorem	istpsi 20746	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
Theorem	eltpsg 20747	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 )
Theorem	eltpsi 20748	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
12.1.2  Topological bases
Syntax	ctb 20749	Syntax for the class of topological bases.
class TopBases
Definition	df-bases 20750*	Define the class of topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 20752). Note that "bases" is the plural of "basis." (Contributed by NM, 17-Jul-2006.)
|- TopBases = { x | A. y e. x A. z e. x ( y i^i z ) C_ U. ( x i^i ~P ( y i^i z ) ) }
Theorem	isbasisg 20751*	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 ) ) ) )
Theorem	isbasis2g 20752*	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 ) ) ) )
Theorem	isbasis3g 20753*	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 ) ) ) ) )
Theorem	basis1 20754	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 ) ) )
Theorem	basis2 20755*	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 ) ) )
Theorem	fiinbas 20756*	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 )
Theorem	basdif0 20757	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 )
Theorem	baspartn 20758*	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 )
Theorem	tgval 20759*	The topology generated by a basis. See also tgval2 20760 and tgval3 20767. (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 ) } )
Theorem	tgval2 20760*	Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 20773) that ( topGen ` B ) is indeed a topology (on U. B, see unitg 20771). See also tgval 20759 and tgval3 20767. (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 ) ) } )
Theorem	eltg 20761	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 ) ) )
Theorem	eltg2 20762*	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 ) ) ) )
Theorem	eltg2b 20763*	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 ) ) )
Theorem	eltg4i 20764	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 ) )
Theorem	eltg3i 20765	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 ) )
Theorem	eltg3 20766*	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 ) ) )
Theorem	tgval3 20767*	Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80. See also tgval 20759 and tgval2 20760. (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 ) } )
Theorem	tg1 20768	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 )
Theorem	tg2 20769*	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 ) )
Theorem	bastg 20770	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 ) )
Theorem	unitg 20771	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.) (Proof shortened by OpenAI, 30-Mar-2020.)
|- ( B e. V -> U. ( topGen ` B ) = U. B )
Theorem	tgss 20772	Subset relation for generated topologies. (Contributed by NM, 7-May-2007.)
|- ( ( C e. V /\ B C_ C ) -> ( topGen ` B ) C_ ( topGen ` C ) )
Theorem	tgcl 20773	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 )
Theorem	tgclb 20774	The property tgcl 20773 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 )
Theorem	tgtopon 20775	A basis generates a topology on U. B. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( B e. TopBases -> ( topGen ` B ) e. (TopOn ` U. B ) )
Theorem	topbas 20776	A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
|- ( J e. Top -> J e. TopBases )
Theorem	tgtop 20777	A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
|- ( J e. Top -> ( topGen ` J ) = J )
Theorem	eltop 20778	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 ) ) )
Theorem	eltop2 20779*	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 ) ) )
Theorem	eltop3 20780*	Membership in a topology. (Contributed by NM, 19-Jul-2006.)
|- ( J e. Top -> ( A e. J <-> E. x ( x C_ J /\ A = U. x ) ) )
Theorem	fibas 20781	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
Theorem	tgdom 20782	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 )
Theorem	tgiun 20783*	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 ) )
Theorem	tgidm 20784	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 ) )
Theorem	bastop 20785	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 ) )
Theorem	tgtop11 20786	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 )
Theorem	0top 20787	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 = { (/) } ) )
Theorem	en1top 20788	{ (/) } is the only topology with one element. (Contributed by FL, 18-Aug-2008.)
|- ( J e. Top -> ( J ~~ 1o <-> J = { (/) } ) )
Theorem	en2top 20789	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 =/= (/) ) ) )
Theorem	tgss3 20790	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 ) ) )
Theorem	tgss2 20791*	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 ) ) ) )
Theorem	basgen 20792	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 )
Theorem	basgen2 20793*	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 )
Theorem	2basgen 20794	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 ) )
Theorem	tgfiss 20795	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 )
Theorem	tgdif0 20796	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 )
Theorem	bastop1 20797*	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 ) ) )
Theorem	bastop2 20798*	A version of bastop1 20797 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 ) ) ) )
12.1.3  Examples of topologies
Theorem	distop 20799	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 )
Theorem	topnex 20800	The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 6968; an alternate proof uses indiscrete topologies (see indistop 20806) and the analogue of pwnex 6968 with pairs { (/) , x } instead of power sets ~P x (that analogue is also a consequence of abnex 6965). (Contributed by BJ, 2-May-2021.)
|- Top e/ _V