P. 128 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12701-12800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bastg 12701	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 12702	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 12703	Subset relation for generated topologies. (Contributed by NM, 7-May-2007.)
|- ( ( C e. V /\ B C_ C ) -> ( topGen ` B ) C_ ( topGen ` C ) )
Theorem	tgcl 12704	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 12705	The property tgcl 12704 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 12706	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 12707	A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
|- ( J e. Top -> J e. TopBases )
Theorem	tgtop 12708	A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
|- ( J e. Top -> ( topGen ` J ) = J )
Theorem	eltop 12709	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 12710*	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 12711*	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	tgdom 12712	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 12713*	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 12714	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 12715	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 12716	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	en1top 12717	{ (/) } is the only topology with one element. (Contributed by FL, 18-Aug-2008.)
|- ( J e. Top -> ( J ~~ 1o <-> J = { (/) } ) )
Theorem	tgss3 12718	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 12719*	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 12720	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 12721*	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	2basgeng 12722	Conditions that determine the equality of two generated topologies. (Contributed by NM, 8-May-2007.) (Revised by Jim Kingdon, 5-Mar-2023.)
|- ( ( B e. V /\ B C_ C /\ C C_ ( topGen ` B ) ) -> ( topGen ` B ) = ( topGen ` C ) )
Theorem	bastop1 12723*	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 12724*	A version of bastop1 12723 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 ) ) ) )
8.1.3  Examples of topologies
Theorem	distop 12725	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 12726	The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 4427. (Contributed by BJ, 2-May-2021.)
|- Top e/ _V
Theorem	distopon 12727	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 ) )
Theorem	sn0topon 12728	The singleton of the empty set is a topology on the empty set. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- { (/) } e. (TopOn ` (/) )
Theorem	sn0top 12729	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
Theorem	epttop 12730*	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 ) )
Theorem	distps 12731	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
8.1.4  Closure and interior
Syntax	ccld 12732	Extend class notation with the set of closed sets of a topology.
class Clsd
Syntax	cnt 12733	Extend class notation with interior of a subset of a topology base set.
class int
Syntax	ccl 12734	Extend class notation with closure of a subset of a topology base set.
class cls
Definition	df-cld 12735*	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 } )
Definition	df-ntr 12736*	Define a function on topologies whose value is the interior function on the subsets of the base set. See ntrval 12750. (Contributed by NM, 10-Sep-2006.)
|- int = ( j e. Top |-> ( x e. ~P U. j |-> U. ( j i^i ~P x ) ) )
Definition	df-cls 12737*	Define a function on topologies whose value is the closure function on the subsets of the base set. See clsval 12751. (Contributed by NM, 3-Oct-2006.)
|- cls = ( j e. Top |-> ( x e. ~P U. j |-> |^| { y e. ( Clsd ` j ) | x C_ y } ) )
Theorem	fncld 12738	The closed-set generator is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
|- Clsd Fn Top
Theorem	cldval 12739*	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 } )
Theorem	ntrfval 12740*	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 ) ) )
Theorem	clsfval 12741*	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 } ) )
Theorem	cldrcl 12742	Reverse closure of the closed set operation. (Contributed by Stefan O'Rear, 22-Feb-2015.)
|- ( C e. ( Clsd ` J ) -> J e. Top )
Theorem	iscld 12743	The predicate "the class 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 ) ) )
Theorem	iscld2 12744	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 ) )
Theorem	cldss 12745	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 )
Theorem	cldss2 12746	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
Theorem	cldopn 12747	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 )
Theorem	difopn 12748	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 )
Theorem	topcld 12749	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 ) )
Theorem	ntrval 12750	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 ) )
Theorem	clsval 12751*	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 } )
Theorem	0cld 12752	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 ) )
Theorem	uncld 12753	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 ) )
Theorem	cldcls 12754	A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.)
|- ( S e. ( Clsd ` J ) -> ( ( cls ` J ) ` S ) = S )
Theorem	iuncld 12755*	A finite indexed union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.) (Revised by Jim Kingdon, 10-Mar-2023.)
|- 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 ) )
Theorem	unicld 12756	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 ) )
Theorem	ntropn 12757	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 )
Theorem	clsss 12758	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 ) )
Theorem	ntrss 12759	Subset relationship for interior. (Contributed by NM, 3-Oct-2007.) (Revised by Jim Kingdon, 11-Mar-2023.)
|- X = U. J   =>    |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( ( int ` J ) ` T ) C_ ( ( int ` J ) ` S ) )
Theorem	sscls 12760	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 ) )
Theorem	ntrss2 12761	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 )
Theorem	ssntr 12762	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 ) )
Theorem	ntrss3 12763	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 )
Theorem	ntrin 12764	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 ) ) )
Theorem	isopn3 12765	A subset is open iff it equals its own interior. (Contributed by NM, 9-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
|- X = U. J   =>    |- ( ( J e. Top /\ S C_ X ) -> ( S e. J <-> ( ( int ` J ) ` S ) = S ) )
Theorem	ntridm 12766	The interior operation is idempotent. (Contributed by NM, 2-Oct-2007.)
|- X = U. J   =>    |- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` ( ( int ` J ) ` S ) ) = ( ( int ` J ) ` S ) )
Theorem	clstop 12767	The closure of a topology's underlying set is the entire set. (Contributed by NM, 5-Oct-2007.) (Proof shortened by Jim Kingdon, 11-Mar-2023.)
|- X = U. J   =>    |- ( J e. Top -> ( ( cls ` J ) ` X ) = X )
Theorem	ntrtop 12768	The interior of a topology's underlying set is the entire set. (Contributed by NM, 12-Sep-2006.)
|- X = U. J   =>    |- ( J e. Top -> ( ( int ` J ) ` X ) = X )
Theorem	clsss2 12769	If a subset is included in a closed set, so is the subset's closure. (Contributed by NM, 22-Feb-2007.)
|- X = U. J   =>    |- ( ( C e. ( Clsd ` J ) /\ S C_ C ) -> ( ( cls ` J ) ` S ) C_ C )
Theorem	clsss3 12770	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 )
Theorem	ntrcls0 12771	A subset whose closure has an empty interior also has an empty interior. (Contributed by NM, 4-Oct-2007.)
|- X = U. J   =>    |- ( ( J e. Top /\ S C_ X /\ ( ( int ` J ) ` ( ( cls ` J ) ` S ) ) = (/) ) -> ( ( int ` J ) ` S ) = (/) )
Theorem	ntreq0 12772*	Two ways to say that a subset has an empty interior. (Contributed by NM, 3-Oct-2007.) (Revised by Jim Kingdon, 11-Mar-2023.)
|- X = U. J   =>    |- ( ( J e. Top /\ S C_ X ) -> ( ( ( int ` J ) ` S ) = (/) <-> A. x e. J ( x C_ S -> x = (/) ) ) )
Theorem	cls0 12773	The closure of the empty set. (Contributed by NM, 2-Oct-2007.) (Proof shortened by Jim Kingdon, 12-Mar-2023.)
|- ( J e. Top -> ( ( cls ` J ) ` (/) ) = (/) )
Theorem	ntr0 12774	The interior of the empty set. (Contributed by NM, 2-Oct-2007.)
|- ( J e. Top -> ( ( int ` J ) ` (/) ) = (/) )
Theorem	isopn3i 12775	An open subset equals its own interior. (Contributed by Mario Carneiro, 30-Dec-2016.)
|- ( ( J e. Top /\ S e. J ) -> ( ( int ` J ) ` S ) = S )
Theorem	discld 12776	The open sets of a discrete topology are closed and its closed sets are open. (Contributed by FL, 7-Jun-2007.) (Revised by Mario Carneiro, 7-Apr-2015.)
|- ( A e. V -> ( Clsd ` ~P A ) = ~P A )
Theorem	sn0cld 12777	The closed sets of the topology { (/) }. (Contributed by FL, 5-Jan-2009.)
|- ( Clsd ` { (/) } ) = { (/) }
8.1.5  Neighborhoods
Syntax	cnei 12778	Extend class notation with neighborhood relation for topologies.
class nei
Definition	df-nei 12779*	Define a function on topologies whose value is a map from a subset to its neighborhoods. (Contributed by NM, 11-Feb-2007.)
|- nei = ( j e. Top |-> ( x e. ~P U. j |-> { y e. ~P U. j | E. g e. j ( x C_ g /\ g C_ y ) } ) )
Theorem	neifval 12780*	Value of the neighborhood function on the subsets of the base set of a topology. (Contributed by NM, 11-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
|- X = U. J   =>    |- ( J e. Top -> ( nei ` J ) = ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) )
Theorem	neif 12781	The neighborhood function is a function from the set of the subsets of the base set of a topology. (Contributed by NM, 12-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
|- X = U. J   =>    |- ( J e. Top -> ( nei ` J ) Fn ~P X )
Theorem	neiss2 12782	A set with a neighborhood is a subset of the base set of a topology. (This theorem depends on a function's value being empty outside of its domain, but it will make later theorems simpler to state.) (Contributed by NM, 12-Feb-2007.)
|- X = U. J   =>    |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ X )
Theorem	neival 12783*	Value of the set of neighborhoods of a subset of the base set of a topology. (Contributed by NM, 11-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
|- X = U. J   =>    |- ( ( J e. Top /\ S C_ X ) -> ( ( nei ` J ) ` S ) = { v e. ~P X | E. g e. J ( S C_ g /\ g C_ v ) } )
Theorem	isnei 12784*	The predicate "the class N is a neighborhood of S". (Contributed by FL, 25-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
|- X = U. J   =>    |- ( ( J e. Top /\ S C_ X ) -> ( N e. ( ( nei ` J ) ` S ) <-> ( N C_ X /\ E. g e. J ( S C_ g /\ g C_ N ) ) ) )
Theorem	neiint 12785	An intuitive definition of a neighborhood in terms of interior. (Contributed by Szymon Jaroszewicz, 18-Dec-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
|- X = U. J   =>    |- ( ( J e. Top /\ S C_ X /\ N C_ X ) -> ( N e. ( ( nei ` J ) ` S ) <-> S C_ ( ( int ` J ) ` N ) ) )
Theorem	isneip 12786*	The predicate "the class N is a neighborhood of point P". (Contributed by NM, 26-Feb-2007.)
|- X = U. J   =>    |- ( ( J e. Top /\ P e. X ) -> ( N e. ( ( nei ` J ) ` { P } ) <-> ( N C_ X /\ E. g e. J ( P e. g /\ g C_ N ) ) ) )
Theorem	neii1 12787	A neighborhood is included in the topology's base set. (Contributed by NM, 12-Feb-2007.)
|- X = U. J   =>    |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> N C_ X )
Theorem	neisspw 12788	The neighborhoods of any set are subsets of the base set. (Contributed by Stefan O'Rear, 6-Aug-2015.)
|- X = U. J   =>    |- ( J e. Top -> ( ( nei ` J ) ` S ) C_ ~P X )
Theorem	neii2 12789*	Property of a neighborhood. (Contributed by NM, 12-Feb-2007.)
|- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> E. g e. J ( S C_ g /\ g C_ N ) )
Theorem	neiss 12790	Any neighborhood of a set S is also a neighborhood of any subset R C_ S. Similar to Proposition 1 of [BourbakiTop1] p. I.2. (Contributed by FL, 25-Sep-2006.)
|- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> N e. ( ( nei ` J ) ` R ) )
Theorem	ssnei 12791	A set is included in any of its neighborhoods. Generalization to subsets of elnei 12792. (Contributed by FL, 16-Nov-2006.)
|- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ N )
Theorem	elnei 12792	A point belongs to any of its neighborhoods. Property Viii of [BourbakiTop1] p. I.3. (Contributed by FL, 28-Sep-2006.)
|- ( ( J e. Top /\ P e. A /\ N e. ( ( nei ` J ) ` { P } ) ) -> P e. N )
Theorem	0nnei 12793	The empty set is not a neighborhood of a nonempty set. (Contributed by FL, 18-Sep-2007.)
|- ( ( J e. Top /\ S =/= (/) ) -> -. (/) e. ( ( nei ` J ) ` S ) )
Theorem	neipsm 12794*	A neighborhood of a set is a neighborhood of every point in the set. Proposition 1 of [BourbakiTop1] p. I.2. (Contributed by FL, 16-Nov-2006.) (Revised by Jim Kingdon, 22-Mar-2023.)
|- X = U. J   =>    |- ( ( J e. Top /\ S C_ X /\ E. x x e. S ) -> ( N e. ( ( nei ` J ) ` S ) <-> A. p e. S N e. ( ( nei ` J ) ` { p } ) ) )
Theorem	opnneissb 12795	An open set is a neighborhood of any of its subsets. (Contributed by FL, 2-Oct-2006.)
|- X = U. J   =>    |- ( ( J e. Top /\ N e. J /\ S C_ X ) -> ( S C_ N <-> N e. ( ( nei ` J ) ` S ) ) )
Theorem	opnssneib 12796	Any superset of an open set is a neighborhood of it. (Contributed by NM, 14-Feb-2007.)
|- X = U. J   =>    |- ( ( J e. Top /\ S e. J /\ N C_ X ) -> ( S C_ N <-> N e. ( ( nei ` J ) ` S ) ) )
Theorem	ssnei2 12797	Any subset M of X containing a neighborhood N of a set S is a neighborhood of this set. Generalization to subsets of Property Vi of [BourbakiTop1] p. I.3. (Contributed by FL, 2-Oct-2006.)
|- X = U. J   =>    |- ( ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) /\ ( N C_ M /\ M C_ X ) ) -> M e. ( ( nei ` J ) ` S ) )
Theorem	opnneiss 12798	An open set is a neighborhood of any of its subsets. (Contributed by NM, 13-Feb-2007.)
|- ( ( J e. Top /\ N e. J /\ S C_ N ) -> N e. ( ( nei ` J ) ` S ) )
Theorem	opnneip 12799	An open set is a neighborhood of any of its members. (Contributed by NM, 8-Mar-2007.)
|- ( ( J e. Top /\ N e. J /\ P e. N ) -> N e. ( ( nei ` J ) ` { P } ) )
Theorem	tpnei 12800	The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 12798. (Contributed by FL, 2-Oct-2006.)
|- X = U. J   =>    |- ( J e. Top -> ( S C_ X <-> X e. ( ( nei ` J ) ` S ) ) )