P. 149 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14801-14900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	distopon 14801	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 14802	The singleton of the empty set is a topology on the empty set. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- { (/) } e. (TopOn ` (/) )
Theorem	sn0top 14803	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 14804*	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 14805	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
9.1.4  Closure and interior
Syntax	ccld 14806	Extend class notation with the set of closed sets of a topology.
class Clsd
Syntax	cnt 14807	Extend class notation with interior of a subset of a topology base set.
class int
Syntax	ccl 14808	Extend class notation with closure of a subset of a topology base set.
class cls
Definition	df-cld 14809*	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 14810*	Define a function on topologies whose value is the interior function on the subsets of the base set. See ntrval 14824. (Contributed by NM, 10-Sep-2006.)
|- int = ( j e. Top |-> ( x e. ~P U. j |-> U. ( j i^i ~P x ) ) )
Definition	df-cls 14811*	Define a function on topologies whose value is the closure function on the subsets of the base set. See clsval 14825. (Contributed by NM, 3-Oct-2006.)
|- cls = ( j e. Top |-> ( x e. ~P U. j |-> |^| { y e. ( Clsd ` j ) | x C_ y } ) )
Theorem	fncld 14812	The closed-set generator is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
|- Clsd Fn Top
Theorem	cldval 14813*	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 14814*	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 14815*	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 14816	Reverse closure of the closed set operation. (Contributed by Stefan O'Rear, 22-Feb-2015.)
|- ( C e. ( Clsd ` J ) -> J e. Top )
Theorem	iscld 14817	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 14818	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 14819	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 14820	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 14821	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 14822	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 14823	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 14824	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 14825*	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 14826	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 14827	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 14828	A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.)
|- ( S e. ( Clsd ` J ) -> ( ( cls ` J ) ` S ) = S )
Theorem	iuncld 14829*	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 14830	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 14831	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 14832	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 14833	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 14834	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 14835	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 14836	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 14837	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 14838	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 14839	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 14840	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 14841	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 14842	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 14843	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 14844	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 14845	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 14846*	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 14847	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 14848	The interior of the empty set. (Contributed by NM, 2-Oct-2007.)
|- ( J e. Top -> ( ( int ` J ) ` (/) ) = (/) )
Theorem	isopn3i 14849	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 14850	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 14851	The closed sets of the topology { (/) }. (Contributed by FL, 5-Jan-2009.)
|- ( Clsd ` { (/) } ) = { (/) }
9.1.5  Neighborhoods
Syntax	cnei 14852	Extend class notation with neighborhood relation for topologies.
class nei
Definition	df-nei 14853*	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 14854*	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 14855	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 14856	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 14857*	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 14858*	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 14859	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 14860*	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 14861	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 14862	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 14863*	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 14864	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 14865	A set is included in any of its neighborhoods. Generalization to subsets of elnei 14866. (Contributed by FL, 16-Nov-2006.)
|- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ N )
Theorem	elnei 14866	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 14867	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 14868*	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 14869	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 14870	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 14871	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 14872	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 14873	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 14874	The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 14872. (Contributed by FL, 2-Oct-2006.)
|- X = U. J   =>    |- ( J e. Top -> ( S C_ X <-> X e. ( ( nei ` J ) ` S ) ) )
Theorem	neiuni 14875	The union of the neighborhoods of a set equals the topology's underlying set. (Contributed by FL, 18-Sep-2007.) (Revised by Mario Carneiro, 9-Apr-2015.)
|- X = U. J   =>    |- ( ( J e. Top /\ S C_ X ) -> X = U. ( ( nei ` J ) ` S ) )
Theorem	topssnei 14876	A finer topology has more neighborhoods. (Contributed by Mario Carneiro, 9-Apr-2015.)
|- X = U. J   &    |- Y = U. K   =>    |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ J C_ K ) -> ( ( nei ` J ) ` S ) C_ ( ( nei ` K ) ` S ) )
Theorem	innei 14877	The intersection of two neighborhoods of a set is also a neighborhood of the set. Generalization to subsets of Property Vii of [BourbakiTop1] p. I.3 for binary intersections. (Contributed by FL, 28-Sep-2006.)
|- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ M e. ( ( nei ` J ) ` S ) ) -> ( N i^i M ) e. ( ( nei ` J ) ` S ) )
Theorem	opnneiid 14878	Only an open set is a neighborhood of itself. (Contributed by FL, 2-Oct-2006.)
|- ( J e. Top -> ( N e. ( ( nei ` J ) ` N ) <-> N e. J ) )
Theorem	neissex 14879*	For any neighborhood N of S, there is a neighborhood x of S such that N is a neighborhood of all subsets of x. Generalization to subsets of Property Viv of [BourbakiTop1] p. I.3. (Contributed by FL, 2-Oct-2006.)
|- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> E. x e. ( ( nei ` J ) ` S ) A. y ( y C_ x -> N e. ( ( nei ` J ) ` y ) ) )
Theorem	0nei 14880	The empty set is a neighborhood of itself. (Contributed by FL, 10-Dec-2006.)
|- ( J e. Top -> (/) e. ( ( nei ` J ) ` (/) ) )
9.1.6  Subspace topologies
Theorem	restrcl 14881	Reverse closure for the subspace topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Proof shortened by Jim Kingdon, 23-Mar-2023.)
|- ( ( J ↾t A ) e. Top -> ( J e. _V /\ A e. _V ) )
Theorem	restbasg 14882	A subspace topology basis is a basis. (Contributed by Mario Carneiro, 19-Mar-2015.)
|- ( ( B e. TopBases /\ A e. V ) -> ( B ↾t A ) e. TopBases )
Theorem	tgrest 14883	A subspace can be generated by restricted sets from a basis for the original topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Proof shortened by Mario Carneiro, 30-Aug-2015.)
|- ( ( B e. V /\ A e. W ) -> ( topGen ` ( B ↾t A ) ) = ( ( topGen ` B ) ↾t A ) )
Theorem	resttop 14884	A subspace topology is a topology. Definition of subspace topology in [Munkres] p. 89. A is normally a subset of the base set of J. (Contributed by FL, 15-Apr-2007.) (Revised by Mario Carneiro, 1-May-2015.)
|- ( ( J e. Top /\ A e. V ) -> ( J ↾t A ) e. Top )
Theorem	resttopon 14885	A subspace topology is a topology on the base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( J ↾t A ) e. (TopOn ` A ) )
Theorem	restuni 14886	The underlying set of a subspace topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
|- X = U. J   =>    |- ( ( J e. Top /\ A C_ X ) -> A = U. ( J ↾t A ) )
Theorem	stoig 14887	The topological space built with a subspace topology. (Contributed by FL, 5-Jan-2009.) (Proof shortened by Mario Carneiro, 1-May-2015.)
|- X = U. J   =>    |- ( ( J e. Top /\ A C_ X ) -> { <. ( Base ` ndx ) , A >. , <. (TopSet ` ndx ) , ( J ↾t A ) >. } e. TopSp )
Theorem	restco 14888	Composition of subspaces. (Contributed by Mario Carneiro, 15-Dec-2013.) (Revised by Mario Carneiro, 1-May-2015.)
|- ( ( J e. V /\ A e. W /\ B e. X ) -> ( ( J ↾t A ) ↾t B ) = ( J ↾t ( A i^i B ) ) )
Theorem	restabs 14889	Equivalence of being a subspace of a subspace and being a subspace of the original. (Contributed by Jeff Hankins, 11-Jul-2009.) (Proof shortened by Mario Carneiro, 1-May-2015.)
|- ( ( J e. V /\ S C_ T /\ T e. W ) -> ( ( J ↾t T ) ↾t S ) = ( J ↾t S ) )
Theorem	restin 14890	When the subspace region is not a subset of the base of the topology, the resulting set is the same as the subspace restricted to the base. (Contributed by Mario Carneiro, 15-Dec-2013.)
|- X = U. J   =>    |- ( ( J e. V /\ A e. W ) -> ( J ↾t A ) = ( J ↾t ( A i^i X ) ) )
Theorem	restuni2 14891	The underlying set of a subspace topology. (Contributed by Mario Carneiro, 21-Mar-2015.)
|- X = U. J   =>    |- ( ( J e. Top /\ A e. V ) -> ( A i^i X ) = U. ( J ↾t A ) )
Theorem	resttopon2 14892	The underlying set of a subspace topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ A e. V ) -> ( J ↾t A ) e. (TopOn ` ( A i^i X ) ) )
Theorem	rest0 14893	The subspace topology induced by the topology J on the empty set. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 1-May-2015.)
|- ( J e. Top -> ( J ↾t (/) ) = { (/) } )
Theorem	restsn 14894	The only subspace topology induced by the topology { (/) }. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
|- ( A e. V -> ( { (/) } ↾t A ) = { (/) } )
Theorem	restopnb 14895	If B is an open subset of the subspace base set A, then any subset of B is open iff it is open in A. (Contributed by Mario Carneiro, 2-Mar-2015.)
|- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> ( C e. J <-> C e. ( J ↾t A ) ) )
Theorem	ssrest 14896	If K is a finer topology than J, then the subspace topologies induced by A maintain this relationship. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 1-May-2015.)
|- ( ( K e. V /\ J C_ K ) -> ( J ↾t A ) C_ ( K ↾t A ) )
Theorem	restopn2 14897	If A is open, then B is open in A iff it is an open subset of A. (Contributed by Mario Carneiro, 2-Mar-2015.)
|- ( ( J e. Top /\ A e. J ) -> ( B e. ( J ↾t A ) <-> ( B e. J /\ B C_ A ) ) )
Theorem	restdis 14898	A subspace of a discrete topology is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.)
|- ( ( A e. V /\ B C_ A ) -> ( ~P A ↾t B ) = ~P B )
9.1.7  Limits and continuity in topological spaces
Syntax	ccn 14899	Extend class notation with the class of continuous functions between topologies.
class Cn
Syntax	ccnp 14900	Extend class notation with the class of functions between topologies continuous at a given point.
class CnP