P. 146 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14501-14600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	restopnb 14501	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 14502	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 14503	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 14504	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 14505	Extend class notation with the class of continuous functions between topologies.
class Cn
Syntax	ccnp 14506	Extend class notation with the class of functions between topologies continuous at a given point.
class CnP
Syntax	clm 14507	Extend class notation with a function on topological spaces whose value is the convergence relation for limit sequences in the space.
class ~~> t
Definition	df-cn 14508*	Define a function on two topologies whose value is the set of continuous mappings from the first topology to the second. Based on definition of continuous function in [Munkres] p. 102. See iscn 14517 for the predicate form. (Contributed by NM, 17-Oct-2006.)
|- Cn = ( j e. Top , k e. Top |-> { f e. ( U. k ^m U. j ) | A. y e. k ( `' f " y ) e. j } )
Definition	df-cnp 14509*	Define a function on two topologies whose value is the set of continuous mappings at a specified point in the first topology. Based on Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 17-Oct-2006.)
|- CnP = ( j e. Top , k e. Top |-> ( x e. U. j |-> { f e. ( U. k ^m U. j ) | A. y e. k ( ( f ` x ) e. y -> E. g e. j ( x e. g /\ ( f " g ) C_ y ) ) } ) )
Definition	df-lm 14510*	Define a function on topologies whose value is the convergence relation for sequences into the given topological space. Although f is typically a sequence (a function from an upperset of integers) with values in the topological space, it need not be. Note, however, that the limit property concerns only values at integers, so that the real-valued function ( x e. RR |-> ( sin ` ( pi x. x ) ) ) converges to zero (in the standard topology on the reals) with this definition. (Contributed by NM, 7-Sep-2006.)
|- ~~> t = ( j e. Top |-> { <. f , x >. | ( f e. ( U. j ^pm CC ) /\ x e. U. j /\ A. u e. j ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u ) ) } )
Theorem	lmrcl 14511	Reverse closure for the convergence relation. (Contributed by Mario Carneiro, 7-Sep-2015.)
|- ( F ( ~~> t ` J ) P -> J e. Top )
Theorem	lmfval 14512*	The relation "sequence f converges to point y " in a metric space. (Contributed by NM, 7-Sep-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- ( J e. (TopOn ` X ) -> ( ~~> t ` J ) = { <. f , x >. | ( f e. ( X ^pm CC ) /\ x e. X /\ A. u e. J ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u ) ) } )
Theorem	lmreltop 14513	The topological space convergence relation is a relation. (Contributed by Jim Kingdon, 25-Mar-2023.)
|- ( J e. Top -> Rel ( ~~> t ` J ) )
Theorem	cnfval 14514*	The set of all continuous functions from topology J to topology K. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( J Cn K ) = { f e. ( Y ^m X ) | A. y e. K ( `' f " y ) e. J } )
Theorem	cnpfval 14515*	The function mapping the points in a topology J to the set of all functions from J to topology K continuous at that point. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( J CnP K ) = ( x e. X |-> { f e. ( Y ^m X ) | A. w e. K ( ( f ` x ) e. w -> E. v e. J ( x e. v /\ ( f " v ) C_ w ) ) } ) )
Theorem	cnovex 14516	The class of all continuous functions from a topology to another is a set. (Contributed by Jim Kingdon, 14-Dec-2023.)
|- ( ( J e. Top /\ K e. Top ) -> ( J Cn K ) e. _V )
Theorem	iscn 14517*	The predicate "the class F is a continuous function from topology J to topology K". Definition of continuous function in [Munkres] p. 102. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( F e. ( J Cn K ) <-> ( F : X --> Y /\ A. y e. K ( `' F " y ) e. J ) ) )
Theorem	cnpval 14518*	The set of all functions from topology J to topology K that are continuous at a point P. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) -> ( ( J CnP K ) ` P ) = { f e. ( Y ^m X ) | A. y e. K ( ( f ` P ) e. y -> E. x e. J ( P e. x /\ ( f " x ) C_ y ) ) } )
Theorem	iscnp 14519*	The predicate "the class F is a continuous function from topology J to topology K at point P". Based on Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) -> ( F e. ( ( J CnP K ) ` P ) <-> ( F : X --> Y /\ A. y e. K ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ ( F " x ) C_ y ) ) ) ) )
Theorem	iscn2 14520*	The predicate "the class F is a continuous function from topology J to topology K". Definition of continuous function in [Munkres] p. 102. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- X = U. J   &    |- Y = U. K   =>    |- ( F e. ( J Cn K ) <-> ( ( J e. Top /\ K e. Top ) /\ ( F : X --> Y /\ A. y e. K ( `' F " y ) e. J ) ) )
Theorem	cntop1 14521	Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- ( F e. ( J Cn K ) -> J e. Top )
Theorem	cntop2 14522	Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- ( F e. ( J Cn K ) -> K e. Top )
Theorem	iscnp3 14523*	The predicate "the class F is a continuous function from topology J to topology K at point P". (Contributed by NM, 15-May-2007.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) -> ( F e. ( ( J CnP K ) ` P ) <-> ( F : X --> Y /\ A. y e. K ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ x C_ ( `' F " y ) ) ) ) ) )
Theorem	cnf 14524	A continuous function is a mapping. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- X = U. J   &    |- Y = U. K   =>    |- ( F e. ( J Cn K ) -> F : X --> Y )
Theorem	cnf2 14525	A continuous function is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( J Cn K ) ) -> F : X --> Y )
Theorem	cnprcl2k 14526	Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.) (Revised by Jim Kingdon, 28-Mar-2023.)
|- ( ( J e. (TopOn ` X ) /\ K e. Top /\ F e. ( ( J CnP K ) ` P ) ) -> P e. X )
Theorem	cnpf2 14527	A continuous function at point P is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.) (Revised by Jim Kingdon, 28-Mar-2023.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( ( J CnP K ) ` P ) ) -> F : X --> Y )
Theorem	tgcn 14528*	The continuity predicate when the range is given by a basis for a topology. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
|- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> K = ( topGen ` B ) )   &    |- ( ph -> K e. (TopOn ` Y ) )   =>    |- ( ph -> ( F e. ( J Cn K ) <-> ( F : X --> Y /\ A. y e. B ( `' F " y ) e. J ) ) )
Theorem	tgcnp 14529*	The "continuous at a point" predicate when the range is given by a basis for a topology. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
|- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> K = ( topGen ` B ) )   &    |- ( ph -> K e. (TopOn ` Y ) )   &    |- ( ph -> P e. X )   =>    |- ( ph -> ( F e. ( ( J CnP K ) ` P ) <-> ( F : X --> Y /\ A. y e. B ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ ( F " x ) C_ y ) ) ) ) )
Theorem	ssidcn 14530	The identity function is a continuous function from one topology to another topology on the same set iff the domain is finer than the codomain. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) -> ( ( _I |` X ) e. ( J Cn K ) <-> K C_ J ) )
Theorem	icnpimaex 14531*	Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.) (Revised by Jim Kingdon, 28-Mar-2023.)
|- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) /\ ( F e. ( ( J CnP K ) ` P ) /\ A e. K /\ ( F ` P ) e. A ) ) -> E. x e. J ( P e. x /\ ( F " x ) C_ A ) )
Theorem	idcn 14532	A restricted identity function is a continuous function. (Contributed by FL, 27-Dec-2006.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
|- ( J e. (TopOn ` X ) -> ( _I |` X ) e. ( J Cn J ) )
Theorem	lmbr 14533*	Express the binary relation "sequence F converges to point P " in a topological space. Definition 1.4-1 of [Kreyszig] p. 25. The condition F C_ ( CC X. X ) allows us to use objects more general than sequences when convenient; see the comment in df-lm 14510. (Contributed by Mario Carneiro, 14-Nov-2013.)
|- ( ph -> J e. (TopOn ` X ) )   =>    |- ( ph -> ( F ( ~~> t ` J ) P <-> ( F e. ( X ^pm CC ) /\ P e. X /\ A. u e. J ( P e. u -> E. y e. ran ZZ>= ( F |` y ) : y --> u ) ) ) )
Theorem	lmbr2 14534*	Express the binary relation "sequence F converges to point P " in a metric space using an arbitrary upper set of integers. (Contributed by Mario Carneiro, 14-Nov-2013.)
|- ( ph -> J e. (TopOn ` X ) )   &    |- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   =>    |- ( ph -> ( F ( ~~> t ` J ) P <-> ( F e. ( X ^pm CC ) /\ P e. X /\ A. u e. J ( P e. u -> E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. u ) ) ) ) )
Theorem	lmbrf 14535*	Express the binary relation "sequence F converges to point P " in a metric space using an arbitrary upper set of integers. This version of lmbr2 14534 presupposes that F is a function. (Contributed by Mario Carneiro, 14-Nov-2013.)
|- ( ph -> J e. (TopOn ` X ) )   &    |- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F : Z --> X )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )   =>    |- ( ph -> ( F ( ~~> t ` J ) P <-> ( P e. X /\ A. u e. J ( P e. u -> E. j e. Z A. k e. ( ZZ>= ` j ) A e. u ) ) ) )
Theorem	lmconst 14536	A constant sequence converges to its value. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)
|- Z = ( ZZ>= ` M )   =>    |- ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) -> ( Z X. { P } ) ( ~~> t ` J ) P )
Theorem	lmcvg 14537*	Convergence property of a converging sequence. (Contributed by Mario Carneiro, 14-Nov-2013.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> P e. U )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F ( ~~> t ` J ) P )   &    |- ( ph -> U e. J )   =>    |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( F ` k ) e. U )
Theorem	iscnp4 14538*	The predicate "the class F is a continuous function from topology J to topology K at point P " in terms of neighborhoods. (Contributed by FL, 18-Jul-2011.) (Revised by Mario Carneiro, 10-Sep-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) -> ( F e. ( ( J CnP K ) ` P ) <-> ( F : X --> Y /\ A. y e. ( ( nei ` K ) ` { ( F ` P ) } ) E. x e. ( ( nei ` J ) ` { P } ) ( F " x ) C_ y ) ) )
Theorem	cnpnei 14539*	A condition for continuity at a point in terms of neighborhoods. (Contributed by Jeff Hankins, 7-Sep-2009.)
|- X = U. J   &    |- Y = U. K   =>    |- ( ( ( J e. Top /\ K e. Top /\ F : X --> Y ) /\ A e. X ) -> ( F e. ( ( J CnP K ) ` A ) <-> A. y e. ( ( nei ` K ) ` { ( F ` A ) } ) ( `' F " y ) e. ( ( nei ` J ) ` { A } ) ) )
Theorem	cnima 14540	An open subset of the codomain of a continuous function has an open preimage. (Contributed by FL, 15-Dec-2006.)
|- ( ( F e. ( J Cn K ) /\ A e. K ) -> ( `' F " A ) e. J )
Theorem	cnco 14541	The composition of two continuous functions is a continuous function. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- ( ( F e. ( J Cn K ) /\ G e. ( K Cn L ) ) -> ( G o. F ) e. ( J Cn L ) )
Theorem	cnptopco 14542	The composition of a function F continuous at P with a function continuous at ( F ` P ) is continuous at P. Proposition 2 of [BourbakiTop1] p. I.9. (Contributed by FL, 16-Nov-2006.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
|- ( ( ( J e. Top /\ K e. Top /\ L e. Top ) /\ ( F e. ( ( J CnP K ) ` P ) /\ G e. ( ( K CnP L ) ` ( F ` P ) ) ) ) -> ( G o. F ) e. ( ( J CnP L ) ` P ) )
Theorem	cnclima 14543	A closed subset of the codomain of a continuous function has a closed preimage. (Contributed by NM, 15-Mar-2007.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- ( ( F e. ( J Cn K ) /\ A e. ( Clsd ` K ) ) -> ( `' F " A ) e. ( Clsd ` J ) )
Theorem	cnntri 14544	Property of the preimage of an interior. (Contributed by Mario Carneiro, 25-Aug-2015.)
|- Y = U. K   =>    |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> ( `' F " ( ( int ` K ) ` S ) ) C_ ( ( int ` J ) ` ( `' F " S ) ) )
Theorem	cnntr 14545*	Continuity in terms of interior. (Contributed by Jeff Hankins, 2-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( F e. ( J Cn K ) <-> ( F : X --> Y /\ A. x e. ~P Y ( `' F " ( ( int ` K ) ` x ) ) C_ ( ( int ` J ) ` ( `' F " x ) ) ) ) )
Theorem	cnss1 14546	If the topology K is finer than J, then there are more continuous functions from K than from J. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- X = U. J   =>    |- ( ( K e. (TopOn ` X ) /\ J C_ K ) -> ( J Cn L ) C_ ( K Cn L ) )
Theorem	cnss2 14547	If the topology K is finer than J, then there are fewer continuous functions into K than into J from some other space. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- Y = U. K   =>    |- ( ( L e. (TopOn ` Y ) /\ L C_ K ) -> ( J Cn K ) C_ ( J Cn L ) )
Theorem	cncnpi 14548	A continuous function is continuous at all points. One direction of Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
|- X = U. J   =>    |- ( ( F e. ( J Cn K ) /\ A e. X ) -> F e. ( ( J CnP K ) ` A ) )
Theorem	cnsscnp 14549	The set of continuous functions is a subset of the set of continuous functions at a point. (Contributed by Raph Levien, 21-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- X = U. J   =>    |- ( P e. X -> ( J Cn K ) C_ ( ( J CnP K ) ` P ) )
Theorem	cncnp 14550*	A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 15-May-2007.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( F e. ( J Cn K ) <-> ( F : X --> Y /\ A. x e. X F e. ( ( J CnP K ) ` x ) ) ) )
Theorem	cncnp2m 14551*	A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Jim Kingdon, 30-Mar-2023.)
|- X = U. J   &    |- Y = U. K   =>    |- ( ( J e. Top /\ K e. Top /\ E. y y e. X ) -> ( F e. ( J Cn K ) <-> A. x e. X F e. ( ( J CnP K ) ` x ) ) )
Theorem	cnnei 14552*	Continuity in terms of neighborhoods. (Contributed by Thierry Arnoux, 3-Jan-2018.)
|- X = U. J   &    |- Y = U. K   =>    |- ( ( J e. Top /\ K e. Top /\ F : X --> Y ) -> ( F e. ( J Cn K ) <-> A. p e. X A. w e. ( ( nei ` K ) ` { ( F ` p ) } ) E. v e. ( ( nei ` J ) ` { p } ) ( F " v ) C_ w ) )
Theorem	cnconst2 14553	A constant function is continuous. (Contributed by Mario Carneiro, 19-Mar-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ B e. Y ) -> ( X X. { B } ) e. ( J Cn K ) )
Theorem	cnconst 14554	A constant function is continuous. (Contributed by FL, 15-Jan-2007.) (Proof shortened by Mario Carneiro, 19-Mar-2015.)
|- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ ( B e. Y /\ F : X --> { B } ) ) -> F e. ( J Cn K ) )
Theorem	cnrest 14555	Continuity of a restriction from a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- X = U. J   =>    |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> ( F |` A ) e. ( ( J ↾t A ) Cn K ) )
Theorem	cnrest2 14556	Equivalence of continuity in the parent topology and continuity in a subspace. (Contributed by Jeff Hankins, 10-Jul-2009.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
|- ( ( K e. (TopOn ` Y ) /\ ran F C_ B /\ B C_ Y ) -> ( F e. ( J Cn K ) <-> F e. ( J Cn ( K ↾t B ) ) ) )
Theorem	cnrest2r 14557	Equivalence of continuity in the parent topology and continuity in a subspace. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Jun-2014.)
|- ( K e. Top -> ( J Cn ( K ↾t B ) ) C_ ( J Cn K ) )
Theorem	cnptopresti 14558	One direction of cnptoprest 14559 under the weaker condition that the point is in the subset rather than the interior of the subset. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 31-Mar-2023.)
|- ( ( ( J e. (TopOn ` X ) /\ K e. Top ) /\ ( A C_ X /\ P e. A /\ F e. ( ( J CnP K ) ` P ) ) ) -> ( F |` A ) e. ( ( ( J ↾t A ) CnP K ) ` P ) )
Theorem	cnptoprest 14559	Equivalence of continuity at a point and continuity of the restricted function at a point. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 5-Apr-2023.)
|- X = U. J   &    |- Y = U. K   =>    |- ( ( ( J e. Top /\ K e. Top /\ A C_ X ) /\ ( P e. ( ( int ` J ) ` A ) /\ F : X --> Y ) ) -> ( F e. ( ( J CnP K ) ` P ) <-> ( F |` A ) e. ( ( ( J ↾t A ) CnP K ) ` P ) ) )
Theorem	cnptoprest2 14560	Equivalence of point-continuity in the parent topology and point-continuity in a subspace. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 6-Apr-2023.)
|- X = U. J   &    |- Y = U. K   =>    |- ( ( ( J e. Top /\ K e. Top ) /\ ( F : X --> B /\ B C_ Y ) ) -> ( F e. ( ( J CnP K ) ` P ) <-> F e. ( ( J CnP ( K ↾t B ) ) ` P ) ) )
Theorem	cndis 14561	Every function is continuous when the domain is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
|- ( ( A e. V /\ J e. (TopOn ` X ) ) -> ( ~P A Cn J ) = ( X ^m A ) )
Theorem	cnpdis 14562	If A is an isolated point in X (or equivalently, the singleton { A } is open in X), then every function is continuous at A. (Contributed by Mario Carneiro, 9-Sep-2015.)
|- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ { A } e. J ) -> ( ( J CnP K ) ` A ) = ( Y ^m X ) )
Theorem	lmfpm 14563	If F converges, then F is a partial function. (Contributed by Mario Carneiro, 23-Dec-2013.)
|- ( ( J e. (TopOn ` X ) /\ F ( ~~> t ` J ) P ) -> F e. ( X ^pm CC ) )
Theorem	lmfss 14564	Inclusion of a function having a limit (used to ensure the limit relation is a set, under our definition). (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
|- ( ( J e. (TopOn ` X ) /\ F ( ~~> t ` J ) P ) -> F C_ ( CC X. X ) )
Theorem	lmcl 14565	Closure of a limit. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
|- ( ( J e. (TopOn ` X ) /\ F ( ~~> t ` J ) P ) -> P e. X )
Theorem	lmss 14566	Limit on a subspace. (Contributed by NM, 30-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)
|- K = ( J ↾t Y )   &    |- Z = ( ZZ>= ` M )   &    |- ( ph -> Y e. V )   &    |- ( ph -> J e. Top )   &    |- ( ph -> P e. Y )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F : Z --> Y )   =>    |- ( ph -> ( F ( ~~> t ` J ) P <-> F ( ~~> t ` K ) P ) )
Theorem	sslm 14567	A finer topology has fewer convergent sequences (but the sequences that do converge, converge to the same value). (Contributed by Mario Carneiro, 15-Sep-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) -> ( ~~> t ` K ) C_ ( ~~> t ` J ) )
Theorem	lmres 14568	A function converges iff its restriction to an upper integers set converges. (Contributed by Mario Carneiro, 31-Dec-2013.)
|- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> F e. ( X ^pm CC ) )   &    |- ( ph -> M e. ZZ )   =>    |- ( ph -> ( F ( ~~> t ` J ) P <-> ( F |` ( ZZ>= ` M ) ) ( ~~> t ` J ) P ) )
Theorem	lmff 14569*	If F converges, there is some upper integer set on which F is a total function. (Contributed by Mario Carneiro, 31-Dec-2013.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F e. dom ( ~~> t ` J ) )   =>    |- ( ph -> E. j e. Z ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X )
Theorem	lmtopcnp 14570	The image of a convergent sequence under a continuous map is convergent to the image of the original point. (Contributed by Mario Carneiro, 3-May-2014.) (Revised by Jim Kingdon, 6-Apr-2023.)
|- ( ph -> F ( ~~> t ` J ) P )   &    |- ( ph -> K e. Top )   &    |- ( ph -> G e. ( ( J CnP K ) ` P ) )   =>    |- ( ph -> ( G o. F ) ( ~~> t ` K ) ( G ` P ) )
Theorem	lmcn 14571	The image of a convergent sequence under a continuous map is convergent to the image of the original point. (Contributed by Mario Carneiro, 3-May-2014.)
|- ( ph -> F ( ~~> t ` J ) P )   &    |- ( ph -> G e. ( J Cn K ) )   =>    |- ( ph -> ( G o. F ) ( ~~> t ` K ) ( G ` P ) )
9.1.8  Product topologies
Syntax	ctx 14572	Extend class notation with the binary topological product operation.
class tX
Definition	df-tx 14573*	Define the binary topological product, which is homeomorphic to the general topological product over a two element set, but is more convenient to use. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- tX = ( r e. _V , s e. _V |-> ( topGen ` ran ( x e. r , y e. s |-> ( x X. y ) ) ) )
Theorem	txvalex 14574	Existence of the binary topological product. If R and S are known to be topologies, see txtop 14580. (Contributed by Jim Kingdon, 3-Aug-2023.)
|- ( ( R e. V /\ S e. W ) -> ( R tX S ) e. _V )
Theorem	txval 14575*	Value of the binary topological product operation. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 30-Aug-2015.)
|- B = ran ( x e. R , y e. S |-> ( x X. y ) )   =>    |- ( ( R e. V /\ S e. W ) -> ( R tX S ) = ( topGen ` B ) )
Theorem	txuni2 14576*	The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 31-Aug-2015.)
|- B = ran ( x e. R , y e. S |-> ( x X. y ) )   &    |- X = U. R   &    |- Y = U. S   =>    |- ( X X. Y ) = U. B
Theorem	txbasex 14577*	The basis for the product topology is a set. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- B = ran ( x e. R , y e. S |-> ( x X. y ) )   =>    |- ( ( R e. V /\ S e. W ) -> B e. _V )
Theorem	txbas 14578*	The set of Cartesian products of elements from two topological bases is a basis. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- B = ran ( x e. R , y e. S |-> ( x X. y ) )   =>    |- ( ( R e. TopBases /\ S e. TopBases ) -> B e. TopBases )
Theorem	eltx 14579*	A set in a product is open iff each point is surrounded by an open rectangle. (Contributed by Stefan O'Rear, 25-Jan-2015.)
|- ( ( J e. V /\ K e. W ) -> ( S e. ( J tX K ) <-> A. p e. S E. x e. J E. y e. K ( p e. ( x X. y ) /\ ( x X. y ) C_ S ) ) )
Theorem	txtop 14580	The product of two topologies is a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( R e. Top /\ S e. Top ) -> ( R tX S ) e. Top )
Theorem	txtopi 14581	The product of two topologies is a topology. (Contributed by Jeff Madsen, 15-Jun-2010.)
|- R e. Top   &    |- S e. Top   =>    |- ( R tX S ) e. Top
Theorem	txtopon 14582	The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 22-Aug-2015.) (Revised by Mario Carneiro, 2-Sep-2015.)
|- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( R tX S ) e. (TopOn ` ( X X. Y ) ) )
Theorem	txuni 14583	The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- X = U. R   &    |- Y = U. S   =>    |- ( ( R e. Top /\ S e. Top ) -> ( X X. Y ) = U. ( R tX S ) )
Theorem	txunii 14584	The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 15-Jun-2010.)
|- R e. Top   &    |- S e. Top   &    |- X = U. R   &    |- Y = U. S   =>    |- ( X X. Y ) = U. ( R tX S )
Theorem	txopn 14585	The product of two open sets is open in the product topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( ( R e. V /\ S e. W ) /\ ( A e. R /\ B e. S ) ) -> ( A X. B ) e. ( R tX S ) )
Theorem	txss12 14586	Subset property of the topological product. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- ( ( ( B e. V /\ D e. W ) /\ ( A C_ B /\ C C_ D ) ) -> ( A tX C ) C_ ( B tX D ) )
Theorem	txbasval 14587	It is sufficient to consider products of the bases for the topologies in the topological product. (Contributed by Mario Carneiro, 25-Aug-2014.)
|- ( ( R e. V /\ S e. W ) -> ( ( topGen ` R ) tX ( topGen ` S ) ) = ( R tX S ) )
Theorem	neitx 14588	The Cartesian product of two neighborhoods is a neighborhood in the product topology. (Contributed by Thierry Arnoux, 13-Jan-2018.)
|- X = U. J   &    |- Y = U. K   =>    |- ( ( ( J e. Top /\ K e. Top ) /\ ( A e. ( ( nei ` J ) ` C ) /\ B e. ( ( nei ` K ) ` D ) ) ) -> ( A X. B ) e. ( ( nei ` ( J tX K ) ) ` ( C X. D ) ) )
Theorem	tx1cn 14589	Continuity of the first projection map of a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
|- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( 1st |` ( X X. Y ) ) e. ( ( R tX S ) Cn R ) )
Theorem	tx2cn 14590	Continuity of the second projection map of a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
|- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( 2nd |` ( X X. Y ) ) e. ( ( R tX S ) Cn S ) )
Theorem	txcnp 14591*	If two functions are continuous at D, then the ordered pair of them is continuous at D into the product topology. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
|- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> K e. (TopOn ` Y ) )   &    |- ( ph -> L e. (TopOn ` Z ) )   &    |- ( ph -> D e. X )   &    |- ( ph -> ( x e. X |-> A ) e. ( ( J CnP K ) ` D ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( ( J CnP L ) ` D ) )   =>    |- ( ph -> ( x e. X |-> <. A , B >. ) e. ( ( J CnP ( K tX L ) ) ` D ) )
Theorem	upxp 14592*	Universal property of the Cartesian product considered as a categorical product in the category of sets. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)
|- P = ( 1st |` ( B X. C ) )   &    |- Q = ( 2nd |` ( B X. C ) )   =>    |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> E! h ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) )
Theorem	txcnmpt 14593*	A map into the product of two topological spaces is continuous if both of its projections are continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Aug-2015.)
|- W = U. U   &    |- H = ( x e. W |-> <. ( F ` x ) , ( G ` x ) >. )   =>    |- ( ( F e. ( U Cn R ) /\ G e. ( U Cn S ) ) -> H e. ( U Cn ( R tX S ) ) )
Theorem	uptx 14594*	Universal property of the binary topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
|- T = ( R tX S )   &    |- X = U. R   &    |- Y = U. S   &    |- Z = ( X X. Y )   &    |- P = ( 1st |` Z )   &    |- Q = ( 2nd |` Z )   =>    |- ( ( F e. ( U Cn R ) /\ G e. ( U Cn S ) ) -> E! h e. ( U Cn T ) ( F = ( P o. h ) /\ G = ( Q o. h ) ) )
Theorem	txcn 14595	A map into the product of two topological spaces is continuous iff both of its projections are continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
|- X = U. R   &    |- Y = U. S   &    |- Z = ( X X. Y )   &    |- W = U. U   &    |- P = ( 1st |` Z )   &    |- Q = ( 2nd |` Z )   =>    |- ( ( R e. Top /\ S e. Top /\ F : W --> Z ) -> ( F e. ( U Cn ( R tX S ) ) <-> ( ( P o. F ) e. ( U Cn R ) /\ ( Q o. F ) e. ( U Cn S ) ) ) )
Theorem	txrest 14596	The subspace of a topological product space induced by a subset with a Cartesian product representation is a topological product of the subspaces induced by the subspaces of the terms of the products. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
|- ( ( ( R e. V /\ S e. W ) /\ ( A e. X /\ B e. Y ) ) -> ( ( R tX S ) ↾t ( A X. B ) ) = ( ( R ↾t A ) tX ( S ↾t B ) ) )
Theorem	txdis 14597	The topological product of discrete spaces is discrete. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( ( A e. V /\ B e. W ) -> ( ~P A tX ~P B ) = ~P ( A X. B ) )
Theorem	txdis1cn 14598*	A function is jointly continuous on a discrete left topology iff it is continuous as a function of its right argument, for each fixed left value. (Contributed by Mario Carneiro, 19-Sep-2015.)
|- ( ph -> X e. V )   &    |- ( ph -> J e. (TopOn ` Y ) )   &    |- ( ph -> K e. Top )   &    |- ( ph -> F Fn ( X X. Y ) )   &    |- ( ( ph /\ x e. X ) -> ( y e. Y |-> ( x F y ) ) e. ( J Cn K ) )   =>    |- ( ph -> F e. ( ( ~P X tX J ) Cn K ) )
Theorem	txlm 14599*	Two sequences converge iff the sequence of their ordered pairs converges. Proposition 14-2.6 of [Gleason] p. 230. (Contributed by NM, 16-Jul-2007.) (Revised by Mario Carneiro, 5-May-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> K e. (TopOn ` Y ) )   &    |- ( ph -> F : Z --> X )   &    |- ( ph -> G : Z --> Y )   &    |- H = ( n e. Z |-> <. ( F ` n ) , ( G ` n ) >. )   =>    |- ( ph -> ( ( F ( ~~> t ` J ) R /\ G ( ~~> t ` K ) S ) <-> H ( ~~> t ` ( J tX K ) ) <. R , S >. ) )
Theorem	lmcn2 14600*	The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 15-May-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> J e. (TopOn ` X ) )   &    |- ( ph -> K e. (TopOn ` Y ) )   &    |- ( ph -> F : Z --> X )   &    |- ( ph -> G : Z --> Y )   &    |- ( ph -> F ( ~~> t ` J ) R )   &    |- ( ph -> G ( ~~> t ` K ) S )   &    |- ( ph -> O e. ( ( J tX K ) Cn N ) )   &    |- H = ( n e. Z |-> ( ( F ` n ) O ( G ` n ) ) )   =>    |- ( ph -> H ( ~~> t ` N ) ( R O S ) )