mmtheorems78 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 7701-7800 - Page 78 of 107

Type	Label	Description
Statement
Theorem	islpi 7701	A point belonging to a set's closure but not the set itself is a limit point.
|- X = U.J   =>   |- (((J e. Top /\ S (_ X) /\ (P e. ((cls` J)` S) /\ -. P e. S)) -> P e. ((limPt` J)` S))
Theorem	cldlp 7702	A subset of a topological space is closed iff it contains all its limit points. Corollary 6.7 of [Munkres] p. 97.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> (S e. (Clsd` J) <-> ((limPt` J)` S) (_ S))
Theorem	qdensere 7703	QQ is dense in the standard topology on RR.
|- ((cls` (topGen` ran (,)))` QQ) = RR
Continuity
Syntax	ccn 7704	Extend class notation with the set of continuous functions between topologies.
class Cn
Syntax	ccnp 7705	Extend class notation with the set of functions between topologies continuous at a point.
class CnP
Definition	df-cn 7706	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 7710 for the predicate form.
|- Cn = {<.<.j, k>., z>. | ((j e. Top /\ k e. Top) /\ z = {f e. (U.k ^m U.j) | A.y e. k (`'f"y) e. j})}
Definition	df-cnp 7707	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.
|- CnP = {<.<.j, k>., z>. | ((j e. Top /\ k e. Top) /\ z = {<.x, y>. | (x e. U.j /\ y = {f e. (U.k ^m U.j) | A.w e. k ((f` x) e. w -> E.v e. j (x e. v /\ (f"v) (_ w))})})}
Theorem	cnfval 7708	The set of all continuous functions from topology J to topology K.
|- X = U.J   &   |- Y = U.K   =>   |- ((J e. Top /\ K e. Top) -> (J Cn K) = {f e. (Y ^m X) | A.y e. K (`'f"y) e. J})
Theorem	cnpfval 7709	The function mapping the points in a topology J to the set of all functions from J to topology K continuous at that point.
|- X = U.J   &   |- Y = U.K   =>   |- ((J e. Top /\ K e. Top) -> (J CnP K) = {<.x, y>. | (x e. X /\ y = {f e. (Y ^m X) | A.w e. K ((f` x) e. w -> E.v e. J (x e. v /\ (f"v) (_ w))})})
Theorem	iscn 7710	The predicate "F is a continuous function from topology J to topology K." Definition of continuous function in [Munkres] p. 102.
|- X = U.J   &   |- Y = U.K   =>   |- ((J e. Top /\ K e. Top) -> (F e. (J Cn K) <-> (F:X-->Y /\ A.y e. K (`'F"y) e. J)))
Theorem	cnpval 7711	The set of all functions from topology J to topology K that are continuous at a point P.
|- X = U.J   &   |- Y = U.K   =>   |- ((J e. Top /\ K e. Top /\ 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) (_ y))})
Theorem	iscnp 7712	The predicate "F is a continuous function from topology J to topology K at point P." Based on Theorem 7.2(g) of [Munkres] p. 107.
|- X = U.J   &   |- Y = U.K   =>   |- ((J e. Top /\ K e. Top /\ 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) (_ y)))))
Theorem	iscnp2 7713	The predicate "F is a continuous function from topology J to topology K at point P."
|- X = U.J   &   |- Y = U.K   =>   |- ((J e. Top /\ K e. Top /\ 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 (_ (`'F"y))))))
Theorem	cnf 7714	A continuous function is a mapping. (Contributed by FL, 15-Dec-2006.)
|- X = U.J   &   |- Y = U.K   =>   |- ((J e. Top /\ K e. Top /\ F e. (J Cn K)) -> F:X-->Y)
Theorem	cnpf 7715	A continuous function at point P is a mapping. (Contributed by FL, 31-Dec-2006.)
|- X = U.J   &   |- Y = U.K   =>   |- (((J e. Top /\ K e. Top /\ P e. X) /\ F e. ((J CnP K)` P)) -> F:X-->Y)
Theorem	cnpcl 7716	The value of a continuous function from J to K at point P belongs to the underlying set of topology K. (Contributed by FL, 31-Dec-2006.)
|- X = U.J   &   |- Y = U.K   =>   |- (((J e. Top /\ K e. Top /\ P e. X) /\ (F e. ((J CnP K)` P) /\ A e. X)) -> (F` A) e. Y)
Theorem	cnpimaex 7717	Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.)
|- X = U.J   =>   |- (((J e. Top /\ K e. Top /\ 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) (_ A))
Theorem	idcn 7718	A restricted identity function is a continuous function. (Contributed by FL, 31-Dec-2006.)
|- X = U.J   =>   |- (J e. Top -> (I |` X) e. (J Cn J))
Theorem	cnima 7719	An open subset of the codomain of a continuous function has an open pre-image. (Contributed by FL, 15-Dec-2006.)
|- (((J e. Top /\ K e. Top /\ F e. (J Cn K)) /\ A e. K) -> (`'F"A) e. J)
Theorem	cnco 7720	The composition of two continuous functions is a continuous function. (Contributed by FL, 15-Dec-2006.)
|- (((J e. Top /\ K e. Top /\ L e. Top) /\ (F e. (J Cn K) /\ G e. (K Cn L))) -> (G o. F) e. (J Cn L))
Theorem	cnpco 7721	The composition of two continuous functions at point P is a continuous function at point P. Bourbaki TG I.9. (Contributed by FL, 31-Dec-2006.) Warning: The HTML proof page is 0.5MB in size.
|- X = U.J   =>   |- (((J e. Top /\ L e. Top /\ P e. X) /\ (K e. Top /\ F e. ((J CnP K)` P) /\ G e. ((K CnP L)` (F` P)))) -> (G o. F) e. ((J CnP L)` P))
Theorem	iscncl 7722	A definition of a continuous function using closed sets. Bourbaki TG I.9 Th. 1 (d). (Contributed by FL, 30-Jan-2007.)
|- X = U.J   &   |- Y = U.K   =>   |- ((J e. Top /\ K e. Top) -> (F e. (J Cn K) <-> (F:X-->Y /\ A.y e. (Clsd` K)(`'F"y) e. (Clsd` J))))
Theorem	cnclima 7723	A closed subset of the codomain of a continuous function has a closed pre-image.
|- (((J e. Top /\ K e. Top /\ F e. (J Cn K)) /\ A e. (Clsd` K)) -> (`'F"A) e. (Clsd` J))
Theorem	cnsscnp 7724	The set of continuous functions is a subset of the set of continuous functions at a point. (Contributed by Raph Levien, 21-Oct-2006.)
|- X = U.J   &   |- Y = U.K   =>   |- ((J e. Top /\ K e. Top /\ P e. X) -> (J Cn K) (_ ((J CnP K)` P))
Theorem	cncnpi 7725	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.)
|- X = U.J   &   |- Y = U.K   =>   |- (((J e. Top /\ K e. Top) /\ (F e. (J Cn K) /\ A e. X)) -> F e. ((J CnP K)` A))
Theorem	cncnplem1 7726	Lemma for cncnp2 7731.
Theorem	cncnplem2 7727	Lemma for cncnp2 7731.
Theorem	cncnplem3 7728	Lemma for cncnp2 7731.
Theorem	cncnplem4 7729	Lemma for cncnp2 7731.
Theorem	cncnp 7730	A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107.
|- X = U.J   &   |- Y = U.K   =>   |- ((J e. Top /\ K e. Top /\ F:X-->Y) -> (F e. (J Cn K) <-> A.x e. X F e. ((J CnP K)` x)))
Theorem	cncnp2 7731	A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.)
|- X = U.J   &   |- Y = U.K   =>   |- ((J e. Top /\ K e. Top /\ X =/= (/)) -> (F e. (J Cn K) <-> A.x e. X F e. ((J CnP K)` x)))
Theorem	cnconst 7732	A constant function is continuous. (Contributed by FL, 25-Jan-2007.)
|- X = U.J   &   |- Y = U.K   =>   |- (((J e. Top /\ K e. Top) /\ (B e. Y /\ F:X-->{B})) -> F e. (J Cn K))
Hausdorff spaces
Syntax	cha 7733	Extend class notation with the class of all Hausdorf spaces.
class Haus
Definition	df-haus 7734	Define the class of all Hausdorff spaces. A Hausdorff space is a topology in which distinct points have disjoint open neighborhoods. Definition of Hausdorff space in [Munkres] p. 98.
|- Haus = {j e. Top | A.x e. U.jA.y e. U.j(x =/= y -> E.n e. j E.m e. j (x e. n /\ y e. m /\ (n i^i m) = (/)))}
Theorem	ishaus 7735	Express the predicate "J is a Hausdorff space."
|- X = U.J   =>   |- (J e. Haus <-> (J e. Top /\ A.x e. X A.y e. X (x =/= y -> E.n e. J E.m e. J (x e. n /\ y e. m /\ (n i^i m) = (/)))))
Theorem	hausnei 7736	Neighborhood property of a Hausdorff space.
|- X = U.J   =>   |- ((J e. Haus /\ (P e. X /\ Q e. X /\ P =/= Q)) -> E.n e. J E.m e. J (P e. n /\ Q e. m /\ (n i^i m) = (/)))
Theorem	ishausi 7737	Properties that determine a Hausdorff space.
|- X = U.J   &   |- J e. Top   &   |- ((x e. X /\ y e. X /\ x =/= y) -> E.n e. J E.m e. J (x e. n /\ y e. m /\ (n i^i m) = (/)))   =>   |- J e. Haus
Theorem	haustop 7738	A Hausdorff space is a topology.
|- (J e. Haus -> J e. Top)
Theorem	sncld 7739	A singleton is closed in a Hausdorff space.
|- X = U.J   =>   |- ((J e. Haus /\ P e. X) -> {P} e. (Clsd` J))
Theorem	dnsconst 7740	If a continuous mapping to a Hausdorff space is constant on a dense subset, it is constant on the entire space. Note that ((cls` J)` A) = X means "A is dense in X" and