mmtheorems80 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 7901-8000 - Page 80 of 123

Type	Label	Description
Statement
Theorem	clsss3 7901	The closure of a subset of a topological space is included in the space.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> ((cls` J)` S) (_ X)
Theorem	ntrss3 7902	The interior of a subset of a topological space is included in the space.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> ((int` J)` S) (_ X)
Theorem	cmclsopn 7903	The complement of a closure is open.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> (X \ ((cls` J)` S)) e. J)
Theorem	cmntrcld 7904	The complement of an interior is closed.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> (X \ ((int` J)` S)) e. (Clsd` J))
Theorem	iscld3 7905	A subset is closed iff it equals its own closure.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> (S e. (Clsd` J) <-> ((cls` J)` S) = S))
Theorem	iscld4 7906	A subset is closed iff it contains its own closure.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> (S e. (Clsd` J) <-> ((cls` J)` S) (_ S))
Theorem	isopn3 7907	A subset is open iff it equals its own interior.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> (S e. J <-> ((int` J)` S) = S))
Theorem	clsidm 7908	The closure operation is idempotent.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> ((cls` J)` ((cls` J)` S)) = ((cls` J)` S))
Theorem	ntridm 7909	The interior operation is idempotent.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> ((int` J)` ((int` J)` S)) = ((int` J)` S))
Theorem	clstop 7910	The closure of a topology's underlying set is entire set.
|- X = U.J   =>   |- (J e. Top -> ((cls` J)` X) = X)
Theorem	ntrtop 7911	The interior of a topology's underlying set is entire set.
|- X = U.J   =>   |- (J e. Top -> ((int` J)` X) = X)
Theorem	0ntr 7912	A subset with an empty interior cannot cover a whole (nonempty) topology.
|- X = U.J   =>   |- (((J e. Top /\ X =/= (/)) /\ (S (_ X /\ ((int` J)` S) = (/))) -> (X \ S) =/= (/))
Theorem	clsss2 7913	If a subset is included in a closed set, so is the subset's closure.
|- X = U.J   =>   |- ((J e. Top /\ C e. (Clsd` J) /\ S (_ C) -> ((cls` J)` S) (_ C)
Theorem	elcls 7914	Membership in a closure. Theorem 6.5(a) of [Munkres] p. 95.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X /\ P e. X) -> (P e. ((cls` J)` S) <-> A.x e. J (P e. x -> (x i^i S) =/= (/))))
Theorem	elcls2 7915	Membership in a closure.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> (P e. ((cls` J)` S) <-> (P e. X /\ A.x e. J (P e. x -> (x i^i S) =/= (/)))))
Theorem	clsndisj 7916	Any open set containing a point that belongs to the closure of a subset intersects the subset. One direction of Theorem 6.5(a) of [Munkres] p. 95.
|- X = U.J   =>   |- (((J e. Top /\ S (_ X /\ P e. ((cls` J)` S)) /\ (U e. J /\ P e. U)) -> (U i^i S) =/= (/))
Theorem	ntrcls0 7917	A subset whose closure has an empty interior also has an empty interior.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X /\ ((int` J)` ((cls` J)` S)) = (/)) -> ((int` J)` S) = (/))
Theorem	ntreq0 7918	Two ways to say that a subset has an empty interior.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> (((int` J)` S) = (/) <-> A.x e. J (x (_ S -> x = (/))))
Theorem	cls0 7919	The closure of the empty set.
|- (J e. Top -> ((cls` J)` (/)) = (/))
Theorem	ntr0 7920	The interior of the empty set.
|- (J e. Top -> ((int` J)` (/)) = (/))
Theorem	elcls3 7921	Membership in a closure in terms of the members of a basis. Theorem 6.5(b) of [Munkres] p. 95.
|- J = (topGen` B)   &   |- X = U.J   =>   |- ((B e. Bases /\ S (_ X /\ P e. X) -> (P e. ((cls` J)` S) <-> A.x e. B (P e. x -> (x i^i S) =/= (/))))
Neighborhoods
Syntax	cnei 7922	Extend class notation with neighborhood relation for topologies.
class nei
Definition	df-nei 7923	Define a function on topologies whose value is a map from a subset to its neighborhoods.
|- nei = {<.j, y>. | (j e. Top /\ y = {<.z, v>. | (z (_ U.j /\ v = {w | (w (_ U.j /\ E.g e. j (z (_ g /\ g (_ w))})})}
Theorem	neifval 7924	The neighborhood function on the subsets of a topology's base set.
|- X = U.J   =>   |- (J e. Top -> (nei` J) = {<.z, w>. | (z (_ X /\ w = {v | (v (_ X /\ E.g e. J (z (_ g /\ g (_ v))})})
Theorem	neif 7925	The neighborhood function is a function of the subsets of a topology's base set.
|- X = U.J   =>   |- (J e. Top -> (nei` J) Fn P~X)
Theorem	neiss2 7926	A set with a neighborhood is a subset of the topology's base set. (This theorem depends on a function's value being empty outside of its domain, but it will make later theorems simpler to state.)
|- X = U.J   =>   |- ((J e. Top /\ N e. ((nei` J)` S)) -> S (_ X)
Theorem	neival 7927	The set of neighborhoods of a subset of the base set of a topology.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> ((nei` J)` S) = {v | (v (_ X /\ E.g e. J (S (_ g /\ g (_ v))})
Theorem	isnei 7928	The predicate "N is a neighborhood of S." (Contributed by FL, 25-Sep-2006.)
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> (N e. ((nei` J)` S) <-> (N (_ X /\ E.g e. J (S (_ g /\ g (_ N))))
Theorem	neiint 7929	An intuitive definition of a neighborhood in terms of interior. (Contributed by Szymon Jaroszewicz, 18-Dec-2007.)
|- X = U.J   =>   |- ((J e. Top /\ S (_ X /\ N (_ X) -> (N e. ((nei` J)` S) <-> S (_ ((int` J)` N)))
Theorem	isneip 7930	The predicate "N is a neighborhood of point P."
|- X = U.J   =>   |- ((J e. Top /\ P e. X) -> (N e. ((nei` J)` {P}) <-> (N (_ X /\ E.g e. J (P e. g /\ g (_ N))))
Theorem	neii1 7931	A neighborhood is included in the topology's base set.
|- X = U.J   =>   |- ((J e. Top /\ N e. ((nei` J)` S)) -> N (_ X)
Theorem	neii2 7932	Property of a neighborhood.
|- ((J e. Top /\ N e. ((nei` J)` S)) -> E.g e. J (S (_ g /\ g (_ N))
Theorem	neiss 7933	Any neighborhood of a set S is also a neighborhood of any subset R (_ S. Bourbaki TG I.2. (Contributed by FL, 25-Sep-2006.)
|- ((J e. Top /\ N e. ((nei` J)` S) /\ R (_ S) -> N e. ((nei` J)` R))
Theorem	ssnei 7934	A set is included in its neighborhoods. Based on Bourbaki TG I.3 Viii. (Contributed by FL, 16-Nov-2006.)
|- ((J e. Top /\ N e. ((nei` J)` S)) -> S (_ N)
Theorem	elnei 7935	A point belongs to any of its neighborhoods. Based on Bourbaki TG I.3 Viii. (Contributed by FL, 28-Sep-2006.)
|- ((J e. Top /\ P e. A /\ N e. ((nei` J)` {P})) -> P e. N)
Theorem	0nnei 7936	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	neips 7937	A neighborhood of a set is a neighborhood of every point in the set. Bourbaki TG I.2. (Contributed by FL, 16-Nov-2006.)
|- X = U.J   =>   |- ((J e. Top /\ S (_ X /\ S =/= (/)) -> (N e. ((nei` J)` S) <-> A.p e. S N e. ((nei` J)` {p})))
Theorem	opnneissb 7938	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 (_ X) -> (S (_ N <-> N e. ((nei` J)` S)))
Theorem	opnssneib 7939	Any superset of an open set is a neighborhood of it.
|- X = U.J   =>   |- ((J e. Top /\ S e. J /\ N (_ X) -> (S (_ N <-> N e. ((nei` J)` S)))
Theorem	ssnei2 7940	Any subset of X containing a neigborhood of a set is a neighborhood of this set. Based on Bourbaki TG I.3 Vi. (Contributed by FL, 2-Oct-2006.)
|- X = U.J   =>   |- (((J e. Top /\ N e. ((nei` J)` S)) /\ (N (_ M /\ M (_ X)) -> M e. ((nei` J)` S))
Theorem	neindisj 7941	Any neighborhood of an element in the closure of a subset intersects the subset. Part of proof of Theorem 6.6 of [Munkres] p. 97.
|- X = U.J   =>   |- (((J e. Top /\ S (_ X) /\ (P e. ((cls` J)` S) /\ N e. ((nei` J)` {P}))) -> (N i^i S) =/= (/))
Theorem	opnneiss 7942	An open set is a neighborhood of any of its subsets.
|- ((J e. Top /\ N e. J /\ S (_ N) -> N e. ((nei` J)` S))
Theorem	opnneip 7943	An open set is a neighborhood of any of its members.
|- ((J e. Top /\ N e. J /\ P e. N) -> N e. ((nei` J)` {P}))
Theorem	tpnei 7944	The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 7942. (Contributed by FL, 2-Oct-2006.)
|- X = U.J   =>   |- (J e. Top -> (S (_ X <-> X e. ((nei` J)` S)))
Theorem	unnei 7945	The union of the neighborhoods of a set equals the topology's underlying set. (Contributed by FL, 18-Sep-2007.)
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> U.((nei` J)` S) = X)
Theorem	innei 7946	The intersection of two neighborhoods of a set is also a neighborhood of the set. Based on Bourbaki TG I.3 Vii. (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 7947	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 7948	For any neighborhood N of S, there is a neighborhood x of S such that N is a neighborhood of all subsets of x. Based on Bourbaki TG I.3 Viv. (Contributed by FL, 2-Oct-2006.)
|- ((J e. Top /\ N e. ((nei` J)` S)) -> E.x e. ((nei` J)` S)A.y(y (_ x -> N e. ((nei` J)` y)))
Theorem	0nei 7949	The empty set is a neighborhood of itself. (Contributed by FL, 10-Dec-2006.)
|- (J e. Top -> (/) e. ((nei` J)` (/)))
Limit points
Syntax	clp 7950	Extend class notation with the limit point function for topologies.
class limPt
Definition	df-lp 7951	Define a function on topologies whose value is the set of limit points of the subsets of the base set. See lpval 7953.
|- limPt = {<.z, w>. | (z e. Top /\ w = {<.x, y>. | (x (_ U.z /\ y = {v | v e. ((cls` J)` (x \ {v}))})})}
Theorem	lpfval 7952	The limit point function on the subsets of a topology's base set.
|- X = U.J   =>   |- (J e. Top -> (limPt` J) = {<.x, y>. | (x (_ X /\ y = {v | v e. ((cls` J)` (x \ {v}))})})
Theorem	lpval 7953	The set of limit points of a subset of the base set of a topology. Alternate definition of limit point in [Munkres] p. 97.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> ((limPt` J)` S) = {x | x e. ((cls` J)` (S \ {x}))})
Theorem	islp 7954	The predicate "P is a limit point of S."
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> (P e. ((limPt` J)` S) <-> P e. ((cls` J)` (S \ {P}))))
Theorem	lpsscls 7955	The limits points of a subset are included in the subset's closure.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> ((limPt` J)` S) (_ ((cls` J)` S))
Theorem	lpss 7956	The limits points of a subset are included in the base set.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> ((limPt` J)` S) (_ X)
Theorem	islp2 7957	The predicate "P is a limit point of S," in terms of neighborhoods. Definition of limit point in [Munkres] p. 97. Although Munkres uses open neighborhoods, it also works for our more general neighborhoods.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X /\ P e. X) -> (P e. ((limPt` J)` S) <-> A.n e. ((nei` J)` {P})(n i^i (S \ {P})) =/= (/)))
Theorem	clslp 7958	The closure of a subset of a topological space is the subset together with its limit points. Theorem 6.6 of [Munkres] p. 97.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> ((cls` J)` S) = (S u. ((limPt` J)` S)))
Theorem	islpi 7959	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 7960	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 7961	QQ is dense in the standard topology on RR.
|- ((cls` (topGen` ran (,)))` QQ) = RR
Continuity
Syntax	ccn 7962	Extend class notation with the set of continuous functions between topologies.
class Cn
Syntax	ccnp 7963	Extend class notation with the set of functions between topologies continuous at a point.
class CnP
Definition	df-cn 7964	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 7968 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 7965	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 7966	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 7967	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 7968	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 7969	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 7970	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 7971	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 7972	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 7973	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 7974	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 7975	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 7976	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 7977	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 7978	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 7979	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 7980	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 7981	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 7982	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 7983	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 7984	Lemma for cncnp2 7989.
Theorem	cncnplem2 7985	Lemma for cncnp2 7989.
Theorem	cncnplem3 7986	Lemma for cncnp2 7989.
Theorem	cncnplem4 7987	Lemma for cncnp2 7989.
Theorem	cncnp 7988	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 7989	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 7990	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 7991	Extend class notation with the class of all Hausdorf spaces.
class Haus
Definition	df-haus 7992	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 7993	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 7994	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 7995	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 7996	A Hausdorff space is a topology.
|- (J e. Haus -> J e. Top)
Theorem	sncld 7997	A singleton is closed in a Hausdorff space.
|- X = U.J   =>   |- ((J e. Haus /\ P e. X) -> {P} e. (Clsd` J))
Theorem	dnsconst 7998	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 A (_ (`'F"{P}) means "F is constant on A" (see funconstss 3922).
|- X = U.J   &   |- Y = U.K   =>   |- (((J e. Top /\ K e. Haus /\ F e. (J Cn K)) /\ (P e. Y /\ A (_ (`'F"{P}) /\ ((cls` J)` A) = X)) -> F:X-->{P})
Metric spaces
Basic metric space properties
Syntax	cme 7999	Extend class notation with the class of all metrics.
class Met
Syntax	cmt 8000	Extend class notation with the class of all metric spaces.
class MetSp