mmtheorems79 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 7801-7900 - Page 79 of 123

Type	Label	Description
Statement
Syntax	ctps 7801	Extend class notation with the class of all topological spaces.
class TopSp
Syntax	ctb 7802	Extend class notation with the class of all topological bases.
class Bases
Syntax	ctg 7803	Extend class notation with a function that converts a basis to its corresponding topology.
class topGen
Definition	df-top 7804	Define the (proper) class of all topologies. See istop2g 7809 for an alternate way to express finite intersection and istps5 7822 for a standard definition in terms of both members of a topological space.
|- Top = {x | (A.y(y (_ x -> U.y e. x) /\ A.y e. x A.z e. x (y i^i z) e. x)}
Definition	df-topsp 7805	Define the class of all topological spaces, each of which is an ordered pair the second of which is a topology on the first. See istps5 7822 for a standard way to express a topological space.
|- TopSp = {<.x, y>. | (y e. Top /\ x = U.y)}
Definition	df-bases 7806	Define the class of all topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 7824). Note that "bases" is the plural of "basis."
|- Bases = {x | A.y e. x A.z e. x (y i^i z) (_ U.(x i^i P~(y i^i z))}
Definition	df-topgen 7807	Define a function that converts a basis to its corresponding topology. Equivalent to the definition of a topology generated by a basis in [Munkres] p. 78 (see tgval2 7829). See tgval3 7837 for an alternate expression for the value.
|- topGen = {<.x, y>. | (x e. Bases /\ y = {z | z (_ U.(x i^i P~z)})}
Theorem	istopg 7808	Express the predicate "J is a topology." Note: In the literature, a topology is often represented by a script letter T, which resembles the letter J. This confusion has led to J being used by some authors - e.g. K. D. Joshi, Introduction to General Topology (1983), p. 114 - and it is convenient for us since we later use T to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.)
|- (J e. A -> (J e. Top <-> (A.x(x (_ J -> U.x e. J) /\ A.x e. J A.y e. J (x i^i y) e. J)))
Theorem	istop2g 7809	Express the predicate "J is a topology," using "the intersection of the elements of any finite subcollection" instead of the intersection of any two elements.
|- (J e. A -> (J e. Top <-> (A.x(x (_ J -> U.x e. J) /\ A.x((x (_ J /\ x =/= (/) /\ x e. Fin) -> |^|x e. J))))
Theorem	uniopn 7810	The union of a subset of a topology is an open set. (Contributed by Stefan Allan, 27-Feb-2006.)
|- ((J e. Top /\ A (_ J) -> U.A e. J)
Theorem	iunopn 7811	The indexed union of a subset of a topology is an open set.
|- ((J e. Top /\ A.x e. A B e. J) -> U_x e. A B e. J)
Theorem	inopn 7812	The intersection of two open sets of a topology is also an open set.
|- ((J e. Top /\ A e. J /\ B e. J) -> (A i^i B) e. J)
Theorem	0opn 7813	The empty set is an open subset of a topology. (Contributed by Stefan Allan, 27-Feb-2006.)
|- (J e. Top -> (/) e. J)
Theorem	topopn 7814	The underlying set of a topology is an open set.
|- X = U.J   =>   |- (J e. Top -> X e. J)
Theorem	eltopss 7815	A member of a topology is a subset of its underlying set.
|- X = U.J   =>   |- ((J e. Top /\ A e. J) -> A (_ X)
Theorem	eltopsp 7816	Construct a topological space from a topology and vice-versa. We say that A is a topology on U.A. (This could be proved more efficiently from istps 7818, but the proof here does not require the Axiom of Regularity.)
|- (<.U.J, J>. e. TopSp <-> J e. Top)
Theorem	tpsex 7817	Existence implied by membership in a topological space. This technical lemma, which can help eliminate unnecessary antecedents, uses the Axiom of Regularity (via elirr 4742) along with definitional tricks.
|- (<.A, J>. e. TopSp -> (A e. V /\ J e. V))
Theorem	istps 7818	Express the predicate "is a topological space."
|- (<.A, J>. e. TopSp <-> (J e. Top /\ A = U.J))
Theorem	istps2 7819	Express the predicate "is a topological space."
|- (<.A, J>. e. TopSp <-> ((J e. Top /\ J (_ P~A) /\ ((/) e. J /\ A e. J)))
Theorem	istps3 7820	A standard textbook definition of a topological space.
|- (<.A, J>. e. TopSp <-> ((J (_ P~A /\ (/) e. J /\ A e. J) /\ (A.x(x (_ J -> U.x e. J) /\ A.x e. J A.y e. J (x i^i y) e. J)))
Theorem	istps4 7821	A standard textbook definition of a topological space.
|- (<.A, J>. e. TopSp <-> ((J (_ P~A /\ (/) e. J /\ A e. J) /\ (A.x(x (_ J -> U.x e. J) /\ A.x((x (_ J /\ x =/= (/) /\ x e. Fin) -> |^|x e. J))))
Theorem	istps5 7822	A standard textbook definition of a topological space <.A, J>.: a topology on A is a collection J of subsets of A such that (/) and A are in J and that is closed under union and finite intersection. Definition of topological space in [Munkres] p. 76.
|- (<.A, J>. e. TopSp <-> ((A.x e. J x (_ A /\ (/) e. J /\ A e. J) /\ (A.x(x (_ J -> U.x e. J) /\ A.x((x (_ J /\ x =/= (/) /\ x e. Fin) -> |^|x e. J))))
Bases for topologies
Theorem	isbasisg 7823	Express the predicate "B is a basis for a topology."
|- (B e. C -> (B e. Bases <-> A.x e. B A.y e. B (x i^i y) (_ U.(B i^i P~(x i^i y))))
Theorem	isbasis2g 7824	Express the predicate "B is a basis for a topology."
|- (B e. C -> (B e. Bases <-> A.x e. B A.y e. B A.z e. (x i^i y)E.w e. B (z e. w /\ w (_ (x i^i y))))
Theorem	isbasis3g 7825	Express the predicate "B is a basis for a topology." Definition of basis in [Munkres] p. 78.
|- (B e. C -> (B e. Bases <-> (A.x e. B x (_ U.B /\ A.x e. U.BE.y e. B x e. y /\ A.x e. B A.y e. B A.z e. (x i^i y)E.w e. B (z e. w /\ w (_ (x i^i y)))))
Theorem	basis1 7826	Property of a basis.
|- ((B e. Bases /\ C e. B /\ D e. B) -> (C i^i D) (_ U.(B i^i P~(C i^i D)))
Theorem	basis2 7827	Property of a basis.
|- (((B e. Bases /\ C e. B) /\ (D e. B /\ A e. (C i^i D))) -> E.x e. B (A e. x /\ x (_ (C i^i D)))
Theorem	tgval 7828	The topology generated by a basis.
|- (B e. Bases -> (topGen` B) = {x | x (_ U.(B i^i P~x)})
Theorem	tgval2 7829	Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 7836) that (topGen` B) is indeed a topology (on U.B; see unitg 7835).
|- (B e. Bases -> (topGen` B) = {x | (x (_ U.B /\ A.y e. x E.z e. B (y e. z /\ z (_ x))})
Theorem	eltg 7830	Membership in a topology generated by a basis.
|- (B e. Bases -> (A e. (topGen` B) <-> A (_ U.(B i^i P~A)))
Theorem	eltg2 7831	Membership in a topology generated by a basis.
|- (B e. Bases -> (A e. (topGen` B) <-> (A (_ U.B /\ A.x e. A E.y e. B (x e. y /\ y (_ A))))
Theorem	tg1 7832	Property of a member of a topology generated by a basis.
|- ((B e. Bases /\ A e. (topGen` B)) -> A (_ U.B)
Theorem	tg2 7833	Property of a member of a topology generated by a basis.
|- ((B e. Bases /\ A e. (topGen` B) /\ C e. A) -> E.x e. B (C e. x /\ x (_ A))
Theorem	bastg 7834	A member of a basis is a subset of the topology it generates.
|- (B e. Bases -> B (_ (topGen` B))
Theorem	unitg 7835	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 Bases completely specifies the basis it corresponds to.
|- (B e. Bases -> U.(topGen` B) = U.B)
Theorem	tgcl 7836	Show that a basis generates a topology. Remark in [Munkres] p. 79.
|- (B e. Bases -> (topGen` B) e. Top)
Theorem	tgval3 7837	Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80.
|- (B e. Bases -> (topGen` B) = {x | E.y(y (_ B /\ x = U.y)})
Theorem	eltg3 7838	Membership in a topology generated by a basis.
|- (B e. Bases -> (A e. (topGen` B) <-> E.x(x (_ B /\ A = U.x)))
Theorem	topbas 7839	A topology is its own basis.
|- (J e. Top -> J e. Bases)
Theorem	tgtop 7840	A topology is its own basis.
|- (J e. Top -> (topGen` J) = J)
Theorem	eltop 7841	Membership in a topology, expressed without quantifiers.
|- (J e. Top -> (A e. J <-> A (_ U.(J i^i P~A)))
Theorem	eltop2 7842	Membership in a topology.
|- (J e. Top -> (A e. J <-> (A (_ U.J /\ A.x e. A E.y e. J (x e. y /\ y (_ A))))
Theorem	eltop3 7843	Membership in a topology.
|- (J e. Top -> (A e. J <-> E.x(x (_ J /\ A = U.x)))
Theorem	tgidm 7844	The topology generator function is idempotent.
|- (B e. Bases -> (topGen` (topGen` B)) = (topGen` B))
Theorem	bastop 7845	Two ways to express that a basis is a topology.
|- (B e. Bases -> (B e. Top <-> (topGen` B) = B))
Theorem	tgtop11 7846	The topology generation function is one-to-one when applied to completed topologies.
|- ((J e. Top /\ K e. Top /\ (topGen` J) = (topGen` K)) -> J = K)
Theorem	0top 7847	The singleton of the empty set is the only topology possible for an empty underlying set.
|- (J e. Top -> (U.J = (/) <-> J = {(/)}))
Theorem	tgss 7848	Subset relation for generated topologies.
|- ((B e. Bases /\ C e. Bases /\ B (_ C) -> (topGen` B) (_ (topGen` C))
Theorem	tgss2 7849	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.
|- ((B e. Bases /\ C e. Bases /\ U.B = U.C) -> ((topGen` B) (_ (topGen` C) <-> A.x e. U.BA.y e. B (x e. y -> E.z e. C (x e. z /\ z (_ y))))
Theorem	tgss3 7850	A criterion for determining whether one topology is finer than another. Lemma 2.2 of [Munkres] p. 80 using abbreviations.
|- ((B e. Bases /\ C e. Bases /\ U.B = U.C) -> ((topGen` B) (_ (topGen` C) <-> A.x e. B (U.B i^i x) (_ U.(C i^i P~x)))
Theorem	basgen2 7851	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.
|- ((J e. Top /\ B (_ J /\ A.x e. J A.y e. x E.z e. B (y e. z /\ z (_ x)) -> (B e. Bases /\ (topGen` B) = J))
Theorem	basgen 7852	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.
|- ((J e. Top /\ B (_ J /\ A.x e. J x (_ U.(B i^i P~x)) -> (B e. Bases /\ (topGen` B) = J))
Theorem	2basgen 7853	Conditions that determine the equality of two generated topologies.
|- (((B e. Bases /\ C e. Bases) /\ (B (_ C /\ C (_ (topGen` B))) -> (topGen` B) = (topGen` C))
Theorem	bastop1 7854	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 "B e. Bases /\ (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.
|- ((J e. Top /\ B (_ J) -> ((B e. Bases /\ (topGen` B) = J) <-> A.x e. J E.y(y (_ B /\ x = U.y)))
Theorem	bastop2 7855	A version of bastop1 7854 that doesn't have B (_ J in the antecedent.
|- (J e. Top -> ((B e. Bases /\ (topGen` B) = J) <-> (B (_ J /\ A.x e. J E.y(y (_ B /\ x = U.y))))
Subbases for topologies
Theorem	subbas 7856	The collection of finite intersections of the elements of any set A is a basis for a topology (on U.A by subbas2 7857). The set A is called a subbasis for that topology. Theorem for subbases in [Munkres] p. 82. See abfii1 4704 through abfii5 4708 for other ways to express the collection of finite intersections.
|- A e. V   &   |- B = {x | E.y(y (_ A /\ y e. Fin /\ x = |^|y)}   =>   |- B e. Bases
Theorem	subbas2 7857	The collection of finite intersections of any set (subbasis) A is a basis for a topology on U.A. Justifies the definition of subbasis in [Munkres] p. 82 (after using unitg 7835).
|- B = {x | E.y(y (_ A /\ y e. Fin /\ x = |^|y)}   =>   |- U.B = U.A
Examples of topologies
Theorem	subtop 7858	A subspace topology is a topology. Definition of subspace topology in [Munkres] p. 89. Y is normally a subset of the base set of J. (Contributed by FL, 15-Apr-2007.)
|- (J e. Top -> {x | E.y e. J x = (y i^i Y)} e. Top)
Theorem	sn0top 7859	The singleton of the empty set is a topology. (Contributed by Stefan Allan, 3-Mar-2006.)
|- {(/)} e. Top
Theorem	indistop 7860	The indiscrete topology on a set A. Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 16-Jul-2006; hypothesis eliminated by Stefan Allan, 6-Nov-2008.)
|- {(/), A} e. Top
Theorem	distop 7861	The discrete topology on a set A. Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 18-Jul-2006.)
|- A e. V   =>   |- P~A e. Top
Theorem	cctop 7862	The countable complement topology on a set A. Example 4 in [Munkres] p. 77. (Contributed by FL, 29-Aug-2006.)
|- A e. V   =>   |- {x | (x (_ A /\ ((A \ x) ~<_ om \/ x = (/)))} e. Top
Theorem	indistps 7863	The indiscrete topology on a set A expressed as a topological space. (Contributed by FL, 19-Jul-2006.)
|- A e. V   =>   |- <.A, {(/), A}>. e. TopSp
Theorem	distps 7864	The discrete topology on a set A expressed as a topological space. (Contributed by FL, 20-Aug-2006.)
|- A e. V   =>   |- <.A, P~A>. e. TopSp
Theorem	retopbas 7865	A basis for the standard topology on the reals.
|- ran (,) e. Bases
Theorem	retop 7866	The standard topology on the reals. (Contributed by FL, 4-Jun-2007.)
|- (topGen` ran (,)) e. Top
Theorem	uniretop 7867	The underlying set of the standard topology on the reals is the reals. (Contributed by FL, 4-Jun-2007.)
|- U.(topGen` ran (,)) = RR
Theorem	retps 7868	The standard topological space on the reals.
|- <.RR, (topGen` ran (,))>. e. TopSp
Theorem	iooretop 7869	Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.)
|- (A(,)B) e. (topGen` ran (,))
Closure and interior
Syntax	ccld 7870	Extend class notation with the set of closed sets of a topology.
class Clsd
Syntax	cnt 7871	Extend class notation with interior of a subset of a topology base set.
class int
Syntax	ccl 7872	Extend class notation with closure of a subset of a topology base set.
class cls
Definition	df-cld 7873	Define a function on topologies whose value is the set of closed sets of the topology.
|- Clsd = {<.z, w>. | (z e. Top /\ w = {x | (x (_ U.z /\ (U.z \ x) e. z)})}
Definition	df-ntr 7874	Define a function on topologies whose value is the interior function on the subsets of the base set. See ntrval 7886.
|- int = {<.z, w>. | (z e. Top /\ w = {<.x, y>. | (x (_ U.z /\ y = U.{v e. z | v (_ x})})}
Definition	df-cls 7875	Define a function on topologies whose value is the closure function on the subsets of the base set. See clsval 7887.
|- cls = {<.z, w>. | (z e. Top /\ w = {<.x, y>. | (x (_ U.z /\ y = |^|{v e. (Clsd` z) | x (_ v})})}
Theorem	cldval 7876	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.)
|- X = U.J   =>   |- (J e. Top -> (Clsd` J) = {x | (x (_ X /\ (X \ x) e. J)})
Theorem	ntrfval 7877	The interior function on the subsets of a topology's base set.
|- X = U.J   =>   |- (J e. Top -> (int` J) = {<.x, y>. | (x (_ X /\ y = U.{v e. J | v (_ x})})
Theorem	clsfval 7878	The closure function on the subsets of a topology's base set.
|- X = U.J   =>   |- (J e. Top -> (cls` J) = {<.x, y>. | (x (_ X /\ y = |^|{v e. (Clsd` J) | x (_ v})})
Theorem	iscld 7879	The predicate "S is a closed set."
|- X = U.J   =>   |- (J e. Top -> (S e. (Clsd` J) <-> (S (_ X /\ (X \ S) e. J)))
Theorem	iscld2 7880	A subset of the underlying set of a topology is closed iff its complement is open.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> (S e. (Clsd` J) <-> (X \ S) e. J))
Theorem	cldss 7881	A closed set is a subset of the underlying set of a topology.
|- X = U.J   =>   |- ((J e. Top /\ S e. (Clsd` J)) -> S (_ X)
Theorem	cldopn 7882	The complement of a closed set is open.
|- X = U.J   =>   |- ((J e. Top /\ S e. (Clsd` J)) -> (X \ S) e. J)
Theorem	isopn2 7883	A subset of the underlying set of a topology is open iff its complement is closed.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> (S e. J <-> (X \ S) e. (Clsd` J)))
Theorem	opncld 7884	The complement of an open set is closed.
|- X = U.J   =>   |- ((J e. Top /\ S e. J) -> (X \ S) e. (Clsd` J))
Theorem	topcld 7885	The underlying set of a topology is closed. Part of Theorem 6.1(1) of [Munkres] p. 93.
|- X = U.J   =>   |- (J e. Top -> X e. (Clsd` J))
Theorem	ntrval 7886	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.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> ((int` J)` S) = U.{x e. J | x (_ S})
Theorem	clsval 7887	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.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> ((cls` J)` S) = |^|{x e. (Clsd` J) | S (_ x})
Theorem	0cld 7888	The empty set is closed. Part of Theorem 6.1(1) of [Munkres] p. 93.
|- (J e. Top -> (/) e. (Clsd` J))
Theorem	iincld 7889	The indexed intersection of a collection B(x) of closed sets is closed. Theorem 6.1(2) of [Munkres] p. 93.
|- ((J e. Top /\ A =/= (/) /\ A.x e. A B e. (Clsd` J)) -> |^|_x e. A B e. (Clsd` J))
Theorem	intcld 7890	The intersection of a set of closed sets is closed.
|- ((J e. Top /\ A =/= (/) /\ A (_ (Clsd` J)) -> |^|A e. (Clsd` J))
Theorem	uncld 7891	The union of two closed sets is closed. Equivalent to Theorem 6.1(3) of [Munkres] p. 93.
|- ((J e. Top /\ A e. (Clsd` J) /\ B e. (Clsd` J)) -> (A u. B) e. (Clsd` J))
Theorem	cldcls 7892	A closed subset equals its own closure.
|- ((J e. Top /\ S e. (Clsd` J)) -> ((cls` J)` S) = S)
Theorem	clscld 7893	The closure of a subset of a topology's underlying set is closed.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> ((cls` J)` S) e. (Clsd` J))
Theorem	ntropn 7894	The interior of a subset of a topology's underlying set is open.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> ((int` J)` S) e. J)
Theorem	clsval2 7895	Express closure in terms of interior.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> ((cls` J)` S) = (X \ ((int` J)` (X \ S))))
Theorem	ntrval2 7896	Interior expressed in terms of closure.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> ((int` J)` S) = (X \ ((cls` J)` (X \ S))))
Theorem	clsss 7897	Subset relationship for closure.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X /\ T (_ S) -> ((cls` J)` T) (_ ((cls` J)` S))
Theorem	ntrss 7898	Subset relationship for interior.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X /\ T (_ S) -> ((int` J)` T) (_ ((int` J)` S))
Theorem	sscls 7899	A subset of a topology's underlying set is included in its closure.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> S (_ ((cls` J)` S))
Theorem	ntrss2 7900	A subset includes its interior.
|- X = U.J   =>   |- ((J e. Top /\ S (_ X) -> ((int` J)` S) (_ S)