A Note on Cauchy sn-Symmetric Spaces

CHEN Nei-ping

(College of Information，Hunan University of Commerce，Changsha 410205，China)

1 Introduction and definitions

sn-Symmetric spaces are an important generalization of symmetric spaces.Recently，Ge and Li［1］investigate sn-symmetric spaces and proved that a space is a weak Cauchy sn-symmetric space if and only if it is a sequentially-quotient π-image of a metric space.By viewing this result，the following question can be raised:

Question 1.1How characterize Cauchy sn-symmetric spaces by means of certain π-images of metric spaces?

In this paper，we prove that a space is a Cauchy sn-symmetric space if and only if it is a sequence-covering π-image of a metric space，which answers the above question.

Throughout this paper，all spaces are Hausdroff and all mappings are continuous and surjective.N denotes the set of all natural numbers.For a collection P of subsets of a space X and a mapping f:X →Y，denotes {f(P):P ∈P}by f(P)，Px={P ∈P:x ∈P)and st(x，P)=∩Px.For the usual product spacedenotes the projective ontoFor a sequence {xn}in a space X，denotes〈xn〉={xn:n ∈N}.

Definition 1.2［2-3］Let f:X →Y be a mapping.

(1)f is called a sequence-covering mapping，if whenever {yn}is a convergent sequence in Y，then there exists a convergent sequence {xn}in X such that each xn∈f－1(yn).

(2)f is called a sequentially-quotient mapping，if for each convergent sequence S in Y，there exists a convergent sequence L in X such that f(L)is a subsequence of S.

Obviously，every sequence-covering mapping is sequentially-quotient.

Definition 1.3［4］Let X be a space，and P ?X.

(1)A sequence {xn}in X is called eventually in P，if {xn}converges to x，and there exists m ∈N such that {x}∪{xn:n ≥m}?P.

(2)P is called a sequential neighborhood of x in X，if x ∈P，and whenever a sequence {xn}in X converges to x，then {xn}is eventually in P.

(3)P is called sequential open in X，if P is a sequential neighborhood of each of its points.

(4)X is called a sequential space，if any sequential open subset of X is open in X.

Definition 1.4［5］Let P be a collection of subsets of a space X and x ∈X.

(1)P is called a network of x in X，if x ∈∩P and for each neighborhood U of x，there exists P ∈P such that P ?U.

(2)P is called a sn-network of x in X，if P is a network of x in X and each element of P is also a sequential neighborhood of x.

(3)P is called a cs-cover for X，if P is a cover for X，and every convergent sequence in X is eventually in some element of P.

(4)P is called a sn-cover for X，if P is a cover for X，every element of P is a sequential neighborhood of some point in X，and for each x ∈X there exists a sequential neighborhood P of x in X such that P ∈P.

Definition 1.5Let {Pn}be a sequence of covers of a space X.

(1){Pn}is called a point-star network for X，if for each x ∈X，〈st(x，Pn)〉is a network of x in X.

(2){Pn}is called a sn-development for X，if for each x ∈X，〈st(x，Pn)〉is a sn-network of x in X.

Definition 1.6Let X be a set.A non-negative real valued function d defined on X× X is called a d-function on X，if d(x，x)=0 and d(x，y)=d(y，x)for any x ∈X.

Let d be a d-function on a space X.In this paper we write B(x，1/n)={y ∈X:d(x，y)＜1/n}，d(A)=sup{d(x，y):x，y ∈A}and d(A，B)=inf{d(x，y):x ∈and y ∈B}，where x ∈X，n ∈N and A，B ?X.

Definition 1.7［6］Let d be a d-function on a space X.(X，d)is called an sn-symmetric space，if d satisfies the condition:{B(x，1/n):n ∈N}is an sn-network of x in X for any x ∈X，where d is called an sn-symmetric on X.

Definition 1.8［7］Let(X，d)be a metric space and let f:X →Y be a mapping.f is called a π-mapping with respect to d，if for each y ∈Y and each open neighborhood V in Y，d(f－1(y)，Mf－1(V))＞0.

Definition 1.9［8］Let(X，d)be an sn-symmetric space，Then，

(1)A sequence {xn}in X is called d-Cauchy，if for each ε ＞0，there exists k ∈N such that d(xm，xn)＜ε for all n，m ＞k.

(2)X is called a Cauchy sn-symmetric space，if each convergent sequence in X is d-Cauchy.

2 Main results

Lemma 2.1［1］Let(X，d)be an sn-symmetric space，n ∈N and x ∈X.Put Pn={A ?X:d(A)＜1/n}，then st(x，Pn)=B(x，1/n).

Theorem 2.2The following are equivalent for a space X:

(1)X is a Cauchy sn-symmetric space.

(2)X has a sn-development consisting of cs-covers.

(3)X has a sn-development consisting of sn-covers.

(4)X is a sequence-covering π-image of a metric space.

Proof(1)?(2).Suppose X is a Cauchy sn-symmetric space.For each n ∈N，put

Pn={A ?X:d(A)＜1/n}

By Lemma 2.1，st(x，Pn)=B(x，1/n)for each x ∈X，so〈st(x，Pn)〉is a network of x in X for each x ∈X.Thus {Pn}is a point-star network for X.

For each n ∈N and each sequence {xi}converging to x ∈X，since {xi}is d-Cauchy，then there exists m1∈N such that d(xi，xj)＜1/(n+1)for all i，j ≥m1Since X is a sn-symmetric space，then {B(x，1/i):i ∈N}is an sn-network of x in X.So B(x，1/(n+1))is a sequential neighborhood of x in X.Thus there exists m2∈N such that d(x，xi)＜1/(n+1)for all i ≥m2.Put P={x}∪{xi:i ≥m}with m=max{m1，m2}，then P ∈Pn.Obviously，{xi}is eventually in P.Hence each Pnis a cs-cover for X.

For each x ∈X and n ∈N，since Pnis a cs-cover for X，then st(x，Pn)is a sequential neighborhood of x in X.So〈st(x，Pn)〉is a sn-network of x in X.Thus，{Pn}is a sn-development for X.

t(x，y)=min{n:x ?st(y，Pn)}(x ≠y).

We define

then a d-function on X.

ClaimFor each x，y ∈X，x ∈st(y，Pn)if and only if t(x，y)＞n.

In fact，the if part is obvious［9-11］.The only if part:Suppose x ∈st(y，Pn)but t(x，y)≤n，since Pnrefine Pt(x，y)，then st(y，Pn)?st(y，Pt(x，y)).Note that x ?st(y，Pt(x，y))，so x ?st(y，Pn)，a contradiction.

For each x ∈X and n ∈N，st(x，Pn)=B(x，1/2n)by the Claim.Because {Pn}is a pointstar network for X，then(X，d)is a sn-symmetric space and d has the following property:for each x ∈X and ε ＞0，there exists δ=δ(x，ε)＞0 such that d(x，y)＜δ and d(x，z)＜δ imply d(y，z)＜ε.Otherwise，there exist ε0＞0 and two sequences {yn}and {zn}in X such that d(yn，zn)≥ε0whenever d(x，yn)＜1/2nand d(x，zn)＜1/2n.From Pnis a point-star network for X，{yn}and {zn}all converge to x.We choose k ∈N such that 1=2k＜ε0.Since Pkis a cs-cover for X，then {ym，zm}?P for some m ∈N and P ∈Pk.Thus ym∈st(zm，Pk).By the Claim，t(ym，zm)＞k.Thus，d(ym，zm)=1/2t(ym，zm)＜1/2k＜ε0，a contradiction.

For each x ∈X and n ∈N，we can pick δ=δ(x，n)such that d(y，z)＜1/n whenever d(x，y)＜δ and d(x，z)＜δ.Let g(n，x)=B(x，δ(x，n)).Since Pnis a cs-cover for X，then st(x，Pn)is a sequential neighborhood of x in X，so g(n，x)is also.Put

Fn={g(n，x):x ∈X}，then every Fnis a sn-cover for X.

If {Fn}is not a point-star network for X，then there exist x ∈G ∈τ(X)and two sequences {xn}and{yn}in X such that x ∈g(n，yn)and xn∈g(n，yn)G.So {xn}does not converge to x，and d(yn，x)＜δ(yn，xn)＜δ(yn，n).By the condition，d(x，xn)＜1/n.This implies that {xn}converges to x，a contradiction.Hence Fnis a point-start network for X［12-13］.

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Since st(x，F(xiàn)n)is a sequential neighborhood of x in X for each x ∈X and n ∈N，then {Fn}is a sn-development for X.

(2)?(1).Suppose {Pi}is a sn-development consisting of cs-covers for X.We can assume that Pn+1refines Pnfor each n ∈N.A similar proof of(2)?(3)，we can define a sn-symmetric d on X such that st(x，Pn)=B(x，1/2n)for each x ∈X and n ∈N.So(X，d)is a sn-symmetric space.For each sequence {xn}in X converging to x ∈X and ε ＞0，there exists k ∈N such that 1/2k＜ε.Since Pkis a cs-cover for X，then there exist P ∈Pkand l ∈N such that {x}∪{xn:n ≥l}?P.If n，m ≥l，then xn，xm∈P，so xn∈st(xm，Pk).Thus t(xn，xm)＞k by the Claim in(2)?(3).Hence d(xn，xm)=1/2t(xn，xm)＜1/2k＜ε whenever n，m ≥l.Therefore {xn}is d-Cauchy.This implies that X is a Cauchy sn-symmetric space.

(3)?(4).Suppose {Pn}is a sn-development consisting of sn-covers for X.For each i ∈N，let Pi={Pα:α ∈Λi}，endow Λiwith the discrete topology，then Λiis a metric space.Put

and endow M with the subspace topology induced from the usual product topology of the collection {Λi:i ∈N}of metric spaces，then M is a metric space.Since X is Hausdroff，xαis unique in X.For each α ∈M.We define f:M →X by f(α)=xα.For each x ∈X and i ∈N，there exists αi∈Λisuch that x ∈Pαi.From {Pi}is a point-star network for X，{Pαi:i ∈N}is a network of x in X.Put α=(αi)，then α ∈M and f(α)=x.Thus f is surjective.Suppose α=(αi)∈M and f(α)=x ∈U ∈τ(X)，then exists n ∈N such that Pαn?U.Put

V={β ∈M:the n-th coordinate of β is αn}.

then α ∈V ∈τ(X)，and f(V)?Pαn?U.Hence f is continuous.

1)f is a π-mapping.For each α，β ∈M，we define

then d is a distance on M.Because the topology of M is the subspace topology induced from the usual product topology of the collection {Λi:i ∈N}of discrete spaces，thus d is metric on M.For each x ∈U ∈τ(X)，note that {Pn}is a point-star network for X，there exists n ∈N such that st(x，Pn)?U.For α ∈f－1(x)，β ∈M，if d(α，β)＜1/n，then πi(α)=πi(β)for all i ≤n.So x ∈Pπn(α)=Pπn(β).Thus

Hence

d(f－1(x)，Mf－1(U))≥1/n.

Therefore f is a π-mapping.

2)f is a sequence-covering mapping.

Suppose {xn}converges to x in X.For each i ∈N，since every Piis a sn-cover for X，then there exists αi∈Λisuch that Pαiis a sequential neighborhood of x in X，so {xn}is eventually in Pαi.From {Pi}is a point-star network for X，〈Pαi〉is a network of x in X.Put，then βx∈f－1(x).For each n ∈N，if xn∈Pαi，let αin=αi;if xn?Pαi，pick αin∈Λisuch that xn∈Pαin.Thus there exists ni∈N such that αin=αifor all n ＞ni.So {αin}converges to αi.For each n ∈N，put

then βn∈f－1(xn)and {βn}converges to βx.Thus f is a sequence-covering mapping.

(4)?(2).Suppose X is a image of a metric space(M，d)under a sequence-covering π-mapping f.For each n ∈N，put Bn={B(z，1/n)):z ∈M)and Pn=f(Bn).Then {Pn}is a point-star network for X.In fact，for each x ∈X and its open neighborhood U，since f is a π-mapping，then there exists n ∈N such that d(f－1(x)，Mf－1(U))＞1/n.We can pick m ∈N such that m ≥2n.If z ∈M with x ∈f(B(z，1/m))，then

f－1(x)∩B(z，1/m)≠?.

d(f－1(x)，Mf－1(U))≤2/m ≤1/n，

a contradiction.Thus B(z，1=m)?f－1(U)，so f(B(z，1/m))?U.Hence st(x，Pm)?U.This implies that{Pn}is a point-star network for X.

For each n ∈N，since Bnis a cs-cover for M and sequence-covering mappings preserve cs-covers，then Pnis a cs-cover for X.

For each x ∈X and n ∈N，since Pnis a cs-cover for X，then st(x，Pn)is a sequential neighborhood of x in X.So〈st(x，Pn)〉is a sn-network of x in X.Thus，{Pn}is a sn-development for X.

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