# AN EXTREME POINT THEOREM ON HYPERSPACE.

1. Introduction

Suppose X is a Banach space equipped with the norm topology (denoted by ||*||) as well as the weak topology (denoted by [T.sub.w]). Let CC (X) ={A [subset.bar] X : A is a non-empty compact convex subset of X}, WCC (X) ={A [subset.bar] X : A is a non-empty weakly compact, convex subset of X} and BCC (X) ={A [subset.bar] X : A is a non-empty bounded, closed, convex subset of X}: Then (CC (X); h) ; (WCC (X); h) and (BCC (X); h) are known as the hyperspaces over the underlying space (X; ||*||). If [bar.X] = {[bar.x] = {[bar.x]} : x [member of] X}; then ([bar.X]; h) is isometrically isomorphic to the underlying space (X; ||*||). Thus every theorem proved on the hyperspaces is a natural extension of its corresponding counterpart of the underlying space (X; ||*||).

Blaschke  proved that every infinite sequence {[A.sub.n]g with [A.sub.n] [member of] K where K is an h-bounded and h-closed subset of the hyperspace (CC ([R.sup.n]); h) contains a convergent subsequence [mathematical expression not reproducible] (i:e:, there exists a subsequence [mathematical expression not reproducible] [subset.bar] K and [A.sub.0] [member of] K such that [mathematical expression not reproducible]. Blaschke's Theorem is an extension of the classical Heine-Borel Theorem which states that every closed and bounded subset K [??] [R.sup.n] is sequentially compact. Many mathematicians have studied convergence of convex sets on different spaces (, , ):

In 1986, De Blasi and Myjak () introduced the concept of weak sequential convergence on the hyperspace WCC (X): Suppose [A.sub.n]; A [member of] WCC (X); they define [A.sub.n] converges to [A.sub.0] weakly ([mathematical expression not reproducible]) if and only if [mathematical expression not reproducible] (x*) [right arrow] [mathematical expression not reproducible] (x*) = sup{x* (a) | a [member of] [A.sub.0]} and proved an infinite dimensional version of Blaschke's Theorem and other results. The notion of weak topology [T.sub.w] has been introduced and investigated by Hu and company (, , , , ). They showed that Browder-Kirk's fixed point theorem can be extended to the hyperspace WCC (X) equipped with Hausdorff metric h as well as a certain weak topology [T.sub.w] and many other results. We remind the readers that many fundamental results that are valid on the underlying space X cannot be extended to hyperspace. For example, it is well-known that every ||*||-closed (originally closed, strongly closed) convex set is also [T.sub.w]-closed (weakly closed). Also for every compact convex set K [subset.bar] X and x [??] K; there exists some x* [member of] X* such that [mathematical expression not reproducible]. Examples have been given in  that these results cannot be extended to hyperspace.

Suppose now X is a locally convex topological vector space and X* its dual space. Let X be equipped with the original topology T as well as the weak topology [T.sub.w], and WCC (X) ={A [subset.bar] X : A is a [T.sub.w]-compact convex subset of X} is the corresponding hyperspace. A topology [T.sub.w] will be introduced on X and the main result of this paper is to show that every [T.sub.w]-compact, convex subset of WCC (X) has an extreme point. This result is an extension of the classical Krein-Milman Theorem.

2. NOTATIONS AND PRELIMINARIES

Let X be a Banach space and X* its topological dual, and BCC (X) is the collection of all non-empty bounded, closed, convex subsets of X: In general we have CC (X) [??] WCC (X) [??] BCC (X): For reflexive Banach space X, we have WCC (X) = BCC (X): If X is finite dimensional, then CC (X) = WCC (X) = BCC (X): To avoid avoid confusion we shall use small letters a; b; c; ... ; z to denote elements of the underlying space X; capital letters A;B;... ; Z to denote elements of the hyperspaces CC (X); WCC (X) and BCC (X) as well as subsets of X; e.g., A;B [subset.bar] X and A;B[member of] BCC (X): We shall use script letters to denote subsets of the corresponding hyperspaces, e.g., K [subset.bar] BCC (X); W [subset.bar] BCC (X): For A; B [member of] BCC (X); let A + B = a + b : a [member of] A; b [member of] B} ; N (A; [epsilon]) ={x [member of] X : d(x;a) = ||x - a|| < [epsilon] for some a [epsilon] A} and h(A;B) = inf{[epsilon] > 0 : A [subset.bar] N(B;[epsilon]);B [subset.bar] N (A; [epsilon])}, equivalently, [mathematical expression not reproducible]. The metric h just defined is known as the Hausdorff metric and (BCC (X); h) is known to be a complete metric space. Since h is induced by the ||*|| of the underlying space X; h is closely related to the norm (||*||) as well as x* [member of] X*: The following lemmas give some elementary properties of the Hausdorff h and its relationship with them.

Lemma 1. Suppose A; B;C; D [member of] WCC (X) and a [member of] C: Then we have

(i) h (A; {0}) = sup f||a|| : a [member of] A} ; (ii) h (A + B; C + D) [less than or equal to] h (A; C) + h (B; D); (iii) h ([alpha]A; [alpha]B) = |[alpha]| h (A; B); (iv) h([[a.sub.1]; [a.sub.2]]; [[b.sub.1]; [b.sub.2]]) = max {|[b.sub.1] [a.sub.1]|; |[b.sub.2] [a.sub.2]|} for[[a.sub.1]; [a.sub.2]] ; [[b.sub.1]; [b.sub.2]] [member of] (CC (R); h):

Lemma 2. Suppose A; B [member of] WCC (X) and x*;y* [member of] X*: Then

(i) x* (A); x* (B) [member of] (CC (C); h);

(ii) A = B if and only if x* (A) = x* (B) for each x* [member of] X*;

(iii) h (x* (A); x* (B)) [less than or equal to] ||x* || h (A; B);

(iv) h (x* (A); y* (A)) [less than or equal to] ||x* - y* || h (A; f0}):

Proof. Since x* : (X;[T.sub.w]) [right arrow] (C; ||*||) is continuous and linear, it follows that x* (A) ;x* (B) are compact, convex subsets of C and (i) is proved.

If A = B; then x* (A) = x* (B) for each x* [member of] X*: Suppose A [not equal to] B; without loss of generality, we may assume there exists some b0 [member of] B such that [b.sub.0] [??] A: It follows then from Hahn-Banach Theorem that there exists some x* [member of] X* which separates b0 from A; i:e:; there exists x* [member of] X* such that sup fRex* (a) : a [member of] A} < Re x* ([b.sub.0]) : That is a contradiction and (ii) is proved.

(iii) and (iv) follow from that

||x* (a) x* (b)|| = ||x* (a b)|| < ||x*|| * ||a b|| ; ||x* (a) y* (a)|| < ||x* y*|| * ||a||

and the definition of Hausdorff metric.

Now, it follows from Lemma 2 (i) that x* maps the space WCC (X) into the space CC (C) or x* : (WCC (X); h) [right arrow] (CC (C); h): Also, by Lemma 2 (iii) that x* : (WCC (X); h) [right arrow] (CC (C); h) is continuous. Note that both the domain and the range are now hyperspaces endowed with corresponding Hausdorff metric h: Now, recall that the weak topology [T.sub.w] on X is defined to be the weakest topology such that each x* : (X;[T.sub.w]) [right arrow] (C; |*|) is continuous. Analogously, we may define the weak topology on WCC (X) as follows:

Definition 1. The weak topology [T.sub.w] on WCC (X) is defined to be the weakest topology on WCC (X) such that each x* : (WCC (X) ;[T.sub.w]) [right arrow] (CC (C) ;h) is continuous. Thus a typical [T.sub.w]-neighborhood of A [member of] WCC (X) is denoted by W (A; [x*.sub.1]; ... ; [x*.sub.n]; [epsilon]) = {B [member of] WCC (X) ; h([x*.sub.i](B); [x*.sub.i](A)) < [epsilon] fori = 1; 2; ... ; n; [epsilon] > 0}:

As mentioned in the introduction, several results have been extended to the hyperspace WCC (X): In the next section, we shall further extend the notion of hyperspace and its corresponding topology [T.sub.w] where the underlying space X is a locally convex topological vector space instead of a Banach space and prove an extreme point theorem which is an extension of the classical Krein-Milman Theorem.

3. Main Results

In this section, X is assumed to be a locally convex topological vector space, X* its dual space and X is endowed with original topology T as well as weak topology [T.sub.w]: Let WCC (X) = {A [subset.bar] X : A is a non-empty weakly compact, convex subset of X}: Since each x* is also weakly continuous (i:e: x* : (X;[T.sub.w]) [right arrow] (C; |*|) is continuous and linear), it follows that for each A [member of] WCC (X); x* (A) is a compact convex subset of the complex plane C: Thus each x* is a mapping from the set WCC (X) into the metric space (CC (C); h): Define [T.sub.w] to be the weakest topology and WCC (X) such that each x* : (WCC (X) ;[T.sub.w]) [right arrow] (CC (C) ;h) is continuous. Denote B(x*(A);[epsilon]) = {B [member of] CC (C) : h(x*(A);B) < [epsilon]} for [epsilon] > 0:

Some basic properties of the weak topology [T.sub.w] are stated in the following lemma.

Lemma 3. (a) The collection fW (A; [x*.sub.1]; [x*.sub.2]; ... ; [x*.sub.n]; [epsilon]) : xi [member of] X* fori = 1; 2; ... ; n; [epsilon] > 0} is a local base at A [member of] WCC (X) where

[mathematical expression not reproducible]

(b) [T.sub.w] is a Hausdorff topology on WCC (X); i:e:; distinct A;B [member of] WCC (X) have disjoint neighborhoods containing them.

Proof. (a) Since x* : (WCC (X) ;[T.sub.w]) [right arrow] (CC (C); h) is continuous, (B([x*.sub.i](A);[epsilon]) is open in (CC (C); h); we have [([x*.sub.i]).sup.-1](B([x*.sub.i](A); [epsilon]) is open in (WCC (X); [T.sub.w]); i:e:; W(A;[x*.sub.i]; [epsilon]) = [([x*.sub.i]).sup.-1](B([x*.sub.i](A);[epsilon]) [member of] [T.sub.w] and W (A; [x*.sub.1];[x*.sub.2]; ... ;[x*.sub.n]; [epsilon]) being the finite intersection of open sets is also open.

(b) Let A;B[member of] WCC (X) with A [not equal to] B: We may assume without loss of generality that there exists some a [member of] A such that a [member of] B: Since B is a [T.sub.w]-compact subset of the locally convex topological vector space (X;[T.sub.w]) ; it follows from the Hahn-Banach Separation Theorem that there exists some x* [member of] X* such that sup Rex* (b) < Rex* (a): Let [delta] = Rex* (a) [mathematical expression not reproducible] (b) > 0: We have Rex* (a) Rex* (b) [greater than or equal to] [delta] for all b [member of] B which in turn implies that

(1) |x* (a) x* (b) | [greater than or equal to] |Re x* (a) Re x* (b)] [greater than or equal to] [delta]:

Suppose 0 < [epsilon] < [delta] is chosen, we have |x* (a) x* (b)\ > [epsilon] for all b [member of] B: Claim that W(A; x*; [epsilon]/2]) [intersection] W(B; x*; [epsilon]/2]) = [??]: Otherwise, there exists some D [member of] W(A; x*; [epsilon]/2])[intersection] W(B; x*; [epsilon]/2]) and we have h(x*(D); x*(A)) < [epsilon]/2]; h(x*(D); x*(B)) < [epsilon]/2]: Consequently x*(A) [subset] N(x*(D); [epsilon]/2]); x*(D) [subset] N(x*(B); [epsilon]/2]): Hence for the given a [member of] A; there exists some d [member of] D such that |x* (a) - x* (d)| < |; and for the d [member of] D; there exists some b [member of] B such that |x* (d) - x* (b)| < [epsilon]/2] which in turn implies that

|x* (a) x* (b)| [less than or equal to] |x* (a) x* (d)| + |x* (d) x* (b)| < [epsilon] < [delta]:

That is a contradiction to the inequality (1) and the proof is complete.

Suppose A;B [member of] WCC (X): Then A; B are weakly compact, convex subsets of X: Since addition and scalar multiplication are continuous operations on (X; [T.sub.w]) ; we have A + B; [alpha]A are weakly compact, convex subsets of X; i:e:; A + B; [alpha]A [member of] WCC (X): Thus we may define, algebraic line segments, convex sets, extremal subsets and extreme points on the hyperspace WCC (X) analogous to their counterparts on the underlying space X:

Definition 2. (a) [A;B] = {[alpha]A + (1 - [alpha])B : A; B [member of] WCC (X); 0 [less than or equal to] [alpha] [less than or equal to] 1} is called the closed line segment joining A and B:

(b) A subset K [subset] WCC (X) is said to be convex if and only if [A.sub.1]; [A.sub.2];... ; [A.sub.n] [member of]

[mathematical expression not reproducible]

(c) A mapping T : WCC (X) [right arrow] WCC (X) is said to be affine if and only if T([alpha]A + (1 - a)B) = [alpha]T(A) + (1 - [alpha])T(B) where 0 [less than or equal to] [alpha] [less than or equal to] 1:

(d) Suppose [K.sub.1];[K.sub.2] [subset] (WCC (X) ;[T.sub.w]) are closed ([T.sub.w]-closed); convex subsets. Then [K.sub.1] is said to be an extremal subset of [K.sub.2] if and only if A; B [member of] [K.sub.2] and [alpha]A + (1 - [alpha])B [member of] [K.sub.1] for some 0 < [alpha] < 1 implies that A; B [member of] [K.sub.1]:

(e) Suppose K is a [T.sub.w]-closed, convex subset of WCC (X): Then P is said to be an extreme point of K if and only if A; B [member of] K; 0 < a < 1; [alpha]A + (1 - [alpha])B = P implies A = B = P:

We state the following lemmas whose proofs are similar as in the underlying space X:

Lemma 4. Suppose K is a [T.sub.w]-closed, convex subset of the hyperspace (WCC (X); [T.sub.w]): Then

(a) If P [member of] K; then P is an extreme point of K if and only if {P} is an extremal subset of K:

(b) If [K.sub.1] [subset] [K.sub.2] [subset] [K.sub.3] are [T.sub.w]-closed, convex subsets, [K.sub.1] is an extremal subset of [K.sub.2]; and [K.sub.2] is an extremal subset of [K.sub.3]; then [K.sub.1] is an extremal subset of [K.sub.3]:

The next lemma is essential in the proof of our main theorem.

Lemma 5. (a) Suppose [[a.sub.1]; [a.sub.2]]; [[b.sub.1]; [b.sub.2]] [member of] (CC ([R.sup.1]) ; h) : Then h([[a.sub.1]; [a.sub.2]]; [[b.sub.1]; [b.sub.2]]) = max{|[b.sub.1] [a.sub.1]| ; |[b.sub.2] [a.sub.2]|}; and the mapping T : (CC [R.sup.1]) ; h) [right arrow] ([R.sup.1]; |*|) defined by T([[a.sub.1];[a.sub.2]]) = [a.sub.2] is a continuous (in fact, nonexpansive), affine mapping.

(b) Suppose [bar.X] = {[bar.x] = {x} : x [member of] X} [subset] WCC (X): Then the mapping T : X [right arrow] [bar.X] [subset] WCC (X) defined by Tx = x is an isomorphic homeomorphism of (X;[T.sub.w]) onto (X;[T.sub.w]):

Proof (a) That h([[a.sub.1]; [a.sub.2]] ; [[b.sub.1]; [b.sub.2]]) = max{|[b.sub.1] [a.sub.1]| ; |[b.sub.2] [a.sub.2]|} follows immediately from the definition of Hausdorff metric. Next, [[a.sub.1]; [a.sub.2]] + [[b.sub.1]; [b.sub.2]] = [[a.sub.1] + [b.sub.1]; [a.sub.2] + [b.sub.2]] ; [alpha] [[a.sub.1]; [a.sub.2]] = [[alpha][a.sub.1]; a[a.sub.2]] implies

T([alpha] [[a.sub.1]; [a.sub.2]] + (1 - [alpha]) [[b.sub.1]; [b.sub.2]]) = T([[alpha][a.sub.1]; [alpha][a.sub.2]] + [(1 - [alpha])[b.sub.1]; (1 - [alpha])[b.sub.2]]) = T([[alpha][a.sub.1] + (1 - [alpha])[b.sub.1]; [alpha][a.sub.2] + (1 - [alpha])[b.sub.2]]) = [alpha][a.sub.2] + (1 - [alpha])[b.sub.2] = [alpha]T([[a.sub.1]; [a.sub.2]]) + (1 - [alpha])T([[b.sub.1]; [b.sub.2]])

where 0 < [alpha] < 1: Consequently T is affine. Finally

|T([[a.sub.1]; [a.sub.2]]) T([[b.sub.1]; [b.sub.2]])| = |[a.sub.2] [b.sub.2]| [greater than or equal to] max{ [[a.sub.1] [b.sub.1]| ; |[a.sub.2] [b.sub.2]| = h([[a.sub.1] ; [a.sub.2]]; [[b.sub.1]; [b.sub.2]])

showing that T is nonexpansive.

(b) Obviously, T : X [right arrow] [bar.X] is one to one and onto. Also T(x + y) = x + y = {x + y} = {x} + {y} = [bar.x] + [bar.y] = T(x) + T(y) and T([alpha]x) = [bar.[alpha]x] = {[alpha]x} = [alpha]{x} = [alpha][bar.x] = [alpha]T(x) showing that T is linear.

y [member of] w(x; [x*.sub.1]; ... ; [x*.sub.n]; [epsilon]) implies that |x*(y) x*(x)| < [epsilon] for i = 1; 2; ... ;n; which in turn implies that h(x*([bar.y]); x*([bar.x])) = h(x*(Ty);[x*.sub.i](Tx) < [epsilon] for i = 1; 2; ... ; n: Hence Ty = [bar.y] [member of] W ([bar.x]; [x*.sub.1]; ... ; [x*.sub.n]; [epsilon]): Similarly [bar.y] [member of] W ([bar.x]; [x*.sub.1]; ... ; [x*.sub.n]; [epsilon]) implies y [member of] w (x; [x*.sub.1]; ... ; [x*.sub.n]; [epsilon]): Consequently T(w (x; [x*.sub.1]; ... ; [x*.sub.n]; [epsilon])) = W ([bar.x] = Tx; [x*.sub.1]; ... ; [x*.sub.n]; [epsilon]) and the proof is complete.

Our main theorem is an extension of the classical Krein-Milman's Extreme Point Theorem to the hyperspace (WCC (X) ;[T.sub.w]):

Theorem 1. Suppose X is a locally convex topological vector space equipped with original topology T as well as weak topology [T.sub.w]; and (WCC (X) ;[T.sub.w]) is the corresponding hyperspace. Suppose K is a [T.sub.w]-compact, convex subset of (WCC (X); [T.sub.w]): Then K has an extreme point in K:

Proof. Let [OHM] denote the collection of all non-empty, [T.sub.w]-closed, convex subsets of K. [OHM] [not equal to] [??] since K [member of] [OHM]: Define a partial order in [OHM] by inverse inclusion, i:e:; [K.sub.2] [less than or equal to] [K.sub.1] if and only if [K.sub.1] [subset] [K.sub.2]: If {[K.sub.i]}i[member of]I [subset] [subset] is a totally ordered subset, we shall show that [mathematical expression not reproducible] is an upper bound of [{[K.sub.i]}.sub.i[member of]I]: Each [K.sub.i] is [T.sub.w]-compact, convex and [{[K.sub.i]}.sub.i[member of]I] has finite intersection implies that [K.sub.0] is a non-empty [T.sub.w]-compact, convex set. Suppose we have A;B [member of] K;0 < [alpha] < 1 and [alpha]A + (1 - [alpha])B [member of] [K.sub.0]: Since [K.sub.0] [subset] Ki for each i; we have [alpha]A + (1 - [alpha])B [member of] [K.sub.i] which in turn implies that A; B [member of] [K.sub.i] because [K.sub.i] is an extremal subset of K: Thus A; B [member of] [K.sub.0] showing that [K.sub.0] is an extremal subset of K and consequently [K.sub.0] is an upper bound of [{[K.sub.i]}.sub.i[member of]I]. It follows now from Zorn's Lemma that Q has a maximal element, denoted by [K.sub.[infinity]]: We claim that [K.sub.[infinity]] is a singleton. Otherwise, there exists [A.sub.0];[B.sub.0] [member of] [K.sub.[infinity]] with [A.sub.0] [not equal to] [B.sub.0] ; without loss of generality, assume there exists some [b.sub.0] [member of] [B.sub.0] such that [b.sub.0] [??] [A.sub.0]: By Hahn-Banach Separation Theorem, there exists some x* [member of] X* such that [mathematical expression not reproducible] Rex* (a) < Rex* ([b.sub.0]) : Let Rex* ([A.sub.0]) = [[a.sub.1]; [a.sub.2]]; Rex* ([B.sub.0]) = [[b.sub.1]; [b.sub.2]] [member of] CC(R); we have [a.sub.2] = sup Rex* (a) < Rex* ([b.sub.0]) [less than or equal to] [b.sub.2]: Define G : (CC(R); h) [right arrow] (R; |*|) by G([[a.sub.1];[a.sub.2]]) = [a.sub.2]: It follows from Lemma 5 (a) that G is a nonexpansive (hence continuous) affine mapping.

Next, let F : (WCC(X);[T.sub.w]) [right arrow] (R; |*|) be defined by F(A) = G(Rex* (A)): F : ([K.sub.[infinity]]; [T.sub.w]) [right arrow] (R; |*|) is continuous implies F attains its maximum on [K.sub.[infinity]]; i:e:; there exists [b.sub.[infinity]] [member of] R and [B.sub.[infinity]] [member of] [K.sub.[infinity]] such that F([B.sub.[infinity]]) = [b.sub.[infinity]] = [mathematical expression not reproducible] = [mathematical expression not reproducible]. Since [a.sub.2] < [b.sub.2] < [b.sub.[infinity]]; [A.sub.0] [??] [F.sup.1]([b.sub.[infinity]]): Claim that [F.sup.1]([b.sub.[infinity]]) is an extremal subset of [K.sub.[infinity]]: For that purpose, we let D; E [member of] [K.sub.[infinity]], 0 < [alpha] < 1 with [alpha]D+(1 - [alpha])E [member of] [F.sup.-1]([b.sub.[infinity]]): Thus F([alpha]D + (1 - [alpha])E) = [b.sub.[infinity]] which implies that [alpha]F(D) + (1 - [alpha])F(E) = [b.sub.[infinity]]: Also D;E [member of] [K.sub.[infinity]] implies that F(D); F(E) [less than or equal to] [b.sub.[infinity]] and consequently, [alpha]F(D) + (1 - [alpha])F(E) [less than or equal to] [b.sub.[infinity]]: Hence F(D);F(E) = [b.sub.[infinity]] that implies D;E [member of] [F.sup.-1]([b.sub.[infinity]]): Otherwise, we would have [alpha]F(D) + (1 - [alpha])F(E) < [b.sub.[infinity]]; contradicting that [alpha]F(D) + (1 - [alpha])F(E) = [b.sub.[infinity]]: Now that [F.sup.-1]([b.sub.[infinity]]) [subset] [K.sub.[infinity]]; and [F.sup.-1]([b.sub.[infinity]]) is an extremal subset of [K.sub.[infinity]] implies [F.sup.-1]([b.sub.[infinity]]) [subset] [K.sub.[infinity]]: But [A.sub.0] [??] [F.sup.-1]([b.sub.[infinity]]) implies [F.sup.-1]([b.sub.[infinity]]) & [K.sub.[infinity]] contradicting that [K.sub.[infinity]] is a maximal element. Hence [K.sub.[infinity]] is a singleton, say [K.sub.[infinity]] = {P} proving that P is an extreme point of K and the proof is complete.

The following corollary is the classical Krein-Milman extreme point theorem.

Corollary 1. Let K be a non-empty compact, convex subset of a locally convex topological vector space X: Then K has an extreme point in K:

Proof. Let X a locally convex topological vector space endowed with original topology T as well as weak topology [T.sub.w]; and (WCC (X) ;[T.sub.w]) is the corresponding hyperspace. K is r-compact implies K is [r.sub.w]-compact. It follows from Lemma 5 (b) that [bar.K] = {[bar.x] = {x} : x [member of] K} is a [T.sub.w]-compact, convex subset of (WCC (X) ;[T.sub.w]) and hence has an extreme point [bar.P] = {p} by Theorem 1. Consequently p is an extreme point of K and the proof is complete.

Remark 1. (a) Since Krein-Milman Theorem has numerous important applications in various branches of mathematics, we hope further investigation on the hyperspace (WCC (X) ;[T.sub.w]) will lead to some useful applications.

(b) The study of convex sets has always been interesting and useful. However, the traditional method has relied heavily on support fucntionals. With the [T.sub.w]-topology defined on WCC (X); we hope it will provide an altermative way to study convex sets.

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JENNIFER SHUEH-INN HU (*)

DEPARTMENT OF APPLIED STATISTICS AND INFORMATION SCIENCE, MING CHUAN UNIVERSITY, GUISHAN, TAOYUAN, TAIWAN

KUEI-LIN TSENG

DEPARTMENT OF APPLIED MATHEMATICS, ALETHEIA UNIVERSITY, TAMSUI, NEW TAIPEI CITY 25103, TAIWAN

Author: Printer friendly Cite/link Email Feedback Hu, Jennifer Shueh-Inn; Tseng, Kuei-Lin Tamsui Oxford Journal of Information and Mathematical Sciences Report 1USA May 1, 2017 4390 Characterization on N(k)-Mixed Quasi-Einstein Manifold. INNERNESS [sigma]-DERIVATIONS AND INNERNESS [sigma]-HIGHER DERIVATIONS. Topological spaces