VARIOUS EXAMPLES OF THE KKM SPACES

Abstract

Recall that a partial KKM space is a topological space satisfying an abstract form of the well-known KKM theorem, and a KKM space is a partial KKM space satisfying the 'open-valued' version of the form. Recently, we have found several new examples of such spaces. As a continuation of the preceding two works, we are going to introduce old and new examples of the KKM spaces, which play a major role in applications of the KKM theory. Finally, we state why we should use triples (X, D; Γ) for abstract convex spaces by giving further examples, and introduce an example of a partial KKM space which is not a KKM space.

Full text

VARIOUS EXAMPLES OF THE KKM SPACES
SEHIE PARK
Abstract. Recall that a partial KKM space is a topological space satisfying an
abstract form of the well-known KKM theorem, and a KKM space is a partial
KKM space satisfying the ‘open-valued’ version of the form. Recently, we have
found several new examples of such spaces. As a continuation of the preceding
two works, we are going to introduce old and new examples of the KKM spaces,
which play a major role in applications of the KKM theory. Finally, we state
why we should use triples (X, D; Γ) for abstract convex spaces by giving further
examples, and introduce an example of a partial KKM space which is not a KKM
space.
1. Introduction
In order to upgrade the KKM theory, in 2006-09, we proposed new concepts of
abstract convex spaces and the (partial) KKM spaces which are proper general-
izations of the well-known G-convex spaces and adequate to establish the KKM
theory.
In our previous works [36, 42, 44, 45], we studied elements or foundations of the
KKM theory on abstract convex spaces and noticed there that many important
results therein are related to KKM spaces.
In our recent work [51], we introduced a logical origin of the Brouwer fixed point
theorem, Sperner’s combinatorial lemma, and the Knaster-Kuratowski-Mazurkiewicz
(KKM) theorem being Ky Fan’s 1952 lemma. Also we noted that these three theo-
rems are mutually equivalent and have nearly one hundred equivalent formulations
and several thousand applications.
Moreover, in a consequent work [52], we introduced our multimap classes in the
frame of the KKM theory; that is, the admissible multimap class Aκ
c, the better
admissible class B, and the KKM admissible classes KC,KO. There we collected
the basic properties of such multimap classes and some mutual relations among
them in general topological spaces or our abstract convex spaces.
Recall that a partial KKM space is a topological space satisfying an abstract form
of the KKM theorem, and a KKM space is the one also satisfying the ‘open-valued’
version of the form. Recently, we have found several new examples of KKM spaces.
As a continuation of the preceding two works, we are going to show old and new
examples of the KKM spaces, which play the major role in applications of the KKM
theory. Some historical remarks are also added.
2010 Mathematics Subject Classification. 47H10, 49J53, 54C60, 54H25, 90A14, 90C76, 91A13,
91A10.
Key words and phrases. Convex space, H-space, G-convex space, φA-space, KKM-space, hyper-
convex space, R-tree, Horvath’s convex space, B-space, extended long line L∗.
1

2 SEHIE PARK
This article is organized as follows: Section 2 deals with minimum amount of
preliminaries on our abstract convex spaces including the KKM spaces. In Section
3, we introduce a metatheorem on (partial) KKM spaces. This metatheorem is
concerned with the common properties of all partial KKM spaces. Section 4 deals
with known examples of classical KKM spaces including convex spaces, H-spaces,
and G-convex spaces. In Section 5, we give basic theory of φA-spaces as a sample of
the KKM spaces. Section 6 concerns with relatively new examples of KKM spaces
mainly founded by ourselves. Finally, in Section 7, we state why we should use
triples (X, D; Γ) for abstract convex spaces instead of pairs (X;Γ) by giving further
examples, and introduce an example of a partial KKM space due to Kulpa and
Szymanski in 2014 which is not a KKM space.
2. Abstract convex spaces
For sets Xand Y, a multimap (or a multifunction or simply a map) F:X(Y
is a function F:X→2Yto the power set of Y.
For the concepts on our abstract convex spaces, KKM spaces and the KKM
admissible classes KC,KO, we follow [45, 46] with some modifications and the
references therein:
Definition 2.1. Let Ebe a topological space, Da nonempty set, hDithe set of all
nonempty finite subsets of D, and Γ : hDi(Ea multimap with nonempty values
ΓA:= Γ(A) for A∈ hDi. The triple (E , D; Γ) is called an abstract convex space
whenever the Γ-convex hull of any D0⊂Dis denoted and defined by
coΓD0:= [{ΓA|A∈ hD0i} ⊂ E.
A subset Xof Eis called a Γ-convex subset of (E, D; Γ) relative to some D0⊂D
if for any N∈ hD0i, we have ΓN⊂X, that is, coΓD0⊂X.
When D⊂E, a subset Xof Eis said to be Γ-convex if coΓ(X∩D)⊂X; in other
words, Xis Γ-convex relative to D0:= X∩D. In case E=D, let (E; Γ) := (E , E; Γ).
Definition 2.2. Let (E, D; Γ) be an abstract convex space and Za topological
space. For a multimap F:E(Zwith nonempty values, if a multimap G:D(Z
satisfies
F(ΓA)⊂G(A) := [
y∈A
G(y) for all A∈ hDi,
then Gis called a KKM map with respect to F. A KKM map G:D(Eis a KKM
map with respect to the identity map 1Eof E.
Definition 2.3. A multimap F:E(Zis called a KC-map [resp. a KO-map] if,
for any closed-valued [resp. open-valued] KKM map G:D(Zwith respect to F,
the family {G(y)}y∈Dhas the finite intersection property. In this case, we denote
F∈KC(E, Z ) [resp. F∈KO(E, Z )].
Definition 2.4. The partial KKM principle for an abstract convex space (E, D; Γ) is
the statement 1E∈KC(E, E); that is, for any closed-valued KKM map G:D(E,
the family {G(y)}y∈Dhas the finite intersection property. The KKM principle is
the statement 1E∈KC(E, E)∩KO(E, E); that is, the same property also holds for
any open-valued KKM map.

VARIOUS EXAMPLES OF THE KKM SPACES 3
An abstract convex space is called a (partial)KKM space if it satisfies the (par-
tial) KKM principle, resp.
Here we give a system of abstract convex spaces (E, D; Γ) :
Simplex =⇒Convex subset of a t.v.s. =⇒Lassonde type convex space
=⇒H-space =⇒G-convex space =⇒φA-space =⇒KKM space
=⇒Partial KKM space =⇒Abstract convex space.
Later the foundations or elements of the KKM theory on abstract convex spaces
were studied in [36, 41, 42, 44, 45, 47], where it was shown that many important
results hold for partial KKM spaces.
For an abstract convex space (E⊃D; Γ) , an extended real-valued function f:
E→Ris said to be quasiconcave [resp. quasiconvex] whenever {x∈E|f(x)> r}
[resp. {x∈E|f(x)< r}] is Γ-convex for any r∈R.
3. A Metatheorem on KKM spaces
In the KKM theory, it is routine to reformulate the KKM principle to the follow-
ing equivalent forms [44, 45]:
Fan type matching property
Another intersection property
Geometric or section properties
Fan-Browder type fixed point property
Existence theorem of maximal elements
Analytic formulations, analytic alternatives
Minimax inequality, and others
Any of these statements characterizes KKM spaces, and any of closed versions of
them characterizes partial KKM spaces; see [45].
For example, the Fan-Browder type fixed point theorem is used for the following:
Theorem 3.1. An abstract convex space (X, D; Γ) is a KKM space if and only if
for any maps S:D(X, T :X(Xsatisfying
(1) S(z)is open [resp. closed ]for each z∈D;
(2) for each y∈X, coΓS−(y)⊂T−(y); and
(3) X=Sz∈MS(z)for some M∈ hDi,
Thas a fixed point x0∈X; that is x0∈T(x0).
Moreover, from the partial KKM principle we have a whole intersection property
of the Fan type. From this, we can deduce the following:
Theorem 3.2. Let (X, D; Γ) be a partial KKM space, Ka nonempty compact subset
of X, and G:D(Xa map such that
(1) Tz∈DG(z) = Tz∈DG(z) [that is, Gis transfer closed-valued ];
(2) Gis a KKM map; and
(3) either
(i) T{G(z)|z∈M} ⊂ Kfor some M∈ hDi; or

4 SEHIE PARK
(ii) for each N∈ hDi, there exists a compact Γ-convex subset LNof X
relative to some D0⊂Dsuch that N⊂D0and
LN∩\{G(z)|z∈D0} ⊂ K.
Then K∩T{G(z)|z∈D} 6=∅.
From this theorem we can deduce its equivalent formulations of the following
forms for partial KKM spaces [41, 45]:
Theorems of Sperner and Alexandroff-Pasynkoff
Fan type matching theorem
Tarafdar type intersection theorem
Geometric or section properties
Fan-Browder type fixed point theorems
Maximal element theorems
Analytic alternatives
Fan type minimax inequalities
Variational inequalities
Horvath type fixed point theorem
Browder type coincidence theorem
von Neumann type minimax theorem
Nash type equilibrium theorem
Analytic alternatives (a basis of various equilibrium problems)
Fan type minimax inequalities
Variational inequalities, and others
Further applications of our theory on partial KKM spaces are given as follows
[45, 46]:
Best approximations (under certain restrictions)
von Neumann type intersection theorem
Nash type equilibrium theorem
Himmelberg fixed point theorem for KKM spaces
Weakly KKM maps [37]
Consequently, we have the following as is suggested in [44]:
Metatheorem. For any partial KKM space, all theorems mentioned in this section
hold.
4. Known subclasses of KKM spaces
4.1. Convex spaces
Definition 4.1. Let Xbe a subset of a vector space and Da nonempty subset of
X. We call (X, D) a convex space if co D⊂Xand Xhas a topology that induces
the Euclidean topology on the convex hulls of any N∈ hDi; see Park [26]. Note
that (X, D) can be represented by (X, D; Γ) where Γ : hDi → Xis the convex hull
operator.
If X=Dis convex, then X= (X, X) becomes a convex space in the sense of
Lassonde [23].

VARIOUS EXAMPLES OF THE KKM SPACES 5
Example 4.2. (1) The original KKM theorem in 1929 is for a triple (∆n⊃V; co),
where Vis the set of vertices of an n-simplex ∆nand co : hVi(∆nis the convex
hull operator. This triple can be regarded as (∆n, N ; Γ), where N:= {0,1, . . . , n}
and ΓA:= co{ei|i∈A}for each A⊂N.
(2) Fan’s celebrated KKM lemma in 1961 is for (E⊃D; co), where Dis a
nonempty subset of a t.v.s. E. He assumed the superfluous Hausdorffness of E.
The above examples are origins of our G-convex spaces (X, D; Γ). It should be
noted that many authors’ KKM type theorems for a pair (X; Γ) can not generalize
the original KKM theorem or Fan’s KKM lemma.
(3) A convexity space (E, C) in the classical sense consists of a non-empty set E
and a family Cof subsets of Esuch that Eitself belongs to Cand any intersection
of a subfamily of Calso belongs to C. See Sortan in 1984; where 283 references
appear. The C-convex hull of any subset X⊂Eis denoted and defined by CoCX:=
T{Y∈ C | X⊂Y}.Xis said to be C-convex whenever X=CoCX. If we define
a multifunction Γ : hEi(Eby ΓA:=CoCAfor each A∈ hEi, then (E , C) is an
abstract convex space (E;Γ) with any topology on E.
(4) Every nonempty convex subset Xof a topological vector space is a convex
space with respect to any nonempty subset Dof X, and the converse is known to
be not true.
4.2. H-spaces
Definition 4.3. A triple (X, D; Γ) is called an H-space by Park [27] if Xis a
topological space, Da nonempty subset of X, and Γ = {ΓA}a family of contractible
(or, more generally, ω-connected) subsets of Xindexed by A∈ hDisuch that
ΓA⊂ΓBwhenever A⊂B∈ hDi.
If D=X, we denote (X; Γ) instead of (X, X ;Γ), which is called a c-space by
Horvath [11] or an H-space by Bardaro and Ceppitelli [1].
Example 4.4. Any convex space Xdue to Lassonde is an H-space (X; Γ) by putting
ΓA= co A, the convex hull of A∈ hXi. Similarly, our convex spaces (X, D; Γ)
becomes H-spaces. Other examples of (X; Γ) are any pseudo-convex space [10],
any homeomorphic image of a convex space, any contractible space, and so on; see
Bardaro and Ceppitelli [1] and Horvath [11].
Horvath noted that a torus, the M¨obius band, or the Klein bottle can be regarded
as c-spaces, and are examples of compact H-spaces without having the fixed point
property. Every n-simplex ∆nis an H-space (∆n, D; Γ), where Dis the set of
vertices and ΓA=co Afor A∈ hDi.
4.3. G-convex spaces
Definition 4.5. Ageneralized convex space or a G-convex space (X, D; Γ) consists
of a topological space X, a nonempty set D, and a map Γ : hDi(Xsuch that for
each A∈ hDiwith the cardinality |A|=n+ 1, there exists a continuous function
φA: ∆n→Γ(A) such that J∈ hAiimplies φA(∆J)⊂Γ(J).
Here, ∆n= co{ei}n
i=0 is the standard n-simplex, and ∆Jthe face of ∆ncorre-
sponding to J∈ hAi; that is, if A={a0, a1,· · · , an}and J={ai0, ai1,· · · , aik} ⊂ A,

6 SEHIE PARK
then ∆J= co{ei0, ei1,· · · , eik}. We may write ΓA= Γ(A) for each A∈ hDiand
(X, Γ) = (X, X ; Γ).
There is a lot of examples of G-convex spaces; see [54, 55].
Example 4.6. (1) If X=Dis a convex subset of a vector space and each ΓAis the
convex hull of A∈ hXiequipped with the Euclidean topology, then (X, Γ) becomes
a convex space due to Lassonde [23].
(2) If X=Dand ΓAis assumed to be contractible or, more generally, infinitely
connected (that is, n-connected for all n≥0) and if for each A, B ∈ hXi,A⊂B
implies ΓA⊂ΓB, than (X, Γ) becomes a c-space (or an H-space) due to Horvath
[11].
(3) For other major examples of G-convex spaces are metric spaces with Michael’s
convex structure, Pasicki’s S-contractible spaces, Horvath’s pseudoconvex spaces,
Komiya’s convex spaces, Bielawski’s simplicial convexities, Joo’s pseudoconvex spaces,
and so on. For the literature, see Park and Kim [54, 55]. Later, we found a num-
ber of new examples of G-convex spaces. Especially, any continuous image of a
G-convex space is a G-convex space; and any almost convex subset of a t.v.s. (see
Himmelberg [9]) is a G-convex space.
(4) Later examples of G-convex spaces were given in [31] as follows: L-spaces
and B’-simplicial convexity of Ben-El-Mechaiekh et al. [3], Verma’s or Stach´o’s
generalized H-spaces, Kulpa’s simplicial structures, P1,1-spaces of Forgo and Jo´o,
mc-spaces of Llinares, hyperconvex metric spaces due to Aronszajn and Panitch-
pakdi, and Takahashi’s convexity in metric spaces.
Remark 4.7. (1) G-convex spaces are actually same to the so-called φA-spaces in
the next section, and these are all KKM spaces.
(2) Most of the families of G-convex spaces mentioned above have some concrete
examples. Some of relatively new ones will be shown in Section 6.
(3) In our previous work [45], we introduced basic results in the KKM theory of
abstract convex spaces and KKM maps. Such results are applied to several variants
of the concepts of G-convex spaces and KKM type maps. We studied the nature
of such variants and criticized other authors’ later ‘generalizations’ of our previous
results.
5. φA-spaces
Since the appearance of G-convex spaces in 1993, many authors have tried to
imitate, modify, or generalize the concept and published a large number of papers.
In fact, there have appeared authors who introduced spaces of the form (X, {ϕA})
having a family {ϕA}of continuous functions defined on simplices. Such example
are L-spaces due to Ben-El-Mechaiekh et al., spaces having property (H) due to
Huang, FC-spaces due to Ding, convexity structures satisfying the H-condition by
Xiang et al., M-spaces and L-spaces due to Gonz´alez et al., and others. Some
authors claimed that such spaces generalize G-convex spaces without giving any
justifications or proper examples. Some authors also tried to generalize the KKM
principle for their own settings. They introduced various types of generalized KKM

VARIOUS EXAMPLES OF THE KKM SPACES 7
maps; for example, generalized KKM maps on L-spaces, generalized R-KKM maps,
and many other artificial terminology. Some of them tried to rewrite certain results
on G-convex spaces by simply replacing Γ(A) by ϕA(∆n) everywhere and claimed
to obtain generalizations without giving any justifications or proper examples. In
2007, we found that most of such spaces are subsumed in the concept of φA-spaces
(X, D;{φA}A∈hDi); see Park [34, 35, 39]. Since then, the spaces became one of the
main theme of the KKM theory; see [34, 35, 37, 39, 40, 43].
Definition 5.1. Aspace having a family {φA}A∈hDior simply a φA-space
(X, D;{φA}A∈hDi) or (X, D;φA)
consists of a topological space X, a nonempty set D, and a family of continuous
functions φA: ∆n→X(that is, singular n-simplices) for A∈ hDiwith the cardi-
nality |A|=n+ 1.
For a φA-space (X, D;{φA}A∈hDi), a subset Cof Xis said to be φA-convex with
respect to a subset D0⊂Dif for each B∈ hD0i, we have Im φB:= φB(∆|B|−1)⊂C.
By putting ΓA:= φA(∆n), any φA-space becomes an abstract convex space, and
we will show that it is a KKM space later.
Note that, when X=D, a φA-space is called an FC-space by Ding [7] or a
simplicial space by Kulpa and Szymanski [21]. Later, a φA-space is called a GFC-
space by Khanh et al. [17, 18].
We collect some known facts on φA-spaces as follows in [33, 38, 42, 46, 47, 52]:
Definition 5.2. For a φA-space (X, D;φA), a KKM map T:D(Xis the one
satisfying
φA(∆J)⊂T(J) for each A∈ hDiand J∈ hAi.
This definition contains many particular cases previously appeared.
Proposition 5.3. A KKM map T:D(Xon a φA-space (X, D;φA)is a KKM
map on the corresponding abstract convex space (X, D; Γ) with ΓA:= φA(∆n)for
all A∈ hDiwith |A|=n+ 1.
Proposition 5.4. A KKM map T:D(Xon a φA-space (X, D;φA)is a KKM
map on a new abstract convex space (X, D; ΓT).
The following is a KKM type theorem for φA-spaces, and it can be proved by
following that for G-convex spaces:
Theorem 5.5. For a φA-space (X, D;φA), let G:D(Xbe a KKM map with
closed values. Then {G(z)}z∈Dhas the finite intersection property. (More precisely,
for each A∈ hDiwith |A|=n+ 1, we have φA(∆n)∩Tz∈AG(z)6=∅.)Further, if
\
z∈M
G(z)is compact for some M∈ hDi
then we have Tz∈DG(z)6=∅.
Theorem 5.6. For a φA-space (X, D;φA), let G:D(Xbe a KKM map with
open values. Then {G(z)}z∈Dhas the finite intersection property.

8 SEHIE PARK
In [37], we applied basic results in the KKM theory on abstract convex spaces
and KKM maps to the class of φA-spaces. These spaces unified G-convex spaces
and many similar variants. Consequently, fundamental theorems on φA-spaces were
used to correct or improve results on the so-called R-KKM maps or L-convex spaces
due to some particular authors.
Example 5.7. There are many examples of φA-spaces; see [34, 35, 36, 39, 40, 43,
47]. The following are some of them:
(1) Since an L-space of Ben-El-Mechaiekh et al. [3] is a G-convex space (X; Γ),
it is a φA-space.
(2) A topological space Yis said to have the property (H) whenever a continuous
function ϕN: ∆n→Yexists for each N={y0, . . . , yn} ∈ hYi.
(3) A pair (Y, {ϕN}) is called an FC-space whenever Yis a topological space and
a continuous function ϕN: ∆n→Yexists for each N={y0, . . . , yn} ∈ hYi(where,
some elements of Nmay be the same).
(4) Similarly, a ψA-space (X, D;{ψA}A∈hDi) is such that, for each A∈ hDiwith
its cardinality |A|=n+ 1, the function ψA: [0,1]n→Xis continuous. Such type
of spaces were treated by Michael [25] and Llinares [24]; see also [31].
For each n≥0, consider a continuous function gn: ∆n→[0,1]ndefined by
gn:u=
n
X
i=0
λi(u)ei7→ (λ0(u), . . . , λn−1(u))
for each u∈∆n. Moreover, by putting φA:= ψA◦gn, a ψA-space becomes a
φA-space.
(5) In a t.v.s. E, consider a neighborhood system Vof the origin O. Let Y
be an almost convex dense subset of a subset Dof E. For any V∈ V and each
A:= {x0, x1, . . . , xn}∈hDi, a subset B:= {y0, y1, . . . , yn}∈hYiis determined
such that yi−xi∈Vfor each i= 0,1, . . . , n and co B⊂Y. Define a continuous
function φA: ∆n→co Bby
φA:u=
n
X
i=0
λi(u)ei7→ φA(u) :=
n
X
i=0
λi(u)yi
for each u∈∆n. Then (Y, D;{φA}A∈hDi) is a φA-space, and can be made into a
G-convex space. Note that Y⊂D.
(6) Any G-convex space is a φA-space, and its converse holds:
Proposition 5.8. AφA-space (X, D;{φA}A∈hDi)can be made into a G-convex
space (X, D; Γ).
Consequently, G-convex spaces and φA-spaces are essentially same.
Theorem 5.9. (i) A KKM map G:D(Xon a G-convex space (X, D; Γ) is a
KKM map on the corresponding φA-space (X, D;{φA}A∈hDi).
(ii) A KKM map T:D(Xon a φA-space (X, D;{φA}A∈hDi)is a KKM map
on a new G-convex space (X, D; Γ).

VARIOUS EXAMPLES OF THE KKM SPACES 9
The following is a KKM theorem for φA-spaces that can be proved by simply
modifying the corresponding ones in [30, 32, 48]:
Theorem 5.10. For a φA-space (X, D;{φA}A∈hDi), let G:D(Xbe a KKM map
with closed [resp. open]values. Then {G(z)}z∈Dhas the finite intersection property.
(More precisely, for each N∈ hDiwith |N|=n+1, we have φN(∆n)∩Tz∈NG(z)6=
∅.)
Further, if
(1) Tz∈MG(z)is compact for some M∈ hDi,
then we have Tz∈DG(z)6=∅.
Theorem 5.10 means that φA-spaces are KKM spaces.
Remark 5.11. (1) Let X= ∆n,Dbe the set of vertices of ∆n, and Γ := co be the
convex hull operator, then Theorem 5.10 reduces to the original KKM theorem in
1929 and its ‘open-valued’ version.
(2) In case Dis a nonempty subset of a (not necessarily Hausdorff) t.v.s. X,
Theorem 5.10 extends Fan’s celebrated KKM lemma in 1961.
(3) Any KKM type theorem for a (X, {ϕA}) type space can extend neither the
original KKM theorem nor Fan’s KKM lemma.
Here we give two KKM type theorems which are improved versions of correspond-
ing ones in previous literature:
Theorem 5.12. Let Xbe a topological space, Da nonempty set, and G:D(X
a map such that
(1) Gis transfer closed-valued [that is, Tz∈DG(z) = Tz∈DG(z)];
(2) there exists z∗∈Dwith G(z∗)compact.
Then, there exists a G-convex space (X, D; Γ) such that Gis a KKM map if and
only if Tz∈DG(z)6=∅.
Theorem 5.13. For a φA-space (Y, D;{φN}N∈hDi), let T:D→2Ybe a map such
that T(z)is nonempty and closed for each z∈D.
(i) If Tis a KKM map, then for each N∈ hDiwith |N|=n+ 1,
φN(∆n)∩\
x∈N
T(x)6=∅.
(ii) If the family {T(z) : z∈D}has the finite intersection property, then Tis a
KKM map.
6. Old and new examples of KKM spaces
A large number of examples of G-convex spaces were already given in Section
4. In this section, we introduce some new important examples of KKM spaces
chronologically, most of them are due to ourselves. For details of some of them, see
[54].
(1) Hyperconvex metric spaces — In a metric space (M, d), for a point x∈M
and any t > 0, let the closed ball be
B(x, t) := {y∈M|d(x, y)≤t}.

10 SEHIE PARK
In 1956, Aronszajn and Panitchpakdi defined as follows:
Definition 6.1. A metric space (H, d) is called a hyperconvex metric space when-
ever, for any arbitrary collection {B(xα, rα)}of closed balls of H,d(xα, xβ)≤rα+rβ
implies TαB(xα, rα)6=∅.
A normed vector space Xis not hyperconvex in general, and spaces (Rn,|| · ||∞),
l∞,L∞, and R-tree is hyperbolic.
Horvath [12] showed the following:
Lemma 6.2. Any hyperconvex metric space H is a c-space (H; Γ), where Γ(A)is
the intersection of all closed balls in H containing A∈ hHi.
Therefore, we have the following by our KKM theory:
Theorem 6.3. Every hyperconvex metric space is a KKM metric space, that is, a
metric space satisfying the KKM principle.
(2) Hyperbolic metric spaces — Kirk [19] in 1982 first considered a wide
class of spaces including convex metric spaces of ‘hyperbolic’ type, and later Reich
and Shafrir [57] introduced hyperbolic metric spaces, which is a particular c-space.
Definition 6.4. ([57]) Let (X, ρ) be a metric space and Rthe real line. We say
that a map c:R→Xis a metric embedding of Rinto Xif
ρ(c(s), c(t)) = |s−t|
for all real sand t. The image of a metric embedding is called a metric line. The
image of a real interval [a, b]:= {t∈R|a≤t≤b}under such a map is called a
metric segment.
Assume that (X, ρ) contains a family Mof metric lines, such that for each pair
of distinct points xand yin Xthere is a unique metric line in Mwhich passes
through xand y. This metric line determines a unique metric segment denoted by
[x, y] joining xand y. For each 0 ≤t≤1 there is a unique point zin [x, y] such
that
ρ(x, z) = tρ(x, y) and ρ(z, y) = (1 −t)ρ(x, y).
This point zis denoted by (1 −t)x⊕ty.
Definition 6.5. ([57]) We say that X, or more precisely (X, ρ, M), is a hyperbolic
metric space if
ρ(1
2x⊕1
2y, 1
2x⊕1
2z)≤1
2ρ(y, z)
for all x, y and zin X.
Examples of such metric spaces are all normed vector spaces, all Hadamard man-
ifolds, and Hilbert balls having hyperbolic distance. Arbitrary cartesian product of
hyperbolic spaces is also hyperbolic.
(3) Transfer FS convex map — Tian [59] defined the following:

VARIOUS EXAMPLES OF THE KKM SPACES 11
Definition 6.6. ([59]) Let Xbe a convex subset of a Hausdorff t.v.s. and Y⊂X
be nonempty. A multimap F:Y(Xis said to be transfer FS convex whenever,
for any {y1, . . . , yn} ⊂ Y, corresponding {x1, . . . , xn} ⊂ Xexists such that, for any
J⊂ {1, . . . , n}, the following holds:
co{xj|j∈J} ⊂ [
j∈J
F(yj).
Here by putting Γ{yj|j∈J}:= co{xj|j∈J}, (X, Y ; Γ) becomes an abstract
convex space, and a transfer FS map becomes a KKM map. So our KKM theory is
applicable.
Transfer FS convexity was obtained already by Chang and Zhang [6] in 1991 and
extended later by Park and Lee [56] in 2001.
(4) Topological semilattices — Horvath and Llinares-Ciscar [14] show that
any topological semilattice (X, ≤) having path-connected intervals satisfies an order
theoretic variant of the KKM principle. We showed that such topological semilat-
tices become G-convex spaces; see Park [28].
Asemilattice or, more exactly, a sup-semilattice, is a partially ordered set (X, ≤)
for which any pair (x, x0) of elements has a lub x∨x0. Any A∈ hXihas a lub
denoted by sup A. If x≤x0, then the set [x, x0] = {y∈X|x≤y≤x0}is called an
order interval. For details, see [14].
Lemma 6.7. Any topological semilattice (X, ≤)with path-connected intervals is a
G-convex space. More precisely, let D be a nonempty subset of X and Γ : hDi(X
a map such that
ΓA= Γ(A) = [
a∈A
[a, sup A]for A ∈ hDi.
Then (X, D; Γ) is a G-convex space.
(5) Hyperconvex metric spaces — Let Abe a nonempty bounded subset of
a metric space (M, d). Then we define the following as in Khamsi [16]:
(i) BI(A) = ad(A) := T{B⊂M|Bis a closed ball in Msuch that A⊂B}.
(ii) A(M) := {A⊂M|A= BI(A)}, i.e., A∈ A(M) if and only if Ais an
intersection of closed balls. In this case we will say Ais an admissible subset of M.
(iii) Ais called subadmissible, if for each N∈ hAi, BI(N)⊂A. Obviously, if Ais
an admissible subset of M, then Amust be subadmissible.
We introduce new definitions:
Definition 6.8. An abstract convex space (M , D; Γ) is called simply a metric space
if (M, d) is a metric space, D⊂Mis nonempty, and Γ : hDi→A(M) is a map
such that ΓA:= BI(A)∈ A(M) for each A∈ hDi. A multimap G:D(Mis a
KKM map if ΓA⊂G(A) for each A∈ hDi.
A Γ-convex subset of (M, D; Γ) is subadmissible and conversely.
The following is due to Khamsi [16] and Yuan [61]:

12 SEHIE PARK
Definition 6.9. Let (M, d) be a metric space. A subset S⊂Mis said to be
finitely metrically closed [resp. finitely metrically open] if for each F∈ A(M), the
set F∩S= BI(F)∩Sis closed [resp. open]. Note that BI(F) is always defined
and belongs to A(M). Thus if Sis closed [resp. open] in M, it is obviously finitely
metrically closed [resp. open].
The following is also due to Khamsi [16] and Yuan [61]:
Theorem 6.10. [KKM-Map Principle] Let H be a hyperconvex metric space, X an
arbitrary subset of H, and G:X(Ha KKM map such that each G(x)is finitely
metrically closed [resp. finitely metrically open]. Then the family {G(x)|x∈X}
has the finite intersection property.
This shows that any hyperconvex metric space having finitely metric topology is
a KKM space. Hence such space satisfies all results in Section 3 and [45].
In view of the above theorem, we have the following due to Khamsi [16, Theorem
4]:
Theorem 6.11. Let H be a hyperconvex metric space and X⊂Han arbitrary
subset. Let G:X(Hbe a KKM map such that G(x)is closed for any x∈Xand
G(x0)is compact for some x0∈X. Then we have
\
x∈X
G(x)6=∅.
(6) E-convex spaces — Youness [60] introduces the E-convex set and the E-
convex map as follows:
Definition 6.12. A set M⊂Rnis said to be E-convex whenever a function E:
Rn→Rnexists such that (1 −λ)E(x) + λE(y)∈Mholds for any x, y ∈Mand
0<λ<1.
There is an E-convex set that is not convex; see Youness [60]. An E-convex set
is an abstract convex space, and hence our KKM theory is applicable to it.
In fact, let Dbe a nonempty subset of M. By defining Γ : hDi → Mas
Γ{x0, . . . , xn}= co E{x0, . . . , xn}={Σn
i=0λiE(xi)|0≤λi≤1,Σn
i=0λi= 1}
for each A:= {x0, x1, . . . , xn} ∈ hDi, (M, D; Γ) becomes an abstract convex space.
Such Γ has convex values and clearly a continuous map φA: ∆n→ΓAexists.
Therefore, (M, D;Γ) becomes an H-space and φA-space, that is, a KKM space.
(7) Bayoumi’s KKM spaces — Let 0 < p ≤1. Recall the definitions given by
Bayoumi [2]:
Definition 6.13. (p-convex set) A set Ain a vector space Vis said to be p-convex
if, for any x, y ∈A, s, t ≥0,we have
(1 −t)1/px+t1/py∈A, whenever 0 ≤t≤1.
Definition 6.14. (p-convex hull) If Xis a topological vector space and A⊂X,
the closed p-convex hull of Adenoted by Cp(A) is the smallest closed p-convex set
containing A.

VARIOUS EXAMPLES OF THE KKM SPACES 13
Definition 6.15. (p-convex combination) Let Abe p-convex and x1,· · · , xn∈A,
and ti≥0,Pn
1tp
i= 1.Then Pn
1tixiis called a p-convex combination of {xi}. If
Pn
1|ti|p≤1, then Pn
1tixiis called an absolutely p-convex combination. It is easy
to see that Pn
1tixi∈Afor a p-convex set A.
Definition 6.16. (locally p-convex space) A topological vector space is said to
be locally p-convex if the origin has a fundamental set of absolutely p-convex 0-
neighborhoods. This topology can be determined by p-seminorms which are defined
in the obvious way.
Using these concepts, in [8], definitions of almost p-convex sets and the p-convexly
almost fixed point property are introduced as generalizations of almost convex sets
(due to Himmelberg [9]) and the almost fixed point property, resp.
Now we have a new KKM space as in [50]:
Lemma 6.17. Suppose that Xis a subset of a topological vector space Eand Dis
a nonempty subset of Xsuch that Cp(D)⊂X. Let ΓN:= Cp(N)for each N∈ hDi.
Then (X, D; Γ) is a φA-space.
(8) Γ-convex spaces — Zafarani [62] introduce Γ-convex spaces as generaliza-
tions of our G-convex spaces. This concept is actually our abstract convex spaces
without assuming any topology. We recognized the existence of Zafarani’s paper
after we developed sufficiently rich investigations on our abstract convex spaces.
(9) R-tree — Suppose Xis a closed convex subset of a complete R-tree H, and
for each A∈ hXi, ΓA:= convH(A), where convH(A) is the intersection of all closed
convex subsets of Hthat contain A; see Kirk and Panyanak [20]. Later it was
known that (H, X;Γ) is a KKM space; see Park [45].
(10) Horvath’s convex space — According to Horvath [13], a convexity on a
topological space Xis an algebraic closure operator A7→ [[A]] from P(X) to P(X)
such that [[{x}]] = {x}for all x∈X, or equivalently, a family Cof subsets of X, the
convex sets, which contains the whole space and the empty set as well as singletons
and which is closed under arbitrary intersections and updirected unions.
For Horvath’s convex space (X, C) with the weak Van de Vel property, the corre-
sponding abstract convex space (X; Γ) is a KKM space, where ΓA:= [[A]] = T{C∈
C | A⊂C}is metrizable for each A∈ hXi; see Horvath [13].
(11) B-spaces — Briec and Horvath [4] introduced B-spaces, which are also
KKM spaces [4, Corollary 2.2].
(12) Connected linearly ordered spaces — Such a space (X, ≤) can be made
into an abstract convex topological space (X, D; Γ) for any nonempty D⊂Xby
defining ΓA:= [min A, max A] = {x∈X|min A≤x≤max A}for each A∈ hDi.
Further, it is a KKM space; see Park [33, Theorem 5(i)] and [41].
(13) Extended long line L∗— The set L∗can be made into a KKM space
(L∗, D; Γ). In fact, L∗is constructed from the ordinal space D:= [0,Ω] consisting
of all ordinal numbers less than or equal to the first uncountable ordinal Ω, together
with the order topology; see Park [41].

14 SEHIE PARK
Recall that L∗is a generalized arc obtained from [0,Ω] by placing a copy of the
interval (0,1) between each ordinal αand its successor α+ 1 and we give L∗the
order topology. Now let Γ : hDi(L∗be the one as in the above (12). But L∗
is not a G-convex space. In fact, since Γ{0,Ω}=L∗is not path connected, for
A:= {0,Ω}∈hL∗iand ∆1= [0,1], there does not exist a continuous function
φA: [0,1] →ΓAsuch that φA{0} ⊂ Γ{0}={0}and φA{1} ⊂ Γ{Ω}={Ω}.
Therefore (L∗⊃D; Γ) is not G-convex.
(14) R-KKM spaces of Sankar Raj and Somasundaram — The authors
[58] introduced an R-KKM map T:A→2Bfor two nonempty subsets A, B of
a normed space X, the sufficient condition for which the set T{T(x)|x∈A}is
nonempty. Applying such intersection theorem, they show an extended version of
the Fan-Browder fixed point theorem, in a normed linear space setting, by providing
an existence of the best proximity point.
In [49], we show that, when we introduce the finitely generated topology on X, if
(A, B) is a proximal pair and we let Γ{x1, . . . , xn}:= co{y1, . . . , yn}, then we show
that the abstract convex space (B, A; Γ) becomes a partial KKM space. Moreover,
since Γ has convex values, (B, A;Γ) becomes an H-space and hence a KKM space.
So it has many equivalent properties and the Fan-Browder type fixed point theorem
is one of them.
(15) KKM spaces of Chaipunya and Kumam — The authors [5] study
applications of intersection theorems related non-self KKM maps on Hadamard
manifolds. Their results improve corresponding ones of Sanka Raj and Somasun-
daram [58]. They apply their KKM lemma to the Browder fixed point theorem for
nonself maps and the solvability of generalized equilibrium problems.
The pair (A, B) set up by two given nonempty subsets Aand Bof a metric space
(S, d) is called a proximal pair if to each point (x, y)∈A×B, there corresponds a
point (x, y)∈A×Bsuch that
d(x, y) = d(x, y) = dist(A, B),
where dist(A, B) := inf {d(x, y)|x∈A, y ∈B}. In addition, if both Aand Bare
convex, we say that (A, B) is a convex proximal pair.
And then, the authors assumed that Mis a Hadamard manifold with the geodesic
distance d. Given a point x∈Mand two nonempty subsets A, B ⊂M, they write
d(x, A) := infz∈Ad(x, z). For any nonempty subset A⊂M, denoted by co(A) the
geodesically convex hull of A, i.e., the smallest geodesically convex set containing
A. Note that the geodesically convex hull of any finite subset is compact.
Definition 6.18. ([5]) Let (A, B) be a proximal pair in a Hadamard manifold
M. A nonself map T:A(Bis said to be KKM if for each finite subset
D:= {x1, x2,· · · , xm} ⊂ A, there is a subset E:= {y1, y2,· · · , ym} ⊂ Bsuch
that d(xi, yi) = dist(A, B),∀i∈ {1,2,· · · , m}, and
co({yi|i∈I})⊂T({xi|i∈I})
for every ∅ 6=I⊂ {1,2,· · · , m}.

VARIOUS EXAMPLES OF THE KKM SPACES 15
Theorem 6.19. ([5]) Suppose that (A, B)is a proximal pair in a Hadamard man-
ifold Mand T:A(Bis a KKM map with nonempty closed values. Then, the
family {T(x)|x∈A}has the finite intersection property.
Theorem 6.20. ([5]) Suppose that (A, B)is a proximal pair in a Hadamard man-
ifold M and T:A(Bis a KKM map with nonempty closed values. If T(x0)is
compact at some x0∈A, then the intersection T{T(x)|x∈A}is nonempty.
Here Bseems to be better assumed to be geodesically convex.
In [5], an example of a partial KKM space (B, A; Γ) is given and, Aand Bhave
no inclusion relation. Note that, the KKM non-selfmap T:A(Bis a generalized
KKM map in the sense of Chang and Zhang [6]; see also [56].
Finally in this section, we add a partial KKM space:
(16) Vector lattices of Kawasaki et al. — The authors study several fixed
point theorems on vector lattices with units having certain topologies introduced
by Kawasaki himself in 2009. Their main result ([15], Theorem 1) is essentially the
following:
Theorem 6.21. Let Xbe a Hausdorff Archimedean vector lattice with unit having
the Kawasaki topology, Ya compact subset of X, and Za convex subset of Y.
Suppose that two multimaps S:Z(Y, T :Y(Ysatisfy
(1) S(z)is open for each z∈Z;
(2) for each y∈Y, ∅ 6= coΓS−(y)⊂T−(y).
Then Thas a fixed point x0∈Y; that is x0∈T(x0).
Note that (Y⊃Z; co) is a partial KKM space in view of the Theorem 3.1
characterizing such spaces. Therefore the vector lattice in Theorem 6.21 satisfies a
large number of statements in [44] and Section 3.
7. Further examples on abstract convex spaces
In this section, we state why we should use triples (X, D; Γ) for abstract convex
spaces instead of pairs (X; Γ) by recalling several examples of G-convex spaces, and
we introduce an example of a partial KKM space due to Kulpa and Szymanski in
2014 which is not a KKM space.
(I) The original definition of G-convex space (X, D; Γ) due to Park and H. Kim
[53–55] assumed X⊃Dand the following isotonicity condition:
(α) if A, B ∈ hDiand A⊂B, then ΓA⊂ΓB
This isotonicity was removed since 1998, and the restriction X⊃Dwas erased
since 1999; see our articles after 2000. However most of useful examples of G-convex
spaces seems to satisfy (α), some examples not satisfying (α) seem to be artificial:
Example 7.1. Let ∆3= co Vand V={e0, e1, e2, e3}.
(1) As seen in the original KKM theorem, (∆3, V ; co) is a G-convex space, where
co : hVi(∆3is the closure operation. Note that (α) holds in this example.
(2) For the G-convex space (∆3, V ; Γ), let Γ{e0, e1}:= co{e0, e1, e2}, and let
Γ(N) := co Nfor any other N∈ hVi. Then Γ does not satisfy the isotonicity (α).

16 SEHIE PARK
We give another example showing the necessity of using a triple (X, D; Γ) instead
of a pair (E; Γ):
Example 7.2. (1) The well-known Sperner theorem and Alexandorff-Pasynkoff
theorem on n+ 1 closed sets covering an n-simplex were derived by applying the
KKM theorem to the triple (∆n, V ; co). No proofs of such theorems using a pair
(E; Γ) appeared yet.
(2) In Shapley’s generalization of the KKM theorem, a triple (∆n, N ; Γ) appears,
where N:= {0,1, . . . , n}and ΓS:= ∆S= co {ei|i∈S}for each S∈ hNi; see [29]
and the references therein.
(3) Let C:= C[0,1] be the class of continuous real functions defined on [0,1],
and P:= P[0,1] be the subclass of all polynomials p(x), x∈[0,1], having real
coefficients. Choose an ε > 0 and, for each f∈ C, choose a pf∈ P in ε-neighborhood
of f, we have maxx∈[0,1] |f(x)−pf(x)|< ε. Let Γ : hCi (Pbe defined by ΓA:=
co {pfi}n
i=0 ∈ P for each A={fi}n
i=0 ∈ hCi. Moreover, let φA: ∆n→ΓAbe a
linear mapping satisfying ei7→ pfifor each i. Then, (X, D; Γ) := (P,C; Γ) becomes
a G-convex space satisfying the condition (∗) and X(D.
(4) Similarly, by choosing a proper subset Dof C, we obtain G-convex space
(X, D; Γ) satisfying X*Dor X+D.
Such examples are why we did not assume any inclusion relation between Xand
Din the definition of G-convex spaces.
(5) Since there are many forms of the Stone-Weierstrass approximation theorem,
we can make many examples similar to 3. or 4.
(II) For a quite long time, there was a question that whether the class of partial
KKM spaces properly contain that of KKM spaces. At last Kulpa and Szymanski
[22] found an example of a partial KKM space that is not a KKM space as follows:
Example 7.3. ([22]) Define an abstract convex space ([0,1];Γ) by defining Γ :
h[0,1]i([0,1] as follows: for 0 <p<0.5< q < 1, we define
Γ({p}) = {p},Γ({q}) = {q},Γ({p, q}) = [0,1] \ {0.5},
and define Γ(A) = [0,1] for other A∈ h[0,1]i.
Then, ([0,1]; Γ) is a partial KKM space, but not a KKM space.
Added in Proof. Further examples of KKM spaces are obtained our forthcoming
paper entitled Extending the realm of Horvath spaces.
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VARIOUS EXAMPLES OF THE KKM SPACES 19
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(Sehie Park) The National Academy of Sciences, Republic of Korea; Seoul 06579
and Department of Mathematical Sciences, Seoul National University, Seoul 08826,
Korea
E-mail address:park35@snu.ac.kr; sehiepark@gmail.com; http://parksehie.com