Otemon
Economic Studies,
15
16（1983 ）
A NOTE ON THE KNASTER-KURATOWSKIMAZURKIEWICZ THEOREM
Shigeaki
Watanabe
0.
This paper 沁 the supplementary note to S.
Watanabe,
'THE
BROUWER FIXED POINT THEOREM AND RELATED THEOREMS',
(OTEMON ECONOMIC STUDIES, No.
14, 1981). Its purpose is to give
the direct proof of the equivalence of the variant forms of the K.
K. M.
theorem.
1.
We use the following notation.
N={0,
1, …, n｝:the set of integers
’
十 ［ j］ご［a ＼ α＼ ...,
a"］:a closed simpleχ
[si : the - face opposite to the verte
χぶ
Let A be a set.
−
A is the closure of
A' denotes the complement of A.
ん
（I ）［Sperner's lemma ］
Let K
be triangulati
） n of［5］with each vertex of K
integer in A^ such that no vertex in
labelled with an
ト ］;is labelled i. Then there is a
simpleχ in K whose vertices carry all the labels in
χ
(ni ）［the K. K. M. theorem]
If 人 ，（i^N
），are closed sets such that each face［￡］＝［a!。a'', ...,
衣A ，（ ん≦ 戒), of the simplex [s] satisfies
国 ⊂ 爪U
…U ■^in （1 ）
then n 或 ≒6. ▽
（m ）［the Brouwer fiχed point theorem］
If F: ト ］→
ト ］is continuous, then F has a 丘χed point.
The popular method of proof of the Brouwer fi
（I ）。
＝ ⇒（ni ）⇒
2.
The variant forms of （ni ）are following.
( 1 ）
χed point theorem is
16
SHIGEAKI WATANABE
(n2)If Ai
，(μ≡A^)，are closed sets such that
(i)[5] ＝U 人
(ii) 式 ∩[4 ＝φ，
then n 人 ≒ φ
・
(113)If
人 ，(zgN) ，are closed sets such that
(i)[s] ＝UA,(ii)if L( ≒ φ)⊂N and J ニiV 一￡,
thenn[ 迂 ⊂Uye/ λ,，
1 '∈ ￡
then
（n4
n 人 ≒ φ・
）If
几
（ μ≡N
）, are closed sets such that
（i） ［s ］＝U
或
（ii) ［迂 ⊂ 几
then
n
（ii ≡N ）
人 ≒ φ・
These
variant forms are
equivalent to each other.
We will prove it in
the following.
Note.
（ni
If we notice that
）and
（n3
valent to
（n3
（n
…,
where J
diffent here. So,
φ ］is equivalent to
＝{h ， ￡i
， ...， 心>.
if we admit that
（ni
n
[s] j，
But,
we
￥
）is equi-
）, there are three different forms.
1） ⇒
（n
Let's show that
satisfied,
ぶ＼
） are clearly the same,
take these conditions as
［proof]
μ ］＝[ ぷ.
2）. （Kuratowski
（i）, （ii）of
there e χists a point X
（n2
）imply
（1 ）of （ni
）.
in ［z ］that is not included in any
by （i ）of （112 ）,
・"
■
ら. Then,
）
If （1 ）is not
− ・
A, ・．
■
■
・>
there is an index such that
i＝
≒＝h， ■■
■> i
≒ 心 ・ I I
ぶ∈ ふ
On the other hand,
］⊂ ［α＼ ＼ α ＋＼ ..., ］＝［s ］i
ぶ ∈［z ］＝［α'
■ ", ・
・
・
・
・・
・9 α'￣
a"
タ a'''
x G A,-∩［5].-. Since this is inconsistent with
（ii）of （n2 ）. （1 ）
Therefore,
follows from
（i）, （ii) of （n2
（i ）, （ii） ⇒
（n2
）⇒
For
from the above condition,
（l ）⇒
（n3
{A,}
）.
∩ 或 ≒ φ,
Hence
n 人 ≒ φby
i. e.,（ni
）⇒
（n2
（ni
）.
）.
）.
that satisfy the conditions
of （113 ）, we consider {
−
{（ 人 ∩（迂 ）｝satisfy the conditions of
−
）, the result follows.
For,
from （112 ）, n {（ A,- ∩W
If we can show that
（n2
we getn
Now,
show that
In other words.
（A- ∩
−
寸}.
（i）， （ii）of
）} ≒ φ.
Hence,
Ne χt,
we will
人 ≒ φ・
it is easy to see that
（i ）is satisfied. We set J
（ii）of （n2
＝N
（2
−L.
）
）is satisfied.
By elementary set operation.
A
NOTE ON THE KNASTER-KURATOWSKI-MAZURKIEWICZ THEOREM 17
u{(An[5])}
In particular,
り[ 寸 ＝[s],
＝ ∩{(UA)U(U[5];)}. (2)
if L ＝N, then
because Q[
り 人 ＝[ ヰAnd,
ヰ ＝φ. Now,
if ￡= φ, i. e., J=N, then
in general,
by the conditic]n
(ii)of
(n3),
∧ ? 人 ⊃9[
心 ⇒(?
＝(U[
瓦y ⊂(n[sly
寸]
−
⊂(U(
迂)
．
Therefore,
{(U 刄)u(u
−
迂]} ⊃{(U 或)U(U
人)1 ＝[ 球
Hence,
(2) ＝[5] ∩[5]∩ … ∩[s] ＝[s].
This means
that {(A, ∩(s]D} satisfy thecondition of (i)of (112). Consequ- .
ently, by (n2),
∩{( 人 ∩[ 寸]} ≒(j>. Hence, ∩ 瓦≒ φ.
(n3)
⇒(n4).
Set
T;＝( ∩ ん)U ぶ ＝( ∩ ん) ＼J
ぶ
y N-i
by the condition
（ii）of （114 ）, 扁 ⊃ ［ ふ
Then,
Therefore,
（i） U7:. ＝（ ∩ 瓦 ）u（U ぶ ）
￡
＝(∩ん)U
(U A) ⊃∩[5]
j
（ 幻 UT, ＝（ ∩ 刄 ）U（UAY
＝ 閻．
Consequently,
{Ti}satisfy the conditions of (n3). Hence,
by
(n3) ， ∩ 司≒ φ. Since U A,＝[s],
∩T; ＝（ ∩A ）U（ ∩ ぶ ）
Ⅳ
ｙ寸
＝(∩ふ)U(U
人)'
＝（ ∩A ）U［
＝（ ∩ 瓦 ）U φ＝ ∩A,.
N N
人 ≒ φ・
It follows that n
（n4
）
・⇒ ‘ N
（ni ）.
For
{A
｝that satisfy the conditions of
M, ＝U
Then,
）, set
ん.
N-i
（i）[4
Therefore,
（ni
刈
α'￣ α'＋＼ ・
・
・タ タ ・゛
タ ＝ 叉.
=［α°
・
，*
⊂UA
y 一j
（ii） UM,.=
U（U ん ）＝ ［ヰ
N N N-i ・
{Mi} satisfy the conditions of
（3
(n4).
）
Hence,
by (Ⅱ4), ∩ 漏 ≒
18
SHIGEAKI WATANABE
φ. Since
nM, ＝∩（UA
）＝∩A.
it follows that 瓦≒<!>
D
． 十
(
4
）