❧✐➟♥ ❤Ö ❣✐÷❛ ❦❤➠♥❣ ❣✐❛♥ ♠❡tr✐❝ ♠ê ✈í✐
❦❤➠♥❣ ❣✐❛♥ ▼❡♥❣❡r ✈➭ ❦❤➠♥❣ ❣✐❛♥ ♠❡tr✐❝ ①➳❝ s✉✃t
◆❣✉②Ô♥ ❈❤Ý ❚❤➽♥❣
✭❛✮
❚ã♠ t➽t✳
❚r♦♥❣ ❜➭✐ ❜➳♦ ♥➭②✱ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè tÝ♥❤ ❝❤✃t ❝ñ❛ sè ♠ê✱ t❐♣
α
✲♠ø❝✱
♠è✐ ❧✐➟♥ ❤Ö ❣✐÷❛ ❝❤ó♥❣ ✈➭ ➤➢❛ r❛ ❝➳❝ ➤✐Ò✉ ❦✐Ö♥ ➤Ó ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♠ê ❧➭ ❦❤➠♥❣ ❣✐❛♥
♠➟tr✐❝ ①➳❝ s✉✃t✱ ❤♦➷❝ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ▼❡♥❣❡r ✈➭ ♥❣➢î❝ ❧➵✐✳
❑❤➳✐ ♥✐Ö♠ sè ♠ê✱ t❐♣
α
✲♠ø❝ ✈➭ ❝➳❝ tÝ♥❤ ❝❤✃t ❝ñ❛ ♥ã ➤➲ ➤➢î❝ ❝➳❝ t➳❝ ❣✐➯ ❖✳ ❑❛❧❡✈❛ ✈➭
❙✳ ❙❡✐❦❦❛❧❛ ❣✐í✐ t❤✐Ö✉ tr♦♥❣ ❬✸❪✳ ❉ù❛ ✈➭♦ ❝➳❝ ❦❤➳✐ ♥✐Ö♠ ♥➭②✱ ❝➳❝ t➳❝ ❣✐➯ ➤➲ ➤➢❛ r❛ ❦❤➳✐
♥✐Ö♠ ✈➭ ❝➳❝ tÝ♥❤ ❝❤✃t ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♠ê✳ ❚r♦♥❣ ❬✷❪ ❝➳❝ t➳❝ ❣✐➯ ➤➲ ➤➢❛ r❛ ❦❤➳✐
♥✐Ö♠ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ①➳❝ s✉✃t ✈➭ ❦❤➠♥❣ ❣✐❛♥ ▼❡♥❣❡r✳ ▼ét ✈✃♥ ➤Ò ➤➢î❝ ➤➷t r❛ ❧➭ ❝➳❝
❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♠ê ❝ã ♠è✐ ❧✐➟♥ ❤Ö ❣× ✈í✐ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ①➳❝ s✉✃t ✈➭ ❦❤➠♥❣ ❣✐❛♥
▼❡♥❣❡r❄✳ ●✐➯✐ q✉②Õt ❝➞✉ ❤á✐ ♥➭② tr♦♥❣ ♣❤➬♥ ➤➬✉ ❝ñ❛ ❜➭✐ ❜➳♦ ❝❤ó♥❣ t➠✐ ❝❤ø♥❣ ♠✐♥❤
♠ét sè ❧✐➟♥ ❤Ö ❣✐÷❛ sè ♠ê ✈➭ t❐♣
α
✲♠ø❝✳ P❤➬♥ t✐Õ♣ t❤❡♦ ❝ñ❛ ❜➭✐ ❜➳♦ ❝❤ó♥❣ t➠✐ ♥➟✉ r❛
❝➳❝ ➤✐Ò✉ ❦✐Ö♥ ➤Ó ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♠ê trë t❤➭♥❤ ♠ét tr♦♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♥➟✉
tr➟♥ ✈➭ ♥❣➢î❝ ❧➵✐✳
1.
▼ë ➤➬✉
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳
✭❬✶❪✮ ❈❤♦ t❐♣ ❤î♣
X
✳ ▼ét
t❐♣ ♠ê
A
tr➟♥
X
❧➭ ♠ét ➳♥❤ ①➵
µA:X→
[0,1]
tõ
X
✈➭♦ ➤♦➵♥
[0,1]
✈➭ ❦ý ❤✐Ö✉ ❧➭
A={(a, µA(a))|a∈X}
✳ ❍➭♠
µA
➤➢î❝ ❣ä✐ ❧➭
❤➭♠ ❧✐➟♥ t❤✉é❝
✱ ❣✐➳ trÞ
µA(a)∈[0,1]
❝❤Ø
♠ø❝ ➤é ❧✐➟♥ t❤✉é❝
❝ñ❛ ♣❤➬♥ tö
a
✈➭♦ t❐♣ ♠ê
A
✳ ▼✐Ò♥ ❣✐➳ trÞ ❝ñ❛ ❤➭♠
µA
❝❤ø❛ tr♦♥❣ ➤♦➵♥
[0,1]
✱ tr♦♥❣ ➤ã ❣✐➳ trÞ
0
➤➢î❝ ❣ä✐ ❧➭
♠ø❝
➤é ❦❤➠♥❣ ❧✐➟♥ t❤✉é❝ ❤♦➭♥ t♦➭♥
✱ ❝ß♥ ❣✐➳ trÞ
1
❝❤Ø
♠ø❝ ➤é ❧✐➟♥ t❤✉é❝ ❤♦➭♥ t♦➭♥
✳
❚❛ ❝ò♥❣ ❦ý ❤✐Ö✉ t❐♣ ♠ê
A={(a, µA(a))|a∈X}
➤➡♥ ❣✐➯♥ ❧➭
µA
✳
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✷✳
✭❬✶❪✮ ❚❐♣ ♠ê
A
➤➢î❝ ❣ä✐ ❧➭
rç♥❣
♥Õ✉ ❤➭♠ ❧✐➟♥ t❤✉é❝
µA(a) = 0
✱ ✈í✐
♠ä✐
a∈X.
❚❐♣ ♠ê
A
➤➢î❝ ❣ä✐ ❧➭
t♦➭♥ ♣❤➬♥
♥Õ✉ ❤➭♠ ❧✐➟♥ t❤✉é❝
µA(a) = 1,
✈í✐ ♠ä✐
a∈X.
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✸✳
✭❬✶❪✮ ●✐➯ sö
µ
✈➭
ν
❧➭ ❝➳❝ t❐♣ ♠ê tr➟♥
X
✳ ❚❛ ➤Þ♥❤ ♥❣❤Ü❛
µ6ν
✱
µ>ν
✈➭
µ=ν
♥❤➢ s❛✉
µ6ν
❦❤✐ ✈➭ ❝❤Ø ❦❤✐
µ(x)6ν(x),
✈í✐ ♠ä✐
x∈X,(1.0.1)
µ>ν
❦❤✐ ✈➭ ❝❤Ø ❦❤✐
µ(x)>ν(x),
✈í✐ ♠ä✐
x∈X,(1.0.2)
µ=ν
❦❤✐ ✈➭ ❝❤Ø ❦❤✐
µ(x) = ν(x),
✈í✐ ♠ä✐
x∈X.(1.0.3)
1
✲ ◆❤❐♥ ❜➭✐ ♥❣➭② ✷✾✴✵✷✴✷✵✵✽✳ ❙ö❛ ❝❤÷❛ ①♦♥❣ ♥❣➭② ✵✾✴✵✹✴✷✵✵✽✳
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✹✳
✭❬✸❪✮ ▼ét
sè ♠ê
❧➭ t❐♣ ♠ê tr➟♥ trô❝ sè t❤ù❝✳ ◆ã✐ ❝➳❝❤ ❦❤➳❝✱ sè ♠ê ❧➭
♠ét ➳♥❤ ①➵
x:R−→[0,1]
➤➷t t➢➡♥❣ ø♥❣ ♠ç✐ sè t❤ù❝
t∈R
✈í✐ ♣❤➬♥ tö
x(t)
t❤✉é❝ ➤♦➵♥
❬✵✱✶❪✳
❚❛ ♥ã✐ sè ♠ê
x
❧➭
♥ö❛ ❧✐➟♥ tô❝ tr➟♥ t➵✐
t0
♥Õ✉ ➳♥❤ ①➵
x:R→[0,1],
♥ö❛ ❧✐➟♥ tô❝ tr➟♥
t➵✐
t0
✳
❙è ♠ê
x
➤➢î❝ ❣ä✐ ❧➭
❧å✐
♥Õ✉ ✈í✐ ❜✃t ❦ú
s6t6r
✱ t❤× t❛ ❝ã
x(t)>min{x(s), x(r)}.(1.0.4)
◆Õ✉ tå♥ t➵✐
t0∈R
s❛♦ ❝❤♦ sè ♠ê
x
t❤á❛ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥
x(t0) = 1
✱ t❤×
x
➤➢î❝ ❣ä✐ ❧➭
sè ♠ê ❝❤✉➮♥ t➽❝✳
❙è ♠ê
x
➤➢î❝ ❣ä✐ ❧➭
❦❤➠♥❣ ➞♠
♥Õ✉
x(t) = 0
✱ ✈í✐ ♠ä✐
t < 0
✳
◆❤❐♥ ①Ðt ✶✳✺✳
✶✳ ❑ý ❤✐Ö✉
E
❧➭ t❐♣ ❤î♣ ❝➳❝ sè ♠ê ❧å✐✱ ❝❤✉➮♥ t➽❝ ✈➭ ♥ö❛ ❧✐➟♥ tô❝
tr➟♥✱
G
❧➭ t❐♣ ❤î♣ ❝➳❝ sè ♠ê ❧å✐✱ ❝❤✉➮♥ t➽❝✱ ♥ö❛ ❧✐➟♥ tô❝ tr➟♥ ✈➭ ❦❤➠♥❣ ➞♠✳ ❑❤✐ ➤ã t❛
❝ã
G⊆E
✳
✷✳ ❱í✐ ❜✃t ❦ú
x∈R
✱
sè ♠ê
x
➤➢î❝ ①➳❝ ➤Þ♥❤ ♥❤➢ s❛✉
x(t) = (1
♥Õ✉
t=x,
0
♥Õ✉
t6=x.
❑❤✐ ➤ã
♠ç✐ sè t❤ù❝ ❝ã t❤Ó ①❡♠ ♥❤➢ ♠ét sè ♠ê ➤➷❝ ❜✐Öt✳
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✻✳
✭❬✸❪✮ ❈➳❝ ♣❤Ð♣ t♦➳♥ sè ❤ä❝
+,−,·, /
tr➟♥
E×E
➤➢î❝ ➤Þ♥❤ ♥❣❤Ü❛
♥❤➢ s❛✉
(x+y)(t) = sup
s∈R
min{x(s), y(t−s)},∀x, y ∈E, ∀t∈R,(1.0.5)
(x−y)(t) = sup
s∈R
min{x(s), y(s−t)},∀x, y ∈E, ∀t∈R,(1.0.6)
(x·y)(t) = sup
s∈R
min nx(s), yt
so,∀x, y ∈E, ∀t∈R,(1.0.7)
(x/y)(t) = sup
s∈R
min{x(ts), y(s)},∀x, y ∈E, ∀t∈R.(1.0.8)
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✼✳
❱× ♠ç✐
x∈R
❧➭ ♠ét sè ♠ê ①➳❝ ➤Þ♥❤ ♥❤➢ ◆❤❐♥ ①Ðt ✶✳✺✱ t❛ ➤Þ♥❤
♥❣❤Ü❛ ❝➳❝ ♣❤➬♥ tö
0
✱
1
tr♦♥❣
E
♥❤➢ s❛✉
0(t) = (1
♥Õ✉
t= 0,
0
♥Õ✉
t6= 0.
✱
1(t) = (1
♥Õ✉
t= 1,
0
♥Õ✉
t6= 1.
▼Ö♥❤ ➤Ò ✶✳✽✳
❱í✐ ♠ç✐
x∈R
t❛ ❝ã
x(t) = 0(t−x)
✱ ✈í✐ ♠ä✐
t∈R.
❈❤ø♥❣ ♠✐♥❤✳
❚❤❐t ✈❐②✱ ❣✐➯ sö
x∈R
✳ ❑❤✐ ➤ã ✈í✐ ❜✃t ❦ú
t∈R
✱ t❛ ❝ã
0(t−x) = 1
♥Õ✉
t−x= 0,
0
♥Õ✉
t−x6= 0.=1
♥Õ✉
x=t,
0
♥Õ✉
x6=t. =x(t)
✳
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✾✳
✭❬✸❪✮ ●✐➯ sö
x
❧➭ sè ♠ê✳ ❑❤✐ ➤ã✱ t❛ ♥ã✐
❣✐➳ trÞ t✉②Öt ➤è✐ ❝ñ❛
x
✱ ❦ý
❤✐Ö✉ ❧➭
|x|
➤➢î❝ ➤Þ♥❤ ♥❣❤Ü❛ ♥❤➢ s❛✉
|x|(t) = (max{x(t), x(−t))}
♥Õ✉
t>0,
0
♥Õ✉
t < 0.
(1.0.9)
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✵✳
✭❬✸❪✮ ●✐➯ sö
y∈E
✱ t❛ ❦ý ❤✐Ö✉ ♣❤➬♥ tö
0−y∈E
❧➭
−y
✳
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✶✳
✭❬✸❪✮ ●✐➯ sö
x
❧➭ sè ♠ê✳ ❱í✐ ♠ç✐
α∈(0,1]
✱ t❐♣
α
✲♠ø❝ ✭
α
✲❧❡✈❡❧ s❡t✮
❝ñ❛
x
✱ ❦ý ❤✐Ö✉ ❧➭
[x]α
➤➢î❝ ①➳❝ ➤Þ♥❤ ♥❤➢ s❛✉
[x]α={t∈R|x(t)>α}.(1.0.10)
◆❤❐♥ ①Ðt ✶✳✶✷✳
❚❐♣
α
✲♠ø❝ ❝ñ❛ sè ♠ê ❧å✐✱ ❝❤✉➮♥ t➽❝ ✈➭ ♥ö❛ ❧✐➟♥ tô❝ tr➟♥ ❧➭ ➤♦➵♥
aα, bα
✱ ♥❣❤Ü❛ ❧➭
[x]α=aα, bα
✱ tr♦♥❣ ➤ã
aα
❝ã t❤Ó ❧➭ ✲
∞
✈➭
bα
❝ã t❤Ó ❧➭ ✰
∞
✳ ❑❤✐ ➤ã
t❛ ❦ý ❤✐Ö✉
− ∞, bα
❤♦➷❝ ❧➭
aα,+∞
✳
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✸✳
✭❬✸❪✮ ❈❤♦ t❐♣ ❤î♣
X
❦❤➳❝
∅
✳ ❈➳❝ ❤➭♠
λα:X×X→R
tõ
X×X
✈➭♦
R
➤➢î❝ ❣ä✐ ❧➭
❦❤➠♥❣ ❣✐➯♠ t❤❡♦
α∈(0,1]
✱ ♥Õ✉ ✈í✐ ♠ä✐
α1, α2∈(0,1]
✱
α1< α2
t❤×
t❛ ❝ã
λα1(x, y)6λα2(x, y)
✱ ✈í✐ ♠ä✐
(x, y)∈X×X
✳
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✹✳
✭❬✸❪✮ ❈❤♦ t❐♣ ❤î♣
X
❦❤➳❝
∅
✳ ❈➳❝ ❤➭♠
ρα:X×X→R
tõ
X×X
✈➭♦
R
➤➢î❝ ❣ä✐ ❧➭
❦❤➠♥❣ t➝♥❣ t❤❡♦
α∈(0,1]
✱ ♥Õ✉ ✈í✐ ♠ä✐
α1, α2∈(0,1]
✱
α1< α2
t❤× t❛
❝ã
ρα1(x, y)>ρα2(x, y)
✱ ✈í✐ ♠ä✐
(x, y)∈X×X
✳
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✺✳
✭❬✸❪✮ ❈❤♦ t❐♣ ❤î♣
X
❦❤➳❝
∅
✱ ❤➭♠
d:X×X→G
tõ
X×X
✈➭♦
G
✈➭ ❝➳❝ ❤➭♠
L, R : [0,1] ×[0,1] →[0,1]
❧➭ ➤è✐ ①ø♥❣✱ ❦❤➠♥❣ ❣✐➯♠ t❤❡♦ ❝➯ ❤❛✐ ❜✐Õ♥
x
✈➭
y
✱ ➤å♥❣ t❤ê✐ t❤♦➯ ♠➲♥
L(0,0) = 0
✱
R(1,1) = 1
✳ ●✐➯ sö r➺♥❣
d(x, y)α=hλα(x, y), ρα(x, y)i,(1.0.11)
✈í✐
x, y ∈X
❀
λ
✈➭
ρ
♥ã✐ tr♦♥❣ ❝➳❝ ➤Þ♥❤ ♥❣❤Ü❛ ✶✳✶✹ ✈➭ ✶✳✶✺✳ ❑❤✐ ➤ã t❛ ♥ã✐ ❜é
(X, d, L, R)
❧➭
❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♠ê ✭❢✉③③② ♠❡tr✐❝ s♣❛❝❡✮ ✈➭ ❞ ❧➭ ♠➟tr✐❝ ♠ê ✭❢✉③③② ♠❡tr✐❝ ✮
✱ ♥Õ✉ t❤á❛
♠➲♥ ❝➳❝ ➤✐Ò✉ ❦✐Ö♥ s❛✉ ➤➞②
(i.) d(x, y) = 0
❦❤✐ ✈➭ ❝❤Ø ❦❤✐
x=y,
(ii.) d(x, y) = d(y, x)
✱ ✈í✐ ♠ä✐
x, y ∈X,
(iii.)
❱í✐ ♠ä✐
x, y, z ∈X
✱ t❤×
(1.) d(x, y)(s+t)>Ld(x, z)(s), d(z, y)(t)
✈í✐
s6λ1(x, z)
✱
t6λ1(z, y)
✈➭
s+t6λ1(x, y)
✱
(2.) d(x, y)(s+t)6Rd(x, z)(s), d(z, y)(t)
✈í✐
s>λ1(x, z)
✱
t>λ1(z, y)
✈➭
s+t>λ1(x, y)
✳
◆❤❐♥ ①Ðt ✶✳✶✻✳
❛✳ ❚r♦♥❣ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✺ ❝➳❝ ❤➭♠
λα, ρα
❝ã ❝➳❝ tÝ♥❤ ❝❤✃t ❧➭
λα
❦❤➠♥❣
❣✐➯♠ t❤❡♦
α
✱
ρα
❦❤➠♥❣ t➝♥❣ t❤❡♦
α
✳
❜✳ ❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤➠♥❣ t❤➢ê♥❣ ❧➭ ♠ét tr➢ê♥❣ ❤î♣ ➤➷❝ ❜✐Öt ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥
♠➟tr✐❝ ♠ê✳ ❚❤❐t ✈❐②✱ ✈× ❝➳❝ sè t❤ù❝ ➤➢î❝ ①❡♠ ❧➭ ❝➳❝ sè ♠ê ✈➭ ♠➟tr✐❝
d
tr♦♥❣ ❦❤➠♥❣
❣✐❛♥ ♠➟tr✐❝
(X, d)
t❤♦➯ ♠➲♥ ❝➳❝ ➤✐Ò✉ ❦✐Ö♥ ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♠ê ✈í✐
L
✈➭
R
❝❤♦
❜ë✐
L(a, b) = 0
✈í✐ ♠ä✐
a, b ∈[0,1]
✈➭
R(a, b) = (0
♥Õ✉
a=b= 0,
1
tr♦♥❣ ❝➳❝ tr➢ê♥❣ ❤î♣ ❝ß♥ ❧➵✐
.
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✼✳
✭❬✷❪✮ ▼ét ❤➭♠
∆
✿
[0,1] ×[0,1] →[0,1]
➤➢î❝ ❣ä✐ ❧➭
t✲❝❤✉➮♥
✱ ♥Õ✉
∆
t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ò✉ ❦✐Ö♥ s❛✉ ➤➞②
✭❚✲✶✮
∆(a, 1) = a,
✈í✐ ♠ä✐
a∈[0,1];
✭❚✲✷✮
∆(a, b) = ∆(b, a),
✈í✐ ♠ä✐
a, b ∈[0,1];
✭❚✲✸✮
∆(c, d)>∆(a, b),
❦❤✐
c>a
✈➭
d>b
✱ ✈í✐ ♠ä✐
a, b, c, d ∈[0,1];
✭❚✲✹✮
∆(a, ∆(b, c)) = ∆(∆(a, b), c)
✱ ✈í✐ ♠ä✐
a, b, c ∈[0,1].
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✽✳
✭❬✷❪✮ ❍➭♠
F:R−→ R+
➤➢î❝ ❣ä✐ ❧➭
❤➭♠ ♣❤➞♥ ♣❤è✐
♥Õ✉
F
❧➭ ❤➭♠
❦❤➠♥❣ ❣✐➯♠✱ ♥ö❛ ❧✐➟♥ tô❝ tr➟♥✱
inf
t∈RF(t) = 0
✈➭
sup
t∈R
F(t) = 1
✳
❑ý ❤✐Ö✉
D
❧➭ t❐♣ ❤î♣ t✃t ❝➯ ❝➳❝ ❤➭♠ ♣❤➞♥ ♣❤è✐✳
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✾✳
✭❬✷❪✮ ●✐➯ sö
X
❧➭ ♠ét t❐♣ ❤î♣ ❦❤➳❝
∅
✈➭
F:X×X→ D
❧➭
➳♥❤ ①➵ tõ
X×X
✈➭♦ t❐♣ t✃t ❝➯ ❝➳❝ ❤➭♠ ♣❤➞♥ ♣❤è✐
D
✳ ❱í✐ ♠ç✐
x, y ∈X
t❛ ❦ý ❤✐Ö✉
Fxy =F(x, y)
✳ ❑❤✐ ➤ã✱ ❜é
(X, F )
➤➢î❝ ❣ä✐ ❧➭
❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ①➳❝ s✉✃t ✭❤❛② ❝ß♥ ➤➢î❝
❣ä✐ ❧➭ P▼✲❦❤➠♥❣ ❣✐❛♥✮
♥Õ✉ ❤➭♠
Fxy
t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ò✉ ❦✐Ö♥ s❛✉ ➤➞②
✭✶✮✬
Fxy(t) = 1
✱ ✈í✐ ♠ä✐
t > 0
❦❤✐ ✈➭ ❝❤Ø ❦❤✐
x=y,
✭✷✮✬
Fxy(0) = 0
✱ ✈í✐ ♠ä✐
x, y ∈X,
✭✸✮✬
Fxy(t) = Fyx(t)
✱ ✈í✐ ♠ä✐
t∈R
✈➭ ✈í✐ ♠ä✐
x, y ∈X,
✭✹✮✬ ◆Õ✉
Fxz(t) = 1
✈➭
Fzy(s) = 1
✱ t❤×
Fxy(s+t) = 1
✱ ✈í✐ ♠ä✐
x, y, z ∈X
✳
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✷✵✳
✭❬✷❪✮ ●✐➯ sö
X
❧➭ ♠ét t❐♣ ❤î♣ ❦❤➳❝
∅
✳ ❑❤✐ ➤ã✱
(X, F, ∆)
➤➢î❝ ❣ä✐
❧➭
❦❤➠♥❣ ❣✐❛♥ ▼❡♥❣❡r
✱ tr♦♥❣ ➤ã
(X, F )
❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ①➳❝ s✉✃t ✈➭
∆ : [0,1] ×
[0,1] −→ [0,1]
❧➭ ♠ét
t
✲❝❤✉➮♥✱ ♥Õ✉ ✈í✐ ♠ç✐
(x, y)∈X×X
✱ ❤➭♠ ♣❤➞♥ ♣❤è✐
Fxy
♥ö❛
❧✐➟♥ tô❝ tr➟♥✱ t❤á❛ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥
Fxy(0) = 0
✈➭ ➤å♥❣ t❤ê✐ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ò✉ ❦✐Ö♥ s❛✉
➤➞②
(i)′
❱í✐ ♠ä✐
t > 0
✱
Fxy(t) = 1
❦❤✐ ✈➭ ❝❤Ø ❦❤✐
x=y,
(ii)′Fxy =Fyx,
✈í✐ ♠ä✐
x, y ∈X,
(iii)′Fxy(s+r)>∆Fxz (s), Fzy(r),
✈í✐ ♠ä✐
x, y, z ∈X,
✈í✐ ♠ä✐
s, r >0
✳
◆❤❐♥ ①Ðt ✶✳✷✶✳
❑❤✐ ♥❣❤✐➟♥ ❝ø✉ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♠ê ❤♦➷❝ ❦❤➠♥❣ ❣✐❛♥ ▼❡♥❣❡r✱
❝❤ó♥❣ t❛ t❤➢ê♥❣ ❝❤ä♥ ❝➳❝ ❤➭♠
L
✱
R
✱
∆
❧➭ ♠ét tr♦♥❣ ❝➳❝ ❤➭♠ s❛✉ ➤➞②✳
T1(a, b) = max(a+b−1,0) (max(
tæ♥❣
−1,0))
T2(a, b) = ab (
tÝ❝❤
)
T3(a, b) = min(a, b) (min)
T4(a, b) = max(a, b) (max)
T5(a, b) = a+b−ab (
tæ♥❣ ✲ tÝ❝❤
)
T6(a, b) = min(a+b, 1) (min(
tæ♥❣
,1))
❈➳❝ ❤➭♠
Ti, i = 1,2, ..., 6
t➝♥❣ ❞➬♥ t❤❡♦ ❝❤Ø sè ✐✱ ♥❣❤Ü❛ ❧➭✱ ♥Õ✉
i>j
t❤×
Ti(a, b)>
Tj(a, b)
✱ ✈í✐ ♠ä✐
a, b ∈[0,1]
✳
2.
sè ♠ê ✈➭ t❐♣
α
✲♠ø❝
❚r♦♥❣ ♠ô❝ ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè tÝ♥❤ ❝❤✃t ✈Ò sè ♠ê✱ t❐♣
α
✲♠ø❝ ✈➭ ♠è✐ ❧✐➟♥
❤Ö ❣✐÷❛ ❝❤ó♥❣✳
➜Þ♥❤ ❧ý ✷✳✶✳
R
❧➭ t❐♣ ❝➳❝ sè ♠ê ❧å✐✱ ❝❤✉➮♥ t➽❝ ✈➭ ♥ö❛ ❧✐➟♥ tô❝ tr➟♥✳
❈❤ø♥❣ ♠✐♥❤✳
❚❤❐t ✈❐②✱ ✈í✐ ♠ç✐
x∈R,
t❛ ❝ã
x(t) = 1
❦❤✐ ✈➭ ❝❤Ø ❦❤✐
t=x
✈➭
x(t) = 0
❦❤✐
t6=x
✳ ❉♦ ➤ã
x
❧➭ sè ♠ê ❝❤✉➮♥ t➽❝ ✈➭ ♥ö❛ ❧✐➟♥ tô❝ tr➟♥✳ ❇➞② ❣✐ê t❛ ❝❤ø♥❣ ♠✐♥❤
x
❧➭
sè ♠ê ❧å✐✳ ❚❤❐t ✈❐②✱ ✈í✐ ♠ç✐ t s❛♦ ❝❤♦
s6t6r,
tr♦♥❣ ➤ã
s, r ∈R.
❑❤✐ ➤ã ①➮② r❛ ♠ét
tr♦♥❣ ❝➳❝ tr➢ê♥❣ ❤î♣ s❛✉ ➤➞②✳
✶✳ ◆Õ✉
x=t
✱ t❤×
x(t) = 1
✳ ▲ó❝ ➤ã t❛ ❝ã ♥❣❛②
x(t)>min{x(s), x(r)}
✷✳ ◆Õ✉
x6=t,
t❤×
x(t) = 0
✳ ❑❤✐ ➤ã t❛ ❝ã ❝➳❝ tr➢ê♥❣ ❤î♣ s❛✉✿ ❤♦➷❝ ❧➭
x=s
✈➭
x6=r
✱
❤♦➷❝ ❧➭
x=r
✈➭
x6=s
✱ ❤♦➷❝ ❧➭
x6=s
✈➭
x6=r
✳ ❚r♦♥❣ ❝➳❝ tr➢ê♥❣ ❤î♣ ➤ã t❛ ➤Ò✉ ❝ã
x(t)>min{x(s), x(r)}= 0
❱❐②
x
❧➭ sè ♠ê ❧å✐✱ ❤❛②
R
❧➭ t❐♣ ❝➳❝ sè ♠ê ❧å✐✱ ❝❤✉➮♥ t➽❝ ✈➭ ♥ö❛ ❧✐➟♥ tô❝ tr➟♥✳
➜Þ♥❤ ❧ý ✷✳✷✳
✭❬✸❪✮
●✐➯ sö
x, y ∈E
✳ ❑❤✐ ➤ã✱ t❛ ❝ã ❦Õt q✉➯ s❛✉ ➤➞②
(−y)(t) = y(−t)
✱
✈í✐ ♠ä✐
t∈R
✈➭
x−y=x+ (−y)
✳
➜Þ♥❤ ❧ý ✷✳✸✳
✭❬✸❪✮
❙è ♠ê
x
❧➭ ❧å✐ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉✱ ✈í✐ ♠ç✐
α∈(0,1]
t❐♣
α
✲ ♠ø❝
[x]α
❧➭ ♠ét t❐♣ ❧å✐ tr♦♥❣
R
✳