Tietze Extension Theorem for Ordered Fuzzy Gδ-extremally Disconnected Spaces
In this paper, a new class of fuzzy topological spaces called ordered fuzzy G -extremally disconnected spaces is introduced. Tietze extension theorem for ordered fuzzy Gδ-extremally disconnected spaces has been discussed as in [10] besides proving several other propositions and lemmas.
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irk-123456789-1245602017-09-30T03:03:53Z Tietze Extension Theorem for Ordered Fuzzy Gδ-extremally Disconnected Spaces Roja, E. Uma, M.K. Balasubramanian, G. In this paper, a new class of fuzzy topological spaces called ordered fuzzy G -extremally disconnected spaces is introduced. Tietze extension theorem for ordered fuzzy Gδ-extremally disconnected spaces has been discussed as in [10] besides proving several other propositions and lemmas. 2006 Article Tietze Extension Theorem for Ordered Fuzzy Gδ-extremally Disconnected Spaces / E. Roja, M.K. Uma, G. Balasubramanian // Український математичний вісник. — 2006. — Т. 3, № 3. — С. 382-393. — Бібліогр.: 11 назв. — англ. 1810-3200 2000 MSC. 54A40, 03E72. http://dspace.nbuv.gov.ua/handle/123456789/124560 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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In this paper, a new class of fuzzy topological spaces called ordered fuzzy G -extremally disconnected spaces is introduced. Tietze extension theorem for ordered fuzzy Gδ-extremally disconnected spaces has been discussed as in [10] besides proving several other propositions and lemmas. |
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Roja, E. Uma, M.K. Balasubramanian, G. |
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Roja, E. Uma, M.K. Balasubramanian, G. Tietze Extension Theorem for Ordered Fuzzy Gδ-extremally Disconnected Spaces Український математичний вісник |
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Roja, E. Uma, M.K. Balasubramanian, G. |
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Roja, E. |
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Tietze Extension Theorem for Ordered Fuzzy Gδ-extremally Disconnected Spaces |
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Tietze Extension Theorem for Ordered Fuzzy Gδ-extremally Disconnected Spaces |
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Tietze Extension Theorem for Ordered Fuzzy Gδ-extremally Disconnected Spaces |
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Tietze Extension Theorem for Ordered Fuzzy Gδ-extremally Disconnected Spaces |
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Tietze Extension Theorem for Ordered Fuzzy Gδ-extremally Disconnected Spaces |
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tietze extension theorem for ordered fuzzy gδ-extremally disconnected spaces |
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Tietze Extension Theorem for Ordered Fuzzy Gδ-extremally Disconnected Spaces / E. Roja, M.K. Uma, G. Balasubramanian // Український математичний вісник. — 2006. — Т. 3, № 3. — С. 382-393. — Бібліогр.: 11 назв. — англ. |
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AT rojae tietzeextensiontheoremfororderedfuzzygdextremallydisconnectedspaces AT umamk tietzeextensiontheoremfororderedfuzzygdextremallydisconnectedspaces AT balasubramaniang tietzeextensiontheoremfororderedfuzzygdextremallydisconnectedspaces |
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Український математичний вiсник
Том 3 (2006), № 3, 382 – 393
Tietze Extension Theorem for Ordered Fuzzy
Gδ-extremally Disconnected Spaces
Elangovan Roja, Mallasamudram K. Uma,
and
Ganesan Balasubramanian
(Presented by A. I. Stepanets)
Abstract. In this paper, a new class of fuzzy topological spaces called
ordered fuzzy Gδ-extremally disconnected spaces is introduced. Tietze
extension theorem for ordered fuzzy Gδ-extremally disconnected spaces
has been discussed as in [10] besides proving several other propositions
and lemmas.
2000 MSC. 54A40, 03E72.
Key words and phrases. Ordered fuzzy Gδ-extremally disconnected
spaces, ordered fuzzy Gδ-continuous, lower/upper fuzzy Gδ-continuous
functions.
Introduction
The fuzzy concept has invaded almost all branches of mathematics
since the introduction of the concept by L. A. Zadeh [11]. Fuzzy sets
have applications in many fields such as information [7] and control [8].
The theory of fuzzy topological space was introduced and developed by
C. L. Chang [5] and since then various notions in classical topology have
been extended to fuzzy topological space [3, 4]. A new class of fuzzy
topological spaces called ordered fuzzy Gδ-extremally disconnected spaces
is introduced in this paper by using the concepts of fuzzy extremally
disconnected spaces [1], fuzzy Gδ-sets [2] and ordered fuzzy topology [6].
Some interesting properties and characterizations are studied. Tietze
extension theorem for ordered fuzzy Gδ-extremally disconnected spaces
has been discussed as in [10] besides proving several other propositions
and lemmas.
Received 18.07.2005
ISSN 1810 – 3200. c© Iнститут математики НАН України
E. Roja, M. K. Uma, G. Balasubramanian 383
1. Preliminaries
Definition 1.1. Let (X, T ) be a fuzzy topological space and λ be a fuzzy
set in X. λ is called a fuzzy Gδ-set if λ = ∧∞
i=1λi where each λi ∈ T [2].
Definition 1.2. Let (X, T ) be a fuzzy topological space and λ be a fuzzy
set in X. λ is called a fuzzy Fσ-set if λ = ∨∞
i=1λi where each 1− λi ∈ T .
Definition 1.3. Let (X, T ) be any fuzzy topological space. For any fuzzy
set λ in X we define the σ-closure of λ, denote by clσ λ, to be the inter-
section of all fuzzy Fσ-sets containing λ. That is
clσ λ = ∧{µ : µ is a fuzzy Fσ-set and µ ≥ λ}.
Definition 1.4. Let (X, T ) be any fuzzy topological space. For any fuzzy
set λ in X, we define the σ-interior of λ, denote by intσ λ, to be the
union of all fuzzy Gδ-sets contained in λ. That is,
intσ λ = ∨{µ : µ is a fuzzy Gδ-set and µ ≤ λ}.
Definition 1.5. For each t ∈ R, let Lt, Rt : R(I) → I be given by
Lt(λ) = 1 − λ(t−) and Rt(λ) = λ(t+). Define L = {Lt : t ∈ R} ∪ {0, 1}
and R = {Rt|t ∈ R} ∪ {0, 1}. Then L and R are called I-topologies on
R(I) [9].
Definition 1.6. Suppose (X, T ) is a fuzzy topological space. X is said
to be fuzzy extremally disconnected [2] if λ ∈ T implies cl λ ∈ T .
Remark 1.1. The symbol 〈t〉 (t ∈ R) stands for the member of R(L)
containing λ such that λ(t+) = λ(t−)′ = 0 [10].
2. Ordered Fuzzy Gδ-extremally Disconnected Spaces
In this section, the concept of ordered fuzzy Gδ-extremally discon-
nected spaces is introduced. Some interesting properties and characteri-
zations are studied.
Definition 2.1. Let (X, T,≤) be an ordered fuzzy topological space and
let λ be any fuzzy set in (X, T,≤), λ is called fuzzy increasing Gδ/Fσ if
λ = ∧∞
i=1λi/if λ = ∨∞
i=1λi where each λi is fuzzy increasing open/closed
in (X, T,≤). The complement of fuzzy increasing Gδ/Fσ-set is fuzzy
decreasing Fδ/Gσ.
384 Tietze Extension Theorem...
Definition 2.2. Let λ be any fuzzy set in the ordered fuzzy topological
space (X, T,≤). Then we define
Iσ(λ) = fuzzy increasing σ-closure of λ.
= the smallest fuzzy increasing Fσ-set containing λ.
Dσ(λ) = fuzzy decreasing σ-closure of λ.
= the smallest fuzzy decreasing Fσ-set containing λ.
I0
σ(λ) = fuzzy increasing σ-interior of λ.
= the greatest fuzzy increasing Gδ-set contained in λ.
D0
σ(λ) = fuzzy decreasing σ-interior of λ.
= the greatest fuzzy decreasing Gδ-set contained in λ.
Prorosition 2.1. For any fuzzy set λ of an ordered fuzzy topological
space (X, T,≤), the following equalities are valid.
(a) 1 − Iσ(λ) = D0
σ(1 − λ).
(b) 1 − Dσ(λ) = I0
σ(1 − λ).
(c) 1 − I0
σ(λ) = Dσ(1 − λ).
(d) 1 − D0
σ(λ) = Iσ(1 − λ).
Proof. We shall prove (a) only, (b), (c), and (d) can be proved in a similar
manner.
(a) Since Iσ(λ) is a fuzzy increasing Fσ-set containing λ, 1− Iσ(λ) is
a fuzzy decreasing Gδ-set such that 1 − Iσ(λ) ≤ 1 − λ. Let µ be another
fuzzy decreasing Gδ-set such that µ ≤ 1 − λ. Then 1 − µ is a fuzzy
increasing Fσ-set such that 1 − µ ≥ λ. It follows that Iσ(λ) ≤ 1 − µ.
That is, µ ≤ 1 − Iσ(λ). Thus, 1 − Iσ(λ) is the largest fuzzy decreasing
Gδ-set such that 1−Iσ(λ) ≤ 1−λ. That is, 1−Iσ(λ) = 1−D0
σ(1−λ).
Definition 2.3. Let (X, T,≤) be an ordered fuzzy topological space. Let
λ be any fuzzy increasing Gδ-set in (X, T,≤). If Iσ(λ) is fuzzy increasing
Gδ-set in (X, T,≤), then (X, T,≤) is said to be upper fuzzy Gδ-extremally
disconnected. Similarly we can define lower fuzzy Gδ-extremally discon-
nected space. (X, T,≤) is said to be ordered fuzzy Gδ-extremally discon-
nected if it is both upper and lower fuzzy Gδ-extremally disconnected.
Example 2.1. Let X = {a, b, c} and T = {0, 1, λ1, λ2, λ3, λ4} where
λ1 : X → [0, 1] is such that λ1(a) = 0, λ1(b) = 1/4, λ1(c) = 3/4,
λ2 : X → [0, 1] is such that λ2(a) = 1, λ2(b) = 3/4, λ2(c) = 3/4,
E. Roja, M. K. Uma, G. Balasubramanian 385
λ3 : X → [0, 1] is such that λ3(a) = 1, λ3(b) = 3/4, λ3(c) = 1/4,
and
λ4 : X → [0, 1] is such that λ4(a) = 0, λ4(b) = 1/4, λ4(c) = 1/4.
The partial order ”≤“ is defined as a ≤ b, b ≤ c. Then (X, T,≤) is an
ordered fuzzy topological space. It is clear that (X, T,≤) is an ordered
fuzzy Gδ-extremally disconnected space.
Prorosition 2.2. For an ordered fuzzy topological space (X, T,≤), the
following statements are equivalent.
(a) (X, T,≤) is upper fuzzy Gdelta-extremally disconnected.
(b) For each fuzzy decreasing Fσ-set λ, D0
σ(λ) is a decreasing fuzzy
Fσ-set.
(c) For each fuzzy increasing Gδ-set λ, we have
Iσ(λ) + Dσ(1 − Iσ(λ)) = 1.
(d) For each pair of fuzzy increasing Gδ-set λ and a fuzzy decreasing
Gδ-set µ in (X, T,≤) with Iσ(λ) + µ = 1, we have
Iσ(λ) + Dσ(µ) = 1.
Proof. (a) ⇒ (b). Let λ be any fuzzy decreasing Fσ-set. We claim D0
σ(λ)
is a fuzzy decreasing Fσ-set. Now 1− λ is fuzzy increasing Gδ and so by
assumption (a), Iσ(1−λ) is fuzzy increasing Gδ. That is, D0
σ(λ) is fuzzy
decreasing Fσ.
(b) ⇒ (c). Let λ be any fuzzy increasing Gδ-set. Then,
1 − Iσ(λ) = D0
σ(1 − λ). (2.1)
Consider Iσ(λ) + Dσ(1 − Iσ(λ)) = Iσ(λ) + Dσ(D0
σ(1 − λ)). As λ is any
fuzzy increasing Gδ-set, 1 − λ is fuzzy decreasing Fσ and by assumption
(b), D0
σ(1 − λ) is fuzzy decreasing Fσ. Therefore,
Dσ(D0
σ(1 − λ)) = D0
σ(1 − λ).
Now,
Iσ(λ) + Dσ(D0
σ(1 − λ)) = Iσ(λ) + D0
σ(1 − λ) = 1.
That is,
Iσ(λ) + Dσ(1 − Iσ(λ)) = 1.
386 Tietze Extension Theorem...
(c) ⇒ (d). Let λ be any fuzzy increasing Gδ-set and µ be any fuzzy
decreasing Gδ-set such that
Iσ(λ) + µ = 1. (2.2)
By assumption (c),
Iσ(λ) + Dσ(1 − Iσ(λ)) = 1
= Iσ(λ) + µ.
(2.3)
That is, µ = Dσ(1 − Iσ(λ)). Since µ = 1 − Iσ(λ),
Dσ(µ) = Dσ(1 − Iσ(λ)). (2.4)
From (2.3) and (2.4)
Iσ(λ) + Dσ(µ) = 1.
(d) ⇒ (a). Let λ be any fuzzy increasing Gδ-set. Put µ = 1 − Iσ(λ).
Clearly, µ is fuzzy decreasing Gδ-set and from the construction of µ it
follows that Iσ(λ)+µ = 1. By assumption (d), we have Iσ(λ)+Dσ(µ) = 1
and so Iσ(λ) = 1 −Dσ(µ) is fuzzy increasing Gδ. Therefore, (X, T,≤) is
upper fuzzy Gδ-extremally disconnected.
Prorosition 2.3. Let (X, T,≤) be an ordered fuzzy topological space.
Then (X, T,≤) is an upper fuzzy Gδ-extremally disconnected space ⇔
for fuzzy decreasing Gδ-set λ and fuzzy decreasing Fσ-set µ such that
λ ≤ µ, we have Dσ(λ) ≤ D0
σ(µ).
Proof. Suppose (X, T,≤) is an upper fuzzy Gδ-xtremally disconnected
space. Let λ be any fuzzy decreasing Gδ-set such that λ ≤ µ. Then
by (b) of Proposition 2.2, D0
σ(µ) is fuzzy decreasing Fσ. Also, since λ is
fuzzy decreasing Gδ and λ ≤ µ, it follows that λ ≤ D0
σ(µ). Again, since
D0
σ(µ) is fuzzy decreasing Fσ, it follows that Dσ(λ) ≤ D0
σ(µ).
To prove the converse, let µ be any fuzzy decreasing Fσ-set. By
Definition 2.2, D0
σ(µ) is fuzzy decreasing Gδ and it is also clear that
D0
σ(µ) ≤ µ. Therefore by assumption, it follows that Dσ(D0
σ(µ)) ≤
D0
σ(µ). This implies that D0
σ(µ) is fuzzy decreasing Fσ. Hence by (b)
of Proposition 2.2, it follows that (X, T,≤) is upper fuzzy Gδ-extremally
disconnected.
Remark 2.1. Let (X, T,≤) be an upper fuzzy Gδ-extremally discon-
nected space. Let {λi, 1 − µi : i ∈ N} be a collection such that λi, i ∈ N
are fuzzy decreasing Gδ-sets and µi, i ∈ N are fuzzy decreasing Fσ-sets.
E. Roja, M. K. Uma, G. Balasubramanian 387
Let λ, 1 − µ be fuzzy decreasing Gδ-set and fuzzy increasing Gδ-set re-
spectively. If λi ≤ λ ≤ µj and λi ≤ µ ≤ µj for all i, j ∈ N, then there
exists a fuzzy decreasing GδFσ-set γ such that
Dσ(λi) ≤ γ ≤ D0
σ(µj) for all i, j ∈ N.
By Proposition 2.3,
Dσ(λi) ≤ Dσ(λ) ∧ D0
σ(µ) ≤ D0
σ(µj) (i, j ∈ N).
Put γ = Dσ(λ) ∧ D0
σ(µ). Now γ satisfies our required condition.
Prorosition 2.4. Let (X, T,≤) be an ordered fuzzy Gδ-extremally dis-
connected space. Let {λq}q∈Q and {µq}q∈Q be monotone increasing col-
lections of fuzzy decreasing Gδ-sets and fuzzy decreasing Fσ-sets of
(X, T,≤) respectively and suppose that λq1
≤ µq2
whenever q1 < q2 (Q
is the set of rational numbers). Then there exists a monotone increasing
collection {γq}q∈Q of fuzzy decreasing GδFσ-sets of (X, T,≤) such that
Dσ(λq1
) ≤ γq2
and γq1
≤ D0
σ(µq2
) whenever q1 < q2.
Proof. Let us arrange into sequence {qn} of rational numbers without
repetitions. For every n ≥ 2, we shall define inductively a collection
{γqi
: 1 ≤ i ≤ n} ⊂ IX such that
Dσ(λq) ≤ γqi
if q < qi,
γqi
≤ D0
σ(µq) if qi < q,
(Sn)
for all i < n.
By Proposition 2.3, the family {Dσ(λq)} and {D0
σ(µq)} satisfying
Dσ(λq1
) ≤ D0
σ(µq2
) if q1 < q2. By Remark 2.1, there exists fuzzy de-
creasing GδFσ-set δ1 such that
Dσ(λq1
) ≤ δ1 ≤ D0
σ(µq2
).
Setting γq1
= δ1 we get (S2). Assume that fuzzy sets γqi
are already
defined for i < n and satisfy (Sn). Define
Σ = ∨{γqi
: i < n, qi < qn} ∨ λqn
and
Φ = ∧{γqj
: j < n, qj > qn} ∧ µqn .
Then we have that
Dσ(γqi
) ≤ Dσ(Σ) ≤ D0
σ(γqj
)
388 Tietze Extension Theorem...
and
Dσ(γqi
) ≤ Dσ(Φ) ≤ D0
σ(γqj
)
whenever qi < qn < qj (i, j < n) as well as λq ≤ Dσ(Σ) ≤ µq′ and
λq ≤ D0
σ(Φ) ≤ µq′ whenever q < qn < q′. This shows that the countable
collection {γqi
: i < n, qi < qn} ∪ {λq : q < qn} and {γqi
: j < n, qj >
qn} ∪ {µq : q > qn} together with Σ and Φ fulfill all conditions of the
mentioned Remark 2.1. Hence, there exists a fuzzy decreasing GδFsigma-
set δn such that
Dσ(δn) ≤ µq if qn < q,
λq ≤ Dσ0(δn) if q < qn,
Dσ(γqi
) ≤ D0
σ(δn) if qi < qn,
Dσ(δn) ≤ D0
σ(γqj
) if qn < qj ,
where 1 ≤ i, j ≤ n − 1. Now, setting γqn = δn we obtain the fuzzy sets
γq1
, γq2
, . . . , γqn that satisfy (Sn+1). Therefore, the collection {γqi
: i =
1, 2, . . .} has the required property. This completes the proof.
Definition 2.4. Let (X, T,≤) and (Y, S,≤) be ordered fuzzy topological
spaces. A mapping f : (X, T,≤) → (Y, S,≤) is called fuzzy increas-
ing/decreasing Gδ-continuous if f−1(λ) is fuzzy increasing/decreasing
Gδ-set of (X, T,≤) for every fuzzy Gδ-set λ of (Y, S,≤). If f is both
fuzzy increasing and fuzzy decreasing Gδ-continuous, then it is called or-
dered fuzzy Gδ-continuous.
Definition 2.5. Let (X, T,≤) be an ordered fuzzy topological space. A
function f : X → R(I) is called lower fuzzy Gδ-continuous if f−1(Rt) is
fuzzy increasing of fuzzy decreasing Gδ for each t ∈ R and upper fuzzy
Gδ-continuous if f−1(Lt) is fuzzy increasing of fuzzy decreasing Gδ for
each t ∈ R.
Lemma 2.1. Let (X, T,≤) be an ordered fuzzy topological space, let λ ∈
IX , and let f : X → R(I) be such that
f(x)(t) =
1 if t < 0,
λ(x) if 0 ≤ t ≤ 1,
0 if t > 1,
for all x ∈ X. Then f is lower/upper fuzzy Gδ-continuous iff λ is fuzzy
increasing of decreasing Gδ/Fσ-set.
E. Roja, M. K. Uma, G. Balasubramanian 389
Proof. If suffices to observe that
f−1(Rt) =
1 if t < 0,
λ if 0 ≤ t ≤ 1,
0 if t ≥ 1
and
f−1(Lt′) =
1 if t ≤ 0,
λ if 0 < t ≤ 1,
0 if t > 1.
Thus proved.
Definition 2.6. The characteristic function of λ ∈ IX is the map χλ :
X → [0, 1](I) defined by χλ(x) = (λ(x)), x ∈ X [10].
Lemma 2.2. Let (X, T,≤) be an ordered fuzzy topological space, let λ ∈
IX . Then χλ is lower/upper fuzzy Gδ-continuous iff λ is fuzzy increasing
or decreasing Gδ/Fσ-set.
Proof. Proof is similar to Lemma 2.1.
Prorosition 2.5. Let (X, T,≤) be an ordered fuzzy topological space.
Then the following statements are equivalent.
(a) (X, T,≤) is upper fuzzy Gδ-extremally disconnected.
(b) If g, h : X → R(I), g is lower fuzzy Gδ-continuous, h is upper fuzzy
Gδ-continuous and g ≤ h, then there exists an fuzzy increasing Gδ-
continuous function f : (X, T,≤) → R(I) such that g ≤ f ≤ h.
(c) If 1−λ is fuzzy increasing Gδ-set, µ is fuzzy decreasing Gδ-set and
µ ≤ λ, then there exists fuzzy increasing Gδ-continuous function
f : (X, T,≤) → [0, 1](I) such that µ ≤ (1 − L1)f ≤ R0f ≤ λ.
Proof. (a) ⇒ (b). Define Hr = Lrh and Gr = (1−Rr)g, r ∈ Q. Thus we
have two monotone increasing families of respectively fuzzy decreasing
Gδ-sets and fuzzy decreasing Fσ-sets of (X, T,≤). Moreover, Hr ≤ Gs
if r < s. By Proposition 2.4, there exists a monotone increasing family
{Fr}r∈Q of fuzzy decreasing GδFσ-sets of (X, T,≤) such that Dσ(Hr) ≤
Fs and Fr ≤ D0
σ(Gs) whenever r < s. Letting Vt = ∧r<t(1 − Fr) for
all t ∈ R, we define a monotone decreasing family {Vt : t ∈ R} ⊂ IX .
390 Tietze Extension Theorem...
Moreover, we have Iσ(Vt) ≤ I0
σ(Vs), whenever s < t. We have
∨
t∈R
Vt =
∨
t∈R
∧
r<t
(1 − Fr) ≥
∨
t∈R
∧
r<t
(1 − Gr)
=
∨
t∈R
∧
r<t
g−1(Rr) =
∨
t∈R
g−1(Rt) = g−1
(
∨
t∈R
Rt
)
= 1.
Similarly, ∧t∈RVt = 0.
We now define a function f : (X, T,≤) → R(I) satisfying the required
properties. Let f(x)(t) = Vt(x) for all x ∈ X and t ∈ R. By the above
discussion, it follows that f is well defined. To prove f is fuzzy increasing
Gδ-continuous, we observe that
∨
s>t
Vs =
∨
s>t
I0
σ(Vs),
∧
s<t
Vs =
∧
s<t
Iσ(Vs).
Then
f−1(Rt) =
∨
s>t
Vs =
∨
s>t
I0
σ(Vs)
is fuzzy increasing Gδ. Now
f−1(1 − Lt) =
∧
s<t
Vs =
∧
s<t
Iσ(Vs)
is fuzzy increasing Fσ so that f is fuzzy increasing Gδ-continuous. To
conclude the proof it remains to show that g ≤ f ≤ h, that is g−1(1 −
Lt) ≤ f−1(1 − Lt) ≤ h−1(1 − Lt) and g−1(Rt) ≤ f−1(Rt) ≤ h−1(Rt) for
each t ∈ R.
We have
g−1(1 − Lt) =
∧
s<t
g−1(1 − Ls) =
∧
s<t
∧
r<s
g−1(Rr)
=
∧
s<t
∧
r<s
(1 − Gr) ≤
∧
s<t
∧
r<s
(1 − Fr) =
∧
s<t
Vs = f−1(1 − Lt)
and
f−1(1 − Lt) =
∧
s<t
Vs =
∧
s<t
∧
r<s
(1 − Fr) ≤
∧
s<t
∧
r<s
(1 − Hr)
=
∧
s<t
∧
r<s
h−1(1 − Lr) =
∧
s<t
h−1(1 − Ls) = h−1(1 − Lt).
E. Roja, M. K. Uma, G. Balasubramanian 391
Similarly, we obtain
g−1(Rt) =
∨
s>t
g−1(Rs) =
∨
s>t
∨
r>s
g−1(Rr) =
∨
s>t
∨
r>s
(1 − Gr)
≤
∨
s>t
∧
r<s
(1 − Fr) =
∨
s>t
Vs = f−1(Rt)
and
f−1(Rt) =
∨
s>t
Vs =
∨
s>t
∧
r<s
(1 − Fr) ≤
∨
s>t
∨
r>s
(1 − Hr)
=
∨
s>t
∨
r>s
h−1(1 − Lr) =
∨
s>t
h−1(Rs) = h−1(Rt).
Thus, (b) is proved.
(b) ⇒ (c). Suppose 1 − λ is fuzzy increasing Gδ and µ is fuzzy
decreasing Gδ, such that µ ≤ λ. Then χµ ≤ χλ, χµ and χλ are lower and
upper fuzzy Gδ-continuous functions respectively. Hence by (b), there
exists fuzzy increasing Gδ continuous function f : (X, T,≤) → R(I)
such that χµ ≤ f ≤ χλ. Clearly, f(x) ∈ [0, 1](I) for all x ∈ X and
µ = (1 − L1)χµ ≤ (1 − L1)f ≤ R0f ≤ R0χλ = λ.
(c) ⇒ (a). This follows from Proposition 2.3, and the fact that
(1−L1)f and R0f are fuzzy decreasing Fσ and fuzzy decreasing Gδ-sets
respectively. Hence the result.
Remark 2.2. Propositions 2.2–2.5 and Remark 2.1 can be discussed for
other cases also.
3. Tietze Extension Theorem for Ordered
Fuzzy Gδ-extremally Disconnected Spaces
In this section, Tietze extension theorem for ordered fuzzy Gδ-extrem-
ally disconnected space is studied.
Prorosition 3.1 (Tietze Extension Theorem). Let (X, T,≤) be an
upper fuzzy Gδ-extremally disconnected space and let A ⊂ X be such that
χA is fuzzy increasing Gδ in (X, T,≤). Let f : (A, T/A) → [0, 1](I) [6] be
an increasing fuzzy Gδ-continuous function. Then f has an increasing
fuzzy Gδ-continuous extension over (X, T,≤).
Proof. Let g, h : X → [0, 1](I) be such that
g = f = h on A and g(x) = 〈0〉, h(x) = 〈1〉 if x 6∈ A.
392 Tietze Extension Theorem...
We now have
Rtg =
{
µt ∧ χA if t ≥ 0
1 if t < 0
where µt is fuzzy increasing Gδ such that
µt/A = Rtf
and
Lth =
{
λt ∧ χA if t ≤ 1
1 if t > 1
where λt is increasing fuzzy Gδ such that
λt/A = Ltf.
Thus, g is lower fuzzy Gδ-continuous, h is upper fuzzy Gδ-continuous and
g ≤ h. By Proposition 2.5, there exists an increasing fuzzy Gδ-continuous
function F : X → [0, 1](I) such that g ≤ F ≤ h; hence F ≡ f on A.
Remark 3.1. The above proposition can be discussed for other cases
also.
References
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nected spaces // Kybernetica, 33 (1997), 271–277.
[4] G. Balasubramanian and P. Sundaram, On some generalizations of fuzzy contin-
uous functions // Fuzzy Sets and Systems, 86 (1997), 93–100.
[5] C. L. Chang, Fuzzy topological spaces // J. Math. Anal. Appl., 24 (1968), 182–190.
[6] A. K. Katsaras, Ordered fuzzy topological spaces // J. Math. Anal. Appl., 84
(1981), 44–58.
[7] P. Smets, The degree of belief in a fuzzy event // Information Sciences, 25 (1981),
1–19.
[8] M. Sugeno, An intoductory survey of fuzzy control // Information Sciences, 36
(1985), 59–83.
[9] G. Thangaraj and G. Balasubramanian, On fuzzy basically disconnected spaces //
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[10] Tomasz Kubiak, Extending continuous L-real functions // Math. Japonica, 6
(1986), 875–887.
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E. Roja, M. K. Uma, G. Balasubramanian 393
Contact information
Elangovan Roja,
Mallasamudram
Kuppuratnam Uma
Department of Mathematics
Sri Sarada College for Women
Salem–636 016 Tamil Nadu,
India
E-Mail: sudha_nice@yahoo.com
Ganesan
Balasubramanian
University of Madras
Chennai–600 005 Tamil Nadu,
India
E-Mail: rpbalan@sancharnet.in
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