Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 436570, 9 pages
doi:10.1155/2008/436570
Research Article
Fuzzy Inverse Compactness
Halis Aygün,1 A. Arzu Bural,2 and S. R. T. Kudri3
1
Department of Mathematics, Faculty of Arts and Sciences, Kocaeli University, Kocaeli 41380, Turkey
Department of Mathematics, Faculty of Education, Kocaeli University, Kocaeli 41380, Turkey
3
Department of Mathematics, Federal University of Parana, P.O. Box 019081, Curitiba,
PR 81531-990, Brazil
2
Correspondence should be addressed to Halis Aygün, halis@kou.edu.tr
Received 9 February 2007; Revised 25 November 2007; Accepted 14 January 2008
Recommended by Heinz Gumm
We introduce definitions of fuzzy inverse compactness, fuzzy inverse countable compactness, and
fuzzy inverse Lindelöfness on arbitrary L-fuzzy sets in L-fuzzy topological spaces. We prove that
the proposed definitions are good extensions of the corresponding concepts in ordinary topology
and obtain different characterizations of fuzzy inverse compactness.
Copyright q 2008 Halis Aygün et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
In ordinary topology, Matveev 1 has introduced a topological property called inverse compactness which is weaker than compactness and stronger than countable compactness. In 1, 2,
inverse countable compactness and inverse Lindelöfness have been defined and studied.
A topological space X is called inversely compact if and only if for every open cover
β of X, one can select a finite subcover γ of X which consists of the elements of β or their
complements but of course γ is prohibited to contain both U and X \ U for any U ∈ β.
In L-fuzzy topological spaces fuzzy compactness has been introduced by Warner and
McLean 3 and extended to arbitrary L-fuzzy sets by Kudri 4.
In this paper, we initiate fuzzy inverse compactness in L-fuzzy topological spaces which
is weaker than fuzzy compactness and introduce fuzzy inverse countable compactness and
fuzzy inverse Lindelöfness. We prove that proposed definitions are good extensions of the
corresponding notions in ordinary topology.
2. Preliminaries
We assume that the reader is familiar with the usual notations and most of the concepts of
fuzzy topology and lattice theory.
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Throughout this paper X and Y will be nonempty ordinary sets and L L≤, ∨, ∧, will
denote a completely distributive lattice with a smallest element 0 and a largest element 1 0 /
1
and with an order-reversing involution a → a a ∈ L. τ and T will denote L-fuzzy topology
and ordinary topology, respectively.
Definition 2.1 see 5. An element p of a lattice L is called prime if and only if p /
1 and
whenever a, b ∈ L with a ∧ b ≤ p then a ≤ p or b ≤ p.
The set of all prime elements of a lattice L will be denoted by prL.
Definition 2.2 see 5. An element α of a lattice L is called coprime or union irreducible if
and only if α /
0 and whenever a, b ∈ L with α ≤ a ∨ b then α ≤ p or α ≤ b.
The set of all coprime elements of a lattice L will be denoted by ML.
From the definitions we have that p ∈ prL ⇔ p ∈ ML.
Definition 2.3 see 3. Let L be a complete lattice. A subset U of L is called Scott open if and
only if it is an upper set and is inaccessible by directed joins, that is,
a if a ∈ U and a ≤ b, then b ∈ U;
b if D is a directed subset of L with
D ∈ U, then there is a d ∈ D with d ∈ U.
The collection of all Scott-open subsets of L is a topology on L and is called Scott topology of
L. The Scott topology of completely distributive lattice L is generated by the sets of the form
{x ∈ L | x p}, where p ∈ prL. This means that the family {{x ∈ L | x p} | p ∈ prL}
forms a base for Scott topology on L.
A function f from X, T to L with its Scott topology is Scott continuous if and only if
f −1 {x ∈ X | x p} ∈ T for every p ∈ prL.
Definition 2.4 see 6. An L-fuzzy topology on X is a subset τ of LX satisfying the following
properties:
i the constant functions 0 and 1 belong to τ;
ii if f, g ∈ τ, then f ∧ g ∈ τ;
iii if {fi : i ∈ J} ⊆ τ, then i∈J fi ∈ τ.
The pair X, τ is called an L-fuzzy topological space. The elements of τ are called open L-fuzzy
sets. An L-fuzzy set h is said to be closed if and only if h ∈ τ.
Definition 2.5 see 6. Let X, τ and Y, τ ∗ be L-fuzzy topological spaces. A function ϕ :
X, τ → Y, τ ∗ is called fuzzy continuous if and only if ϕ−1 g ∈ τ for every g ∈ τ ∗ . ϕ−1 g gϕx.
Definition 2.6 see 7. Let X, T be an ordinary topological space. The set of all Scott continuous functions from X, T to L with its Scott topology forms an L-fuzzy topology on X, which
will be denoted by ωL T , that is,
ωL T f ∈ LX | f : X, T −→ L is Scott continuous .
2.1
Halis Aygün et al.
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Definition 2.7 see 7. Let X, T be an ordinary topological space. An L-fuzzy topological property Pf is a ”good extension” of a topological property P if and only if the topological space X, T has P if and only if the induced L-fuzzy topological space X, ωL T has Pf .
Definition 2.8 see 4. Let X, τ be an L-fuzzy topological space, g ∈ LX and p ∈ L. A collection β fi i∈J of open L-fuzzy sets is called a p-level open cover of the L-fuzzy set g if and
only if i∈J fi x p for all x ∈ X with gx ≥ p .
If g is the whole space X 1, then β is called a p-level open cover of X.
Thus β is called a p-level open cover of X if and only if f∈β fx p for all
x ∈ X.
Definition 2.9 see 4. Let X, τ be an L-fuzzy topological space and g ∈ LX . The L-fuzzy set
g is said to be fuzzy compact if and only if every p-level open cover of g, where p ∈ prL, has
a finite p-level subcover of g.
If g is the whole space X, then X, τ is called fuzzy compact L-fuzzy topological
space.
3. Proposed definitions
Definition 3.1. Let X be nonempty set and let β, γ ⊆ LX . β is called a partial inversement of γ if
and only if β and γ can be indexed with the same index set, sayJ : β {fi : i ∈ J}, γ {gi : i ∈ J}
so that for every i ∈ J either fi g or fi gi .
Definition 3.2. Let X, τ be an L-fuzzy topological space and g ∈ LX . g is said to be fuzzy
inversely compact if and only if every p-level open cover of g, where p ∈ prL, has a partial
inversement which contains a finite p-level cover of g.
If g is the whole space X, then X, τ is called fuzzy inverse compact L-fuzzy topological
space.
This definition can be restated as follows. g is fuzzy inverse compact if and only if for
every p-level open cover β {fi : i ∈ J} of g, where p ∈ prL, there exists a finite p-level
subcover γ {gi : i ∈ J} of g such that there exist g1 , g2 , . . . , gn ∈ γ : ni1 gi x p for every
x ∈ X with gx ≥ p , where for each i 1, 2, . . . , n, gi fi , or gi fi .
Clearly, every fuzzy compact L-fuzzy set is inversely fuzzy compact.
Definition 3.3. Let X, τ be an L-fuzzy topological space and g ∈ LX . g is said to be fuzzy
inversely countably compact if and only if for every countable p-level open cover of g, where
p ∈ prL, has a partial inversement which contains a finite p-level cover of g.
If g is the whole space, then we say that the L-fuzzy topological space X, τ is fuzzy
inversely countably compact.
Definition 3.4. Let X, τ be an L-fuzzy topological space and g ∈ LX . g is said to be fuzzy
inversely Lindelöf if and only if every p-level open cover of g, where p ∈ prL, has a partial
inversement which contains a countable p-level cover of g.
If g is the whole space, then we say that the L-fuzzy topological space X, τ is fuzzy
inversely Lindelöf.
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From the definitions, it can be easily verified that every fuzzy inversely compact L-fuzzy
set g is fuzzy inversely countably compact and fuzzy inversely Lindelöf.
4. Other characterizations
Definition 4.1. Let α ∈ L, ζ ⊆ LX , and g ∈ LX . ζ is called an α-level centered family of g if and
only if for any h1 , h2 , . . . , hn ∈ ζ there exists x ∈ X with gx ≥ α such that h1 ∧h2 ∧· · ·∧hn x ≥ α.
Theorem 4.2. Let X, τ be an L-fuzzy topological space and g ∈ LX . g is fuzzy compact if and only
if for every α-level centered family ζ of closed L-fuzzy sets, where α ∈ ML, there exists x ∈ X with
gx ≥ α such that h∈ζ hx ≥ α.
Proof
Necessity. Let α ∈ ML and let ζ be α-level centered family of closed L-fuzzy sets such that for
each x ∈ X with gx ≥ α, there exists h ∈ ζ with hx α.
Then, h∈ζ hx α for all x ∈ X with gx ≥ α.
Hence h∈ζ h x p for all x ∈ X with gx ≥ p , where p α .
Thus, β {h : h ∈ ζ} is a p-level open cover of g that has no finite p-level subcover of
g. In fact, if h1 , h2 , . . . , hn ∈ β, then, since ζ is α-level centered, there exists x ∈ X with gx ≥ α
and ni1 hi x ≥ α; hence ni1 hi x ≤ p.
Sufficiency. Suppose that β is a p-level open cover of g with no finite p-level subcover of g,
where p ∈ prL. Then, ζ {f : f ∈ β} is a collection of closed L-fuzzy sets. Moreover, ζ is an
α-level centered family of g, where α p . In fact, if f1 , f2 , . . . , fn ∈ ζ, then there exists x ∈ X
with gx ≥ p α such that ni1 fi x ≤ p; hence ni1 fi x ≥ α.
By the hypothesis, there exists x ∈ X with gx ≥ p with such that f∈β f x ≥ p ;
hence f∈β fx ≤ p which yields a contradiction.
Definition 4.3. Let ζ ⊆ LX , g ∈ LX , and α ∈ L.
ζ is called an α-level independent family of g if and only if for any finite f1 , f2 , . . . , fn ,
g 1 , g2 , . . . , gm ∈ ζ there exists x ∈ X with gx ≥ α such that ni1 fi ∧ m
i1 gi x ≥ α.
In other words, ζ is an α-level independent family of g if and only if for every nonempty
finite partial inversement ζ∗ of ζ, there exists x ∈ X with gx ≥ α and h∈ζ∗ hx ≥ α.
Theorem 4.4. Let X, τ be an L-fuzzy topological space and g ∈ LX . g is fuzzy inversely compact if
and only if for every α-level independent family ζ of closed L-fuzzy sets, where α ∈ ML, there exists
x ∈ X with gx ≥ α and h∈ζ hx ≥ α.
Proof
Necessity. Let α ∈ ML. Suppose that ζ is an α-level independent family of L-fuzzy sets, such
that h∈ζ hx α for every x ∈ X with gx ≥ α.
Then, h∈ζ h x α for all x ∈ X with gx ≥ α.
Hence, β {h : h ∈ ζ} is a p-level open cover of g, where p α .
By the fuzzy inverse compactness of g, there is a partial inversement γ of β which contains a finite p-level subcover of g.
Let γ {gi : i ∈ J} and β {hi : i ∈ J, hi ∈ ζ}.
Then, there exists g1 , g2 , . . . , gn ∈ γ such that ni1 gi x p for all x ∈ X with gx ≥ p ,
where for each i ∈ {1, 2, . . . , n}, gi hi , or gi hi .
Halis Aygün et al.
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Let g1 h1 , g2 h2 , . . . , gk hk , gk1 hk1 , gk2 hk2 , . . . , gn hn . Then g1 , g2 ,
gk1 , . . . , gn ∈ ζ. Since ζ is an α-level independent family of g, there exists x ∈ X with
gx ≥ α such that
. . . , gk ,
k
n
n
gi ∧
gi x gi x ≥ α.
i1
ik1
4.1
i1
n Hence, there exists x ∈ X with gx ≥ p such that
i1 gi x ≤ p, which yields a
contradiction.
Sufficiency. Suppose that β is a p-level open cover of g with no partial inversement which contains a finite p-level subcover of g, where p ∈ prL. Then, ζ {f : f ∈ β} is a collection of
closed L-fuzzy sets. Furthermore, ζ is an α-level independent family of g, where α p . In fact,
if f1 , f2 , . . . , fk , fk1
, . . . , fn ∈ ζ, then by the assumption, there exists x ∈ X with gx ≥ p and
n
k
i1 fi ∨ ik1 fi x ≤ p; hence ki1 fi ∧ nik1 fi x ≥ α.
By the hypothesis, there exists z ∈ X with gz ≥ α and f∈β f z ≥ α; hence
f∈β fz ≤ p, which yields a contradiction.
Theorem 4.5. Let X, τ be an L-fuzzy topological space and g ∈ LX . g is fuzzy inversely countably
compact if and only if for every countable α-level independent family ζ of closed L-fuzzy sets, where
α ∈ ML, there exists x ∈ X with gx ≥ α and h∈β hx ≥ α.
Proof. This is similar to Theorem 4.4.
Definition 4.6. Let ξ ⊆ LX and p ∈ prL. ξ is said to have the finite union property for short
FUP in G {x ∈ X : gx ≥ p } if and only if for any finite f1 , f2 , . . . , fk , g1 , g2 , . . . , gn ∈ ξ, there
exists x ∈ G with ki1 fi ∨ ni1 gi x ≤ p.
Theorem 4.7. Let X, τ be an L-fuzzy topological space and g ∈ LX . g is fuzzy inversely compact if
and only if no ξ ⊆ τ with the FUP in G satisfies f∈ξ fx p for all x ∈ G.
Proof
Necessity. Suppose that ξ ⊆ τ has the FUP in G such that f∈ξ fx p for all x ∈ G.
Then, ζ {f : f ∈ ξ} is an α-level independent family of closed L-fuzzy sets, where
α p . From Theorem 4.4, there exists x ∈ X with gx ≥ α and f∈ζ f x ≥ α; hence
f∈ξ fx ≤ p which yields a contradiction.
Sufficiency. Suppose that g is not fuzzy inverse compact. Then, there is an α-level independent
family ζ of closed L-fuzzy sets such that h∈ζ hx α for all x ∈ X with gx ≥ α, where
α ∈ ML. Hence h∈ζ h x p for all x ∈ X with gx ≥ p , where p α . Moreover,
ξ {h : h ∈ ζ} is a family of open L-fuzzy sets having the FUP in G. This completes the
proof.
5. Some properties
The next theorem shows that fuzzy inverse compactness is a good extension of inverse compactness in general topology.
Theorem 5.1. Let X, T be an ordinary topological space. X, T is inversely compact if and only if the
L-fuzzy topological space X, ωL T is fuzzy inversely compact.
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Proof
Sufficiency. Let α ∈ ML and ζ be an α-level independent family of closed subsets of X. Then,
ζ∗ {χA : A ∈ ζ} is a family of closed L-fuzzy sets. Furthermore, ζ∗ is an α-level independent
family of X. In fact, if χA1 , χA2 , . . . , χAn , χB1 ,..., χBm ∈ ζ∗ , then since ζ is independent, ni1 Ai ∩
m n
m φ. Hence i1 χAi ∧ i1 χBi χni1 Ai ∩mi1 Bi /
0. So, there is x ∈ X such that ni1 χAi ∧
Bi /
i1
m χ x 1 ≥ α. Since X, ωL T is fuzzy inversely compact, there exists z ∈ X with
i1 Bi
A∈ζ χA z ≥ α; hence A∈ζ χA z χA∈ζ A z 1 and therefore z ∈ A∈ζ A. Hence X, T is inversely compact.
Necessity. Let p ∈ prL and β fi i∈J be a p-level open cover of X consisting of basic open
L-fuzzy sets in X, ωL T . Then
⎧
⎨ei ∈ L; x ∈ Ai ∈ T,
fi x ⎩0;
x/
∈ Ai
5.1
for each i ∈ J and i∈J fi x p for all x ∈ X.
Let ζ {Ai ⊆ X : ∃i ∈ J with ei p and fi ∈ β}.
It is clear that ζ is an open cover of X in X, T . Due to inverse compactness of X, T ,
there exists a partial inversement ζ∗ of ζ which contains a finite cover of X.
Let ζ∗ {Bi : i ∈ J}. Then there exist B1 , B2 , . . . , Bn ∈ ζ∗ such that X ni1 Bi , where
Bi Ai and Bi Ai for each i ∈ {1, 2, . . . , n}.
i If Bi Ai for all i ∈ {1, 2, . . . , n}, then β∗ {f1 , f2 , . . . , fn } is a finite partial inversement of β. β∗ is also a p-level subcover of X in X, ωL T . Hence, X, ωL T is fuzzy
inversely compact.
ii Suppose that there exists i0 ∈ {1, 2, . . . , n} such that Bi0 Ai0 .
Let β∗ {f1 , f2 , . . . , fi0 −1 , fi0 , . . . , fn }. Obviously, β∗ is a partial inversement of β. Fur
thermore, β∗ is a p-level subcover of X in X, ωL T . In fact, let x ∈ X. Since X ni1 Bi n
n
Bi0 ∪ i1,i / i0 Bi ⇒ x ∈ Bi0 or x ∈ i1,i / i0 Bi ⇒ x ∈ Ai0 or x ∈ ni1,i / i0 Ai .
1◦ If x ∈ Ai0 , then fi0 x 1 p. Hence fi0 x ∨ ni1,i / i0 fi x p.
2◦ If x /
∈ Ai0 , then there is i ∈ {1, 2, . . . , n} − {i0 } such that x ∈ Ai . Hence fi x ei p and therefore fi0 ∨ ni1,i / i0 fi x p. Consequently, β∗ is p-level subcover of X in
X, ωL T .
Theorem 5.2 the goodness of fuzzy inverse countable compactness. Let X, T be an ordinary
topological space. X, T is inversely countably compact if and only if the L-fuzzy topological space
X, ωL T is fuzzy inversely countably compact.
Proof. This is similar to the proof of the goodness of fuzzy inverse compactness.
Theorem 5.3 the goodness of fuzzy inverse Lindelöfness. Let X, T be an ordinary topological
space. X, T is an inverse Lindelöf space if and only if the L-fuzzy topological space X, ωL T is fuzzy
inverse Lindelöf space.
Proof. This is similar to the proof of the goodness of fuzzy inverse compactness.
Theorem 5.4. Let X, τ be an L-fuzzy topological space. If g ∈ LX is fuzzy inversely compact and
h ∈ LX closed, then g ∧ h is also fuzzy inversely compact.
Halis Aygün et al.
7
Proof. Let p ∈ prL and let β fi i∈J ⊆ LX be a p-level open cover of g ∧h; that is, i∈J fi x p for all x ∈ X with g ∧ hx ≥ p . So β∗ fi i∈J ∪ {h } is p-level family of g. In fact, if
hx ≥ p ⇒
fi x p
∀ x ∈ X with g ∧ hx ≥ p
i∈J
⇒
k x fi x ∨ h x p
k∈β∗
i∈J
hx p ⇒ h x p ⇒
5.2
k x p.
k∈β∗
Since g is fuzzy inversely compact, β∗ has a partial inversement γ ∗ which contains a finite
p-level subcover of g.
Let γ ∗ {gi : i ∈ J} ∪ {h } and β∗ {fi : i ∈ J} ∪ {h }.
Then there exists g1 , g2 , . . . , gn ∈ γ ∗ such that ni1 gi ∨ h x p for all x ∈ X with
gx ≥ p where for each i ∈ {1, 2, . . . , n}, gi fi or gi fi .
So ni1 gi x p for all x ∈ X with g ∧ hx ≥ p . In fact x ∈ X with g ∧ hx ≥ p ⇒
gx ≥ p and hx ≥ p ,
gx ≥ p ⇒
n
gi ∨ h x p
i1
n
gi x ∨ h x p
⇒
5.3
i1
⇒
n
gi x p.
i1
Corollary 5.5. Let X, τ be an L-fuzzy topological space. If g is a fuzzy inversely compact L-fuzzy set
and h is a closed L-fuzzy set with h ≤ g, then h is fuzzy inversely compact as well.
Proof. This follows directly from the previous theorem.
Corollary 5.6. Let X, τ be an L-fuzzy topological space. If X, τ is fuzzy inversely compact, then
every closed L-fuzzy set on X is fuzzy inversely compact.
Proof. This follows from Theorem 5.4.
Proposition 5.7. Let X, τ be an L-fuzzy topological space and g, h ∈ LX with h ≤ g. If g is fuzzy inversely countably compact (fuzzy inversely Lindelöf) and h is closed, then h is fuzzy inversely countably
compact (fuzzy inversely Lindelöf).
Proof. This is similar to Theorem 5.4.
Theorem 5.8. Let X, τ be an L-fuzzy topological space and g, h ∈ LX . If g and h are fuzzy inversely
compact, then g ∨ h is fuzzy inversely compact as well.
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International Journal of Mathematics and Mathematical Sciences
Proof. Suppose that g ∨ h is not fuzzy inverse compact.
Then, there exists an α-level independent family ζ of g ∨ h consisting of closed L-fuzzy
sets such that f∈ζ fx ≥ α for all x ∈ X with g ∨ hx ≥ α, where α ∈ ML.
Since α ∈ ML we have g ∨ hx ≥ α ⇔ gx ≥ α or hx ≥ α.
Let f1 , f2 , . . . , fk , fk1 , . . . , fn ∈ ζ. Since ζ is an α-level independent family of g ∨ h, there
exists x ∈ X with g ∨ hx ≥ α and ki1 fi ∧ nik1 fi x ≥ α. That is, there exists x ∈ X with
n
k
gx ≥ α or hx ≥ α such that i1 fi ∧ ik1 fi x ≥ α.
Assume that gx ≥ α. Then, ζ is α-level independent family of g consisting of closed
L-fuzzy sets. Furthermore, f∈ζ fx α for all x ∈ X with gx ≥ α.
So, g is not fuzzy inversely compact.
Proposition 5.9. Let X, τ be an L-fuzzy topological space and g, h ∈ LX . If g and h are fuzzy
inversely countably compact (fuzzy inversely Lindelöf), then g ∨ h is fuzzy inversely countably compact
(fuzzy inversely Lindelöf).
Proof. This is similar to Theorem 5.8.
Theorem 5.10. Let X, τ and Y, τ ∗ be L-fuzzy topological spaces and let ϕ : X, τ → Y, τ ∗ be a
fuzzy continuous function such that ϕ−1 y is finite for every y ∈ Y . If g ∈ LX is fuzzy inversely com
pact in X, τ, then ϕg ∈ LY is fuzzy inversely compact in Y, τ ∗ , where ϕgy x∈ϕ−1 y gx.
Proof. Let α ∈ ML and ζ is an α-level independent family of ϕg consisting of closed L-fuzzy
sets in Y, τ ∗ . Let ζ∗ {ϕ−1 h : h ∈ ζ}.
Since ϕ is fuzzy continuous, ζ∗ is a family of closed L-fuzzy sets in X, τ. Furthermore, ζ∗ is an α-level independent family of g. In fact, if ϕ−1 h1 , ϕ−1 h2 , . . . , ϕ−1 hk ,
ϕ−1 hk1 , . . . , ϕ−1 hn ∈ ζ∗ , then there exists y ∈ Y with ϕgy ≥ α and ki1 hi ∧ nik1 hi y ≥
α because ζ is an α−level independent family of ϕg. ϕgy x∈ϕ−1 y gx ≥ α implies
that there exists x ∈ X with gx ≥ α and ϕx y because α ∈ ML and ϕ−1 y is finite.
Hence, ki1 ϕ−1 hi ∧ nik1 ϕ−1 hi x ki1 hi ∧ nik1 hi ϕx ≥ α. From the fuzzy in
verse compactness of g, there exists z ∈ X with gz ≥ α and h∈ζ ϕ−1 hz ≥ α. Then,
h∈ζ hϕz ≥ α. This means that ϕg is fuzzy inversely compact in Y, τ ∗ .
Theorem 5.11. Let X, τ and Y, τ ∗ be L-fuzzy topological spaces and let ϕ : X, τ → Y, τ ∗ be
a fuzzy continuous function such that ϕ−1 y is finite for every y ∈ Y . If g ∈ LX is fuzzy inversely
countable compact (fuzzy inversely Lindelöf) in X, τ, then ϕg ∈ LY is fuzzy inversely countably
compact (fuzzy inversely Lindelöf) in Y, τ ∗ .
Proof. This is similar to the proof of Theorem 5.10.
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