9_Breuckmann.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 2, No. 1, 2009, (147-161)

ISSN 1307-5543 – www.ejpam.com

Local Compactness in L-Fuzzy Spaces

T.K. Breuckmann1, S.R.T. Kudri1, and H. Aygün2∗

1 Department of Mathematics, Federal University of Paraná, P. O. Box 019081, Curitiba, PR, 81531-

990, Brazil

2 Department of Mathematics, Kocaeli University, 41380, izmit, Turkey, Fax: +90-262-3032003

Abstract. In an L-topological space we present good definitions for local compactness, weak local

compactness and relative local compactness. We obtain the equivalence of these properties in a Haus-

dorff space and we also obtain a one point compactification theorem.

Key words: Fuzzy lattice, L-topology, local compactness, weak local compactness, relative local com-

pactness

1. Introduction

In general topology there are three usual ways to define local compactness, which ones

we call here local compactness, weak local compactness and relative local compactness.

Definition 1.1. Let 〈X ,δ〉 be a topological space. We say that 〈X ,δ〉 is:

(i) locally compact if and only if for each x ∈ X and V ∈ δ with x ∈ V there exist U ∈ δ and a

compact subset K of X with x ∈ U and U ⊂ K ⊂ V .

(ii) weakly locally compact if and only if for each x ∈ X there exist U ∈ δ and a compact subset

K of X with x ∈ U and U ⊂ K.

∗Corresponding author. Email address: halis�ko
aeli.edu.tr (H. Aygün)

http://www.ejpam.com 147 c© 2009 EJPAM All rights reserved.



T. Breuckmann, S. Kudri, and H. Aygün / Eur. J. Pure Appl. Math, 2 (2009), (147-161) 148

(iii) relatively locally compact if and only if for each x ∈ X there exist U ∈ δ with x ∈ U and U

compact.

In this paper we present a generalization for an L topological spaces of these three proper-

ties. We show the goodness of the proposed definitions, the equivalence in Hausdorff spaces

and present a one point compactification theorem.

2. Preliminaries

Throughout this paper X and Y are assumed nonempty ordinary sets, and L = L



≤,∨,∧,′
�

always will denote a fuzzy lattice with its Scott topology, i.e., a complete completely distribu-

tive lattice with a smallest element 0 and a greatest element 1 (0 6= 1), with an order reversing

involution a→ a′, and the topology is generated by the sets
�

x ∈ L ; x � p
	

where p ∈ pr(L)

is a prime element of L, details in [1]. If A⊂ X we denote by χA the characteristic function of

A in X .

We denote by LX the set of functions f : X → L called L-sets. An L-point in X is an L-set

xp : X → L, where x ∈ X and p ∈ pr(L), defined by: xp(y) = p if y = x , and xp(y) = 1

otherwise. We say that xp ∈ f if and only if f (x)� p, see [6].

Let 〈X ,δ〉 be a topological space. In [7], Warner proved that the set ω(δ) formed by the

continuous functions f : X → L is an L-topology. The base for the space ω(δ) is formed by

the functions

f (x) =







b if x ∈ V ∈ δ

0 if x /∈ V

This provides a ”goodness of extension” criterion for L-topological spaces.

Definition 2.1. [4, Pu and Liu] Let 〈X , T 〉 be an L topological space and let f ∈ LX . The closure

of f , cl( f ) or f , is the L-set defined by:

cl( f ) = ∧
¦

g ∈ LX ; f ≤ g, g′ ∈ T
©



T. Breuckmann, S. Kudri, and H. Aygün / Eur. J. Pure Appl. Math, 2 (2009), (147-161) 149

Definition 2.2. [2, Kudri] Let 〈X , T 〉 be an L-topological space and let g ∈ LX . We say that

g is compact if and only if for each p ∈ pr(L) and each family
¦

f j

©

j∈J
of open L-sets such

that
�

∨ j∈J f j

�

(x) � p for all x ∈ X with g(x) ≥ p′, there exist a finite set J1 of J such that
�

∨ j∈J1
f j

�

(x)� p for all x ∈ X with g(x)≥ p′.

Theorem 2.1. [5, Warner and McLean] Let 〈X ,δ〉 be a topological space. Then: 〈X ,δ〉 is

compact if and only if 〈X ,ω(δ)〉 is compact.

Proposition 2.1. [2, Kudri] Let 〈X , T 〉 be an Hausdorff L-topological space and F ⊂ X . If χF is

compact in 〈X , T 〉 then χF is closed.

Proposition 2.2. [2, Kudri] Let 〈X , T 〉 be an L-topological space. If g ∈ LX is a compact L-set,

then for each closed L-set h ∈ LX , h∧ g is a compact L-set.

Proposition 2.3. [2, Kudri] Let



X , TX

�

and



Y, TY

�

be L-topological spaces and let f : X → Y

be a continuous mapping. If g ∈ LX is a compact L-set, then f (g) ∈ LY is a compact L-set.

Proposition 2.4. [2, Kudri] Let
¦¬

X j, T j

¶©

j∈J
be a family of L-topological spaces and g j ∈ LX j

be a compact L-set for each j ∈ J. Then the product set g = ∧ j∈Jπ
−1
j
(g j) is a compact L-set in

the product space.

Proposition 2.5. [2, Kudri] Let S be a subbase for the L-topology T in X and let g ∈ LX . If for

each p ∈ pr(L) and each family
¦

f j

©

j∈J
of sub basis open L-sets with

�

∨ j∈J f j

�

(x) � p for all

x ∈ X with g(x)≥ p′ there exists a finite subset F of J with
�

∨ j∈F f j

�

(x)� p for all x ∈ X with

g(x)≥ p′, then, g is compact in 〈X , T 〉.

Definition 2.3. [5, Warner and McLean] An L-topological space 〈X , T 〉 is Hausdorff if and only

if for every p,q ∈ pr(L) and every x 6= y in X there exist f , g ∈ T such that f (x)� p, g(y) � q

and, f (z) = 0 or g(z) = 0 for all z ∈ X .

Theorem 2.2. [5, Warner and McLean] Let 〈X ,δ〉 be a topological space. Then: 〈X ,δ〉 is

Hausdorff if and only if 〈X ,ω(δ)〉 is Hausdorff.

Theorem 2.3. [5, Warner and McLean] If 〈X , T 〉 is a compact Hausdorff fully stratified L-

topological space then it’s topological, thats it, there is a topology δ ∈ X such that T =ω(δ).



T. Breuckmann, S. Kudri, and H. Aygün / Eur. J. Pure Appl. Math, 2 (2009), (147-161) 150

Definition 2.4. [5, Warner and McLean] An L-topological space 〈X , T 〉 is regular if and only

if for every p ∈ pr(L), for each x ∈ X and each closed L-set f such that there is y ∈ X with

f (y) ≥ p′ and f (x) = 0, there are u, v ∈ T with u(x) � p, v(z) � p for each z ∈ X with

f (z) ≥ p′, and, u(z) = 0 or v(z) = 0 for each z ∈ X .

Theorem 2.4. [2, Kudri] If 〈X ,δ〉 is a compact Hausdorff L-topological space then 〈X , T 〉 is

regular.

3. Proposed definitions and their goodness theorems

Definition 3.1. An L-topological space 〈X , T 〉 is locally compact if and only if for each x ∈ X ,

p ∈ pr(L) and f ∈ T with f (x) � p there exist g ∈ T and k ∈ LX , with χsupp(k) compact, such

that f (x)� p and g ≤ k ≤ f .

Theorem 3.1. (The goodness of local compactness) Let 〈X ,δ〉 be an topological space. Then:

〈X ,δ〉 is locally compact if and only if 〈X ,ω(δ)〉 is locally compact.

Proof. Necessity: Let x ∈ X , let p ∈ pr(L) and let f ∈ ω(δ) such that f (x) � p. Let

h ∈ω(δ) be an basic open L-set with h(x)� p and h≤ f defined by

h(y) =







e if y ∈ V ∈ δ

0 if y /∈ V

Since 〈X ,δ〉 is locally compact, there exist U ∈ δ and a compact subset J of X such that

x ∈ U and U ⊂ J ⊂ V .

Let g ∈ω(δ) and K ∈ LX defined by:

g(y) =







e if y ∈ U ∈ δ

0 if y /∈ U
k(y) =







e if y ∈ J ∈ δ

0 if y /∈ J

Then g(x) � p, g ≤ k ≤ h ≤ f and χsupp(k) = χJ is compact since J is compact. Hence,

〈X ,ω(δ)〉 is locally compact.



T. Breuckmann, S. Kudri, and H. Aygün / Eur. J. Pure Appl. Math, 2 (2009), (147-161) 151

Suficiency: Let x ∈ X and V ∈ δ such that x ∈ V . Fix p ∈ pr(L). Since 〈X ,ω(δ)〉 is

locally compact, for f = χV , there exist g ∈ ω(δ) and k ∈ LX , with χsupp(k) compact, such

that g(x)� p and g ≤ k ≤ f .

Let U = g−1
�

t ∈ L ; t � p
	

and let K = supp(k), then, U ∈ δ, x ∈ U , K is a compact

subset of X since χsupp(k) is compact and U ⊂ K ⊂ V . Hence, 〈X ,δ〉 is locally compact.

Definition 3.2. An L-topological space 〈X , T 〉 is weakly locally compact if and only if for each

x ∈ X and p ∈ pr(L) there exist f ∈ T and k ∈ LX , with χsupp(k) compact, such that f (x) � p

and f ≤ k.

Theorem 3.2. (The goodness of weak local compactness) Let 〈X ,δ〉 be an topological space.

Then: 〈X ,δ〉 is weakly locally compact if and only if 〈X ,ω(δ)〉 is weakly locally compact.

Proof. Necessity: Let x ∈ X and let p ∈ pr(L). Since 〈X ,δ〉 is weakly locally compact, there

exist U ∈ δ and a compact subset J of X such that x ∈ U and U ⊂ J .

Let g = χU and let K = χJ , then g ∈ ω(δ), g(x) � p, g ≤ k and χsupp(k) = χJ is compact

since J is compact. Hence, 〈X ,ω(δ)〉 is weakly locally compact.

Suficiency: Let x ∈ X and fix p ∈ pr(L). Since 〈X ,ω(δ)〉 is locally compact there exist

g ∈ω(δ) and k ∈ LX , with χsupp(k) compact, such that g(x)� p and g ≤ k.

Let V = g−1
�

t ∈ L ; t � p
	

and let K = supp(k), then, V ∈ δ, x ∈ V , K is a compact

subset of X since χsupp(k) is compact and V ⊂ K . Hence, 〈X ,δ〉 is weakly locally compact.

Definition 3.3. An L-topological space 〈X , T 〉 is relatively locally compact if and only if for each

x ∈ X and p ∈ pr(L) there exists f ∈ T, with χ
supp( f )

compact, such that f (x)� p.

Theorem 3.3. (The goodness of relative local compactness) Let 〈X ,δ〉 be an topological space.

Then: 〈X ,δ〉 is relatively locally compact if and only if 〈X ,ω(δ)〉 is relatively locally compact.

Proof. Necessity: Let x ∈ X an let p ∈ pr(L). Since 〈X ,δ〉 is relatively locally compact

there is V ∈ δ with x ∈ V and V compact. Let f = χV , then f (x) = 1 � p. We also have that

f = χV , hence supp( f ) = V is compact, therefore χ
supp( f )

is compact.



T. Breuckmann, S. Kudri, and H. Aygün / Eur. J. Pure Appl. Math, 2 (2009), (147-161) 152

Suficiency: Let x ∈ X and let p ∈ pr(L) be fixed. Since 〈X ,ω(δ)〉 is relatively locally

compact there is f ∈ ω(δ), with χ
supp( f )

compact, such that f (x) � p, hence supp( f ) is

compact. Let g ∈ LX a basic open L-set, g(x)� p and g ≤ f , defined by

g(y) =







e if y ∈ V ∈ δ

0 if y /∈ V

Since g ≤ f and

g(y) =







e se y ∈ V ∈ δ

0 se y /∈ V

we have g ≤ f and V = supp(g)⊂ supp( f ), thus, V is compact since it is closed and supp( f )

is compact.

4. Some properties and Comparison

Theorem 4.1. Let



X , TX

�

be a locally compact L-topological space and let



Y, TY

�

be an L-

topological space. If h : X → Y is a continuous open surjection then



Y, TY

�

is locally compact.

Proof. Let y ∈ Y with y = h(x), let p ∈ pr(L) and f ∈ TY with f (y) � p. Let j = h−1( f ),

then j ∈ TX since h is continuous and j(x) = f (y) � p. Since



X , TX

�

is locally compact there

exist i ∈ TX and c ∈ LX , withχsupp(c) compact, such that i(x)� p and i � c � j.

Let g = h( j) and let k = h(c). Then g ∈ TY since h is open and g ≤ k ≤ f since i � c � j.

Since h is continuous and χsupp(c) is compact we have h(χsupp(c)), but:

h(χsupp(c)) = χh(supp(c)) = χsupp(h(c)) = χsupp(k)

Hence,



Y, TY

�

is locally compact.

Theorem 4.2. Let



X , TX

�

be a weakly locally compact L-topological space and let



Y, TY

�

be an

L-topological space. If h : X → Y is a continuous open surjection then



Y, TY

�

is weakly locally

compact.



T. Breuckmann, S. Kudri, and H. Aygün / Eur. J. Pure Appl. Math, 2 (2009), (147-161) 153

Proof. Let y ∈ Y with y = h(x) and let p ∈ pr(L). Since



X , TX

�

is weakly locally compact

there exist i ∈ TX and c ∈ LX , with χsupp(c) compact, such that i(x)� p and i � c.

Let g = h( j) and let k = h(c). Then g ∈ TY since h is open and g ≤ k since i � c. Since h

is continuous and χsupp(c) is compact we have h(χsupp(c)), but:

h(χsupp(c)) = χh(supp(c)) = χsupp(h(c)) = χsupp(k)

Hence,



Y, TY

�

is weakly locally compact.

Theorem 4.3. Let



X , TX

�

be a relatively locally compact L-topological space and let



Y, TY

�

be

an L-topological space. If h : X → Y is a continuous open surjection with h(g) ≤ h(g) for every

g ∈ LX , then,



Y, TY

�

is relatively locally compact.

Proof. Let y ∈ Y with y = h(x) and let p ∈ pr(L). Since



X , TX

�

is relatively locally com-

pact there is g ∈ TX , with χsupp(g) compact, such that g(x)≤ p. Let f = h(g), then: f (x)� p,

f ∈ TY since h is open, and h(χsupp(g)) is a compact L-set in LY since h  continuous. But:

h(χsupp(g)) = χh(supp(g)) = χsupp(h(g)) = χsupp(h(g))
= χ

supp( f ))

where the last equality is due to the continuity of h and the condition mention in theorem.

Hence,



Y, TY

�

is relatively locally compact.

Theorem 4.4. Let 〈X , T 〉 be a locally compact L-topological space, then 〈X , T 〉 is weakly locally

compact.

Proof. Let x ∈ X and let p ∈ pr(L). Since 〈X , T 〉 is locally compact, for f = X , there exist

g ∈ T and k ∈ LX , with χsupp(k) compact, such that g(x) � p and g ≤ k ≤ f . So 〈X , T 〉 is

weakly locally compact.

Theorem 4.5. If 〈X , T 〉 is a compact Hausdorff fully stratified L-topological space then 〈X , T 〉 is

locally compact.

Proof. Since 〈X , T 〉 a compact Hausdorff fully stratified L-topological space there is a

topology δ in X such that T = ω(δ). By theorems 2.1 and 2.2 we have that 〈X ,δ〉 is a



T. Breuckmann, S. Kudri, and H. Aygün / Eur. J. Pure Appl. Math, 2 (2009), (147-161) 154

compact Hausdorff topological space, hence it’s locally compact. by theorem 3.1, 〈X , T 〉 is

locally compact.

Theorem 4.6. Let 〈X , T 〉 be a weakly locally compact Hausdorff fully stratified L-topological

space, then 〈X , T 〉 is locally compact.

Proof. Let x ∈ X , let p ∈ pr(L) and let f ∈ T such that f (x)� p. We must show that there

exist g ∈ T and k ∈ LX , with χsupp(k) compact, such that g(x)� p and g ≤ k ≤ f .

Since 〈X , T 〉 is weakly locally compact there exist i ∈ T and j ∈ LX , with χsupp( j) compact,

such that i(x) � p and i ≤ j. Let D = supp( j). Since χsupp( j) is compact and 〈X , T 〉 is Haus-

dorff fully stratified, the subspace



D, TD

�

is a compact Hausdorff fully stratified L-topological

space, then, by theorem 4.5 it’s locally compact, hence for fD = f |D there exist hD ∈ TD and

c ∈ LD, with χsupp(c) compact, such that hD ≤ cD ≤ fD and hD(x)� p.

Let h ∈ T such that h|D = hD and define k ∈ LX by

k(y) =







cD(y) if y ∈ D

0 if y /∈ D

then, h(x)� p and χsupp(k) is compact since supp(k) = supp(cD).

Let g = h∧ j, then g ∈ T and g(x)� p. We proof now that g ≤ k ≤ f , in fact, if y ∈ D then

g(y) ≤ h(y) ≤ k(y) ≤ f (y) since hD ≤ cD ≤ fD, and if y /∈ D then j(y) = 0 and k(y) = 0, so

g(y) = 0= k(y) ≤ f (y).

Theorem 4.7. Let 〈X , T 〉 be a relatively locally compact L-topological space, then 〈X , T 〉 is

weakly locally compact.

Proof. Let x ∈ X and let p ∈ pr(L). Since 〈X , T 〉 is relatively locally compact there exists

g ∈ T , with χsupp(g) compact, such that g(x) � p. Since g ≤ g, 〈X , T 〉 is weakly locally

compact.

Theorem 4.8. Let 〈X , T 〉 be a weakly locally compact Hausdorff fully stratified L-topological

space such that χ
supp( f )

= χsupp( f ), then it’s relatively locally compact.



T. Breuckmann, S. Kudri, and H. Aygün / Eur. J. Pure Appl. Math, 2 (2009), (147-161) 155

Proof. Let x ∈ X and let p ∈ pr(L). Since 〈X , T 〉 is weakly locally compact there exist

f ∈ T and k ∈ LX , with χsupp(k) compact, such that f (x) � p and f ≤ k. Since χsupp(k) is

a compact L-set in a Hausdorff space, it’s closed, by proposition 2.1, so, χsupp(k) = χsupp(k).

Since f ≤ k, χsupp( f ) ≤ χsupp(k) then χsupp( f ) ≤ χsupp(k), hence χsupp( f ) is a compact L-set since

it’s closed and χsupp(k) is compact, by proposition 2.2. But χsupp( f ) = χsupp( f )
, then χ

supp( f )
is

a compact L-set. Therefore 〈X , T 〉 is relatively locally compact.

Theorem 4.9. Let
�

Xλ
	

λ∈J be a family of nonempty fully stratified L-topological spaces. Then:

The product L-topological space
∏

λ∈J Xλ is locally compact if and only if each Xλ is locally

compact and all but finitely many Xλ are compact.

Proof. Necessity: Since the λth projection, πλ :
∏

λ∈J Xλ → Xλ, is a continuous open

surjection and
∏

λ∈J Xλ is locally compact, by theorem 4.1, Xλ is locally compact for each

λ ∈ J . Now, let p ∈ pr(L), x ∈
∏

λ∈J Xλ and let F be an open L-set in
∏

λ∈J Xλ with

f (x) � p. Then by the local compactness of
∏

λ∈J Xλ, there are an open L-set g in
∏

λ∈J Xλ

with g(x)� p and an L-set k in
∏

λ∈J Xλ with χsupp(k) compact such that g ≤ k ≤ f .

Let ∧m
i=1π

−1
λi
(gλi
) be a basic open L-set such that ∧m

i=1π
−1
λi
(gλi
)≤ g ≤ k ≤ f . Then

χsupp(k) ≥ χsupp(∧m
i=1
π−1
λi
(gλi
)) = χ∩m

i=1
supp(π−1

λi
(gλi
)) = ∧

m
i=1χsupp(π−1

λi
(gλi
))

= ∧m
i=1π

−1
λi
(χsupp(gλi

))

Thus πλ(supp(k)) ≥ πλ(∧
m
i=1
π−1
λi
(χsupp(gλi

))) = Xλ for all λ /∈
�

λ1, · · · ,λm

	

. Since πλ is

continuous, χsupp(k) is compact in
∏

λ∈J Xλ and πλ(χsupp(k)) = Xλ, we have by proposition

2.3 that Xλ is compact for each λ except possibly λ ∈
�

λ1, · · · ,λm

	

.

Sufficiency: Let p ∈ pr(L), x ∈
∏

λ∈J Xλ and f ∧m
λ=1
π−1
λi
( fλi
) be a basic open L-set in the

product L-topological space
∏

λ∈J Xλ such that f (x) � p where fλi
is an open L-set in Xλi

.

We assume that
�

λ1, · · · ,λm

	

is expanded to include all λ for which Xλ is not compact.

We have that f (x) � p implies fλi
(xλi
) � p for all i ∈ {1, · · · , m}. From the local com-

pactness of each Xλi
, there are an open L-set gλi

in Xλi
and an L-set kλi

in Xλi
, with χsupp(kλi

)

compact, such that gλi
(xλi
)� p and gλi

≤ kλi
≤ fλi

.



T. Breuckmann, S. Kudri, and H. Aygün / Eur. J. Pure Appl. Math, 2 (2009), (147-161) 156

Let g = ∧m
i=1
π−1
λi
(gλi
) and k = ∧m

i=1
π−1
λi
(kλi
), then, g is an open L-set in

∏

λ∈J Xλ, g ≤

k ≤ f and

g(x) = ∧m
i=1π

−1
λi
(gλi
)(x) = ∧m

i=1 gλi
(xλi
)� p

We also have

χsupp(k) = χsupp(∧m
i=1
π−1
λi
(kλi
)) = ∧

m
i=1π

−1
λi
(χsupp(kλi

)) = ∧λ∈J wλ

where wλi
= π−1

λi
(χsupp(kλi

)) for i ∈ {1, · · · , m} and wλ = Xλ for λ /∈ {1, · · · , m}. Then χsupp(k)

is a compact L-set in
∏

λßJ Xλ by proposition 2.4 since χsupp(kλi
) is compact for each i ∈

{1, · · · , m} and Xλ is compact for each λ /∈ {1, · · · , m}.

Theorem 4.10. Let
�

Xλ
	

λ∈J be a family of nonempty fully stratified L-topological spaces. Then:

The product L-topological space
∏

λ∈J Xλ is weakly locally compact if and only if each Xλ is

weakly locally compact and all but finitely many Xλ are compact.

Proof. The proof is analogous to the theorem 4.9, so we just give the outline for the proof.

Necessity: The weak local compactness of X j is by theorem 4.2. For the rest, use the

weak local compactness to obtain an open L-set g in
∏

j∈J X j and an L-set k in
∏

j∈J X j, with

χsupp(k) compact, such that g(x)� p and g ≤ k.

Sufficiency: For p ∈ pr(L) and x ∈
∏

j∈J X j use the weak local compactness of X j, j /∈
�

j1, · · · jm
	

where the set is the index where X j is not compact, to obtain g ji
in
∏

j∈J X j and

an L-set k ji
in
∏

j∈J X j, with χsupp(k ji
) compact, such that g ji

(x)� p and g ji
≤ k ji

. For the rest

just take g = ∧m
i=1π

−1
λi
(gλi
) and k = ∧m

i=1π
−1
λi
(kλi
).

Theorem 4.11. If 〈X , T 〉 is a Hausdorff weakly locally compact L-topological space then 〈X , T 〉

is regular.

Proof. let x ∈ X , let p ∈ pr(L) and let h be a closed L-set such that h(x) = 0 and there

exists y0 ∈ X with h(y0)≥ p′.

Let’s show that there are u, v ∈ T such that u(x) � p, v(y) � p for each y ∈ X with

h(y)≥ p′, and, u(z) = 0 or v(z) = 0 for each z ∈ X .



T. Breuckmann, S. Kudri, and H. Aygün / Eur. J. Pure Appl. Math, 2 (2009), (147-161) 157

Since 〈X , T 〉 is weakly locally compact there are f ∈ T and h ∈ LX , with χ
supp(k)

compact,

such that f (x)� p and f ≤ k. let D = supp(k), then:

(1) x ∈ D since f (x)� p and f ≤ k.

(2) f (z) = 0 for each z ∈ Dc since f ≤ k and k(z) = 0 for each z ∈ Dc.

(3) Since χD is a compact L-set and 〈X , T 〉 is Hausdorff we have that χD is a closed L-set,

then χ ′D = χDc ∈ T is an open L-set.

since χD is compact and 〈X , T 〉 is Hausdorff, the L-topological space



D, TD

�

is compact

and Hausdorff, hence



D, TD

�

is regular by theorem 2.4.

Case 1: There exist y ∈ D such that h(y) ≥ p′.

In this case, by (1) and by regularity of



D, TD

�

, there are uD, vD ∈ TD such that uD(x)�

p, vD(z) � p for each z ∈ D with h(z) ≤ p, and, uD(z) = 0 or vD(z) = 0 for each z ∈ D.

Let u∗, v∗ ∈ T such that u∗|D = uD and v∗|D = vD, and define u = u∗∨ f and v = v∗∧χDc .

Then we have:

(a) u ∈ T since u∗, f ∈ T , and v ∈ T since v∗ ∈ T an by (3).

(b) Since x ∈ D, u∗(x) = uD(x)� p, then u(x)� p since f (x)� p and p ∈ pr(L).

(c) Let z ∈ X such that h(z) ≥ p′.

z ∈ D ⇒ v(z) = v∗(z) = vD(z) � p

z ∈ Dc ⇒ χDc(z) = 1� p⇒ v(z) = χDc (z)� p

(d) Let z ∈ X such that u(z) 6= 0, then, u∗(z) 6= 0 and f (z) 6= 0. By (2), z ∈ D, then

uD(z) = u∗(z) 6= 0, hence v∗(z) = vD(z) = 0, so

v(z) = v∗(z)∧χDc (z) = 0∧ 0= 0

Case 2: There is not y ∈ D such that h(y)≥ p′.

Let u= f and v = χDc , then u, v ∈ T and:



T. Breuckmann, S. Kudri, and H. Aygün / Eur. J. Pure Appl. Math, 2 (2009), (147-161) 158

(a) u(x) = f (x)� p

(b) let z ∈ X such that h(z) ≥ p′, then z ∈ Dc since in this case there is not z ∈ D with

h(z) ≥ p′, hence χDc(z) = 1, so v(z) = 1� p.

(c) Let z ∈ X such that u(z) = f (z) 6= 0, then by (2), z ∈ D, hence v(z) = χDc (z) = 0.

Therefore 〈X , T 〉 is regular.

It’s immediate that locally compact and relatively locally compact Hausdorff spaces are

regular since these spaces are weakly locally compact by theorems 4.4 and 4.7.

5. The one point compactification

The following is based on [3].

Let



X , TX

�

be a Hausdorff L topologiacal space which is not compact, but weakly locally

compact. Let Y = X∪{∞}with L topology TY generated by the subbase S=
¦

f1 ∈ LY ; f ∈ TX

©

∪
¦

χB∞
∈ LY ; χB ∈ C
©

, where:

(i) f1 ∈ LX is defined by:

f1(y) =







f (y) if y ∈ X

0 if y =∞

(ii) C =
¦

χB ∈ LX ; B ⊂ X , χB compacto
©

(iii) For χB ∈ C, define B∞ = {∞} ∪ (X − B) and:

χB∞
(y) =







1 if y ∈ B∞

0 if y ∈ B.

The L topological space



Y, TY

�

is called the one point compactification of



X , TX

�

.

Theorem 5.1. Let



X , TX

�

be a weakly locally compact Hausdorff L-topological space which is

not compact, and let



Y, TY

�

be their one point compactification. Then,



Y, TY

�

is a compact

Hausdorff L-topological space, cl(X ) = Y and



X , TX

�

is a subspace of



Y, TY

�

.



T. Breuckmann, S. Kudri, and H. Aygün / Eur. J. Pure Appl. Math, 2 (2009), (147-161) 159

Proof.

(i)



X , TX

�

is a subspace of



Y, TY

�

. In fact, given g ∈ TY , g|X ∈ TX .

(ii) cl(X ) = Y . In fact, if cl(X ) 6= Y , then cl(X ) is an L-set of the form

cl(X )(y) =







1 if y ∈ X

l 6= 1 if y =∞

The complement of cl(X ) is the open L-set

cl(X )′(y) =







0 if y ∈ X

l′ 6= 0 if y =∞

Let f = χB1∞
∧ · · · ∧ χBn∞

be a basic open L-set such that f ≤ cl(X )′ where B1, · · · , Bn

are subsets of X with the L-sets χB1
, · · · ,χB1

compacts. We have for y =∞ that f (y) =

1≤ l′, then l′ = 1, so l = 0.

We also have:

f ≤ cl(X )′ ⇒ χB1∞
∧ · · · ∧χBn∞

≤ cl(X )′

⇒ cl(X ) ≤ χ ′B1∞
∨ · · · ∨χ ′Bn∞

.

Since X ≤ cl(X ) and χ ′Bi∞
|X = χBi

we have X ≤ χB1
∨· · ·∨χBn

, thus, X = χB1
∨· · ·∨χBn

.

Hence, X is compact which leads to a contradiction .

(iii)



Y, TY

�

is compact. In fact, let p ∈ pr(L) and B =
¦

f j

©

j∈J
be a family of subbasis open

L-sets with
�

∨ j∈J f j

�

(y) � p for each y ∈ Y . Then there is j ∈ J such that f j = χB∞

with B ⊂ X and χB compact, since in the other side,
�

∨ j∈J f j

�

(∞) = 0≤ p.

Let B1 =
¦

f j |X
©

j∈J1
where J1 =
¦

j ∈ J ; f j 6= χB∞

©

, then B1 is such that
�

∨ j∈J1
f j |X
�

(x)�

p for each x ∈ X with χB(x)≥ p′.

Since χB is compact there is a finite subset J2 of J1 such that
�

∨ j∈J2
f j |X
�

(x) � p for

each x ∈ X with χB(x) ≥ p′. Then,
�

χB∞
∨∨ j∈J2

f j

�

(y) � p for each y ∈ Y . Hence by

proposition 2.5



Y, TY

�

is compact.



REFERENCES 160

(iv)



Y, TY

�

 Hausdorff. In fact, let x � y in Y and p,q ∈ pr(L). If x , y ∈ X , since X is

Hausdorff, there exist f , g ∈ TX with f (x)� p, g(y) � q, and, f (z) = 0 or g(z) = 0 for

each z ∈ X . Then f1 ∈ TY , g1 ∈ TY , f1(x) = f (x)� p, g1(y) = g(y) � q, and, f1(z) = 0

or g1(z) = 0 for each z ∈ Y .

If x ∈ X and y = ∞, since X is weakly locally compact there are f ∈ TX and k ∈ LX ,

with χsupp(k) compact, such that f (x)� p and f ≤ k. Let B = supp(k) and

f1(y) =







f (y) if y ∈ X

0 if y =∞

then f1 ∈ TY and χB∞
∈ TY .

It follow that f1(x) = f (x)� p and χB∞
(y) = 1� q. Also:

(a) z ∈ B⇒ χB∞
(z) = 0

(b) z ∈ X − B = X − supp( f )⇒ f (z) = 0⇒ f1(z) = f (z) = 0

(c) z =∞⇒ f1(z) = 0

hence for each z ∈ Y , f1(z) = 0 or χB∞
(z) = 0.

These conditions proof the theorem.

By an analogous way we can obtain one point compactification theorems for locally com-

pact an relatively locally compact spaces since by theorems 4.4 and 4.7 these space are weakly

locally compact.

References

[1] G. Gierz et al., A compendium of continuous lattices, Springer-Verlag, 1980.

[2] S.R.T. Kudri. “Compacness in L-fuzzy topological spaces”, Fuzzy sets and systems 67, (1994)

329-446.

[3] S.R.T. Kudri. “L-fuzzy local compactness”, Fuzzy sets and systems 67, (1994) 337-345.

[4] P.M. Piu, Y.M. Liu, “Fuzzy topology I, neighbourhood structure of a fuzzy point and Moore-Smith

convergence”, J.Math.Anal.Appl. 76, (1980)



REFERENCES 161

[5] M. W. Warner and R. G. McLean, “On compact hausdorff L-fuzzy spaces”, Fuzzy Sets and Systems

56, (1993) 103-110.

[6] M.W. Warner, Frame-fuzzy points and memberschip, Fuzzy Sets and Systems 42 (1991) 335-344.

[7] M.W. Warner, Fuzzy topology with respect to continuous lattices, Fuzzy Sets and Systems 35

(1990) 85-91.