AndriAGT.dvi


@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 10, No. 1, 2009

pp. 29-37

∗-half completeness in quasi-uniform spaces

Athanasios Andrikopoulos

Abstract. Romaguera and Sánchez-Granero (2003) have intro-
duced the notion of T1

∗-half completion and used it to see when a
quasi-uniform space has a ∗-compactification. In this paper, for any
quasi-uniform space, we construct a ∗-half completion, called stan-
dard ∗-half completion. The constructed ∗-half completion coincides
with the usual uniform completion in the uniform spaces and is the
unique (up to quasi-isomorphism) T1

∗-half completion of a symmetriz-
able quasi-uniform space. Moreover, it constitutes a ∗-compactification
for ∗-Cauchy bounded quasi-uniform spaces. Finally, we give an exam-
ple which shows that the standard ∗-half completion differs from the
bicompletion construction.

2000 AMS Classification: 54E15, 54D35.

Keywords: quasi-uniform, ∗-half completion, ∗-compactification.

1. Introduction and preliminaries

The problem of constructing compactifications of quasi-uniform spaces has
been investigated by several authors ([4, 3.47], [5], [7]). This notion of quasi-
uniform compactification is by definition Hausdorff. Moreover, a point sym-
metric totally bounded T1 quasi-uniform space may have many totally bounded
compactifications (see [5, page 34]) . Contrary to this notion, Romaguera and
Sánchez-Granero have introduced the notion of ∗-compactification of a T1 quasi-
uniform space (see [8], [10] and [11]) and prove that: (a) Each T1 quasi-uniform
space having a T1

∗-compactification has an (up to quasi-isomorphism) unique
T1

∗-compactification ([11, Corollary of Theorem 1]); and (b) All the Wallman-
type compactifications of a T1 topological space can be characterized in terms
of the ∗-compactification of its point symmetric totally transitive compatible
quasi-uniformities ([9, Theorem 1]). The proof of (a) is achieved with the help
of the notion of T1

∗-half completion of a quasi-uniform space, which is intro-
duced in [11]. Following ([11, Theorem 1]), if a quasi-uniform space (X, U) is T1



30 A. Andrikopoulos

∗-half completable (it has a T1
∗-half completion), then any T1

∗-half comple-
tion of (X, U) is unique up to a quasi-isomorphism. In this paper, we prove that
every quasi-uniform space has a ∗-half completion, called standard ∗-half com-
pletion, which in the case of a uniform space coincides with the usual one. We
also give an example which shows that the standard ∗-half completion and the
bicompletion are in general different. While a quasi-uniform space may have
many ∗-half completions, here we prove that a symmetrizable quasi-uniform
space has an (up to a quasi-isomorphism) unique ∗-half completion. We also
prove that the standard ∗-half completion constitutes a ∗-compactification for
∗-Cauchy bounded quasi-uniform spaces.

Let us recall that a quasi-uniformity on a (nonempty) set X is a filter U on
X × X such that for each U ∈ U, (i) ∆(X) = {(x, x)|x ∈ X} ⊆ U , and (ii)
V ◦ V ⊆ U for some V ∈ U, where V ◦ V = {(x, y) ∈ X × X| there is z ∈ X
such that (x, z) ∈ V and (z, y) ∈ V }. The pair (X, U) is called a quasi-uniform
space. If U is a quasi-uniformity on a set X, then U−1 = {U −1|U ∈ U}
is also a quasi-uniformity on X called the conjugate of U. Given a quasi-
uniformity U on X, U⋆ = U

∨
U−1 will denote the coarsest uniformity on

X which is finer than U. If U ∈ U, the entourage U ∩ U −1 of U⋆ will be
denoted by U ⋆. The topology τ (U) induced by the quasi-uniformity U on X
is {G ⊆ X| for each x ∈ G there is U ∈ U such that U (x) ⊆ G } where
U (x) = {y ∈ X|(x, y) ∈ U}. If (X, τ ) is a topological space and U is a quasi-
uniformity on X such that τ = τ (U) we say that U is compatible with τ . Let
(X, U) and (Y, V) be two quasi-uniform spaces. A mapping f : (X, U) → (Y, V)
is said to be quasi-uniformly continuous if for each V ∈ V there is U ∈ U such
that (f (x), f (y)) ∈ V whenever (x, y) ∈ U . A bijection f : (X, U) → (Y, V)
is called a quasi-isomorphism if f and f −1 are quasi-uniformly continuous. In
this case we say that (X, U) and (Y, V) are quasi-isomorphic. A filter B is called
U⋆-Cauchy if and only if for each U ∈ U there exists B ∈ B such that B×B ⊆ U
(see [4, page 48]). A net (xa)a∈A is called U

⋆-Cauchy net if for each U ∈ U
there exists an a

U
∈ A such that (xa, xa′ ) ∈ U whenever a ≥ aU , a

′ ≥ a
U
.

We call a
U

extreme index of (xa)a∈A for U and xa
U

extreme point of (xa)a∈A
for U . A quasi-uniform space (X, U) is half complete if every U⋆-Cauchy filter
is τ (U)-convergent [2]. Following to [11, Theorem 1], a ∗-half completion of a
T1 quasi-uniform space (X, U) is a half complete T1 quasi-uniform space (Y, V)
that has a τ (V⋆)-dense subspace quasi-isomorphic to (X, U). In [11, Definition
3] also the authors introduce and study the notion of a ∗-compactification a T1
quasi-uniform space. A ∗-compactification of a T1 quasi-uniform space (X, U)
is a compact T1 quasi-uniform space (Y, V) that has a τ (V

⋆)-dense subspace
quasi-isomorphic to (X, U).

2. The ∗-half-completion

According to Doitchinov [3, Definition 1], a net (y
β
)

β∈B
is called a conet of

the net (xa)a∈A, if for any U ∈ U there are aU ∈ A and βU ∈ B such that



∗
-half completeness in quasi-uniform spaces 31

(yβ, xa) ∈ U whenever a ≥ aU and β ≥ βU . In this case, we write (yβ , xa ) −→ 0.
We denote (x) the constant net (x

a
)

a∈A
, for which x

a
= x for each a ∈ A.

Definition 2.1 (see [1, Definitions 1.1(3)]). Let (X, U) be a quasi-uniform space.

(1) For every U⋆-Cauchy net (x
a
)

a∈A
we consider a U⋆-Cauchy net (y

β
)

β∈B

which is a conet of (x
a
)

a∈A
, different than (x

a
)

a∈A
. In the following,

we consider all the nets A = {(xia)a∈Ai |i ∈ I} that have (yβ )β∈B as

their conet including (y
β
)

β∈B
itself. In the next, we pick up all the nets

B = {(y
j

β
)

β∈Bj
|j ∈ J} which are conets of all the elements of A. The

ordered couple (A, B) have the following properties:

(a) for every U ∈ U and every (xa
i)a∈Ai ∈ A, (yβ

j

)
β∈Bj

∈ B there

are indices a
U

i, β
U

j

such that (y
β

j

, xia) ∈ U whenever a ≥ aU
i and

β ≥ β
U

j .

We call a
U

i (resp. β
U

j

) extreme index of (xa
i)a∈Ai (resp. (yβ

j

)
β∈Bj

)

for U and xi
a

U
i (resp. y

j

β
U

j ) extreme point of (xa
i)a∈Ai (resp. (yβ

j

)
β∈Bj

)

for U .
(b) B contains all the conets of all the elements of A and conversely

A contains all the nets whose conets are all the elements of B.
We call the ordered pair (A, B) h∗-cut, the nets (xa)a∈A and (yβ )β∈B
generator and co-generator of (A, B) respectively. We also say that
the pair ((y

β
)

β∈B
, (xa)a∈A) generates the h

∗-cut (A, B). It is clear
that different pairs of U⋆-Cauchy nets can generate the same h∗-
cut.
The families A and B are called classes (first and second respec-

tively) of the h∗-cut (A, B). In the following, X̃ denotes the set of
all h∗-cuts in X.

If the above U⋆-Cauchy net (x
a
)

a∈A
has not as conet a U⋆-Cauchy net

different from itself, then we relate to it the h∗-cut which generated by
the pair ((xa)a∈A, (xa)a∈A).

(2) To every x ∈ X we assign an h∗-cut, denoted φ(x) = (A
φ(x)

, B
φ(x)

),

which is generated by the pair ((x), (x)). Clearly, x belongs to both
of A

φ(x)
and B

φ(x)
. Thus the class A

φ(x)
contains all the nets which

converge to x in τ
U

and B
φ(x)

contains nets which converge to x in
τ
U−1

.
(3) Suppose that K = {(x

a
)

a∈A
|(x

a
)

a∈A
is a non τ (U)-convergent U⋆-Cauchy

net}. Let X
K

= {ξ ∈ X̃| the generator of ξ belongs to K}. Then we

put X = φ(X) ∪ X
K

.
(4) We often say for a U⋆-Cauchy net (xa)a∈A with a conet (yβ )β∈B and

U ∈ U that:

“finally ((y
β
)

β
, (xa)a) ∈ U ”or in symbols “τ.((yβ )β , (xa)a) ∈ U ”,

if there are a
U

∈ A and β
U

∈ B such that (y
β
, xa) ∈ U whenever

a ≥ a
U
, β ≥ β

U
.



32 A. Andrikopoulos

Definition 2.2. Let (X, U) be a quasi-uniform space, ξ ∈ X and W ∈ U.

(1) We say that a net (t
γ
)

γ∈Γ
is W -close to ξ, if for each net (xia)a∈Ai ∈ Aξ

there holds τ.((t
γ
)

γ
, (xia)a) ∈ W .

(2) For each U ∈ U denote by U the collection of all pairs (ξ′, ξ′′) for which
a co-generator of ξ′ is U -close to ξ′′.

The proof of the following result is straightforward, so it is omitted.

Proposition 2.3. Let (X, U) be a quasi-uniform space and let (y
β
)

β∈B
be a

co-generator of an h∗-cut ξ in X. Then (y
β
)

β∈B
belongs to both of the classes

Aξ and Bξ.

As an immediate consequence of Definition 2.2 and Proposition 2.3 we obtain
the following proposition.

Proposition 2.4. Let (X, U) be a quasi-uniform space, ξ′, ξ′′ ∈ X and U ∈ U.
If (y

β
)

β∈B
, (y

γ
)

γ∈Γ
are co-generators of ξ′ and ξ′′ respectively, then

(ξ′, ξ′′) ∈ U if and only if τ.((y
β
)

β
, (y

γ
)

γ
) ∈ U .

Corollary 2.5. Let (X, U) be a quasi-uniform space and let ξ′, ξ′′ ∈ X. If
(y

β
)

β∈B
, (y

γ
)

γ∈Γ
are co-generators of ξ′ and ξ′′ respectively, then

ξ′ = ξ′′ if and only if (y
β
, y

γ
) −→ 0 in τ (U⋆).

The following lemma is obvious.

Lemma 2.6. Let U, V ∈ U. Then U ⊆ V if and only if V ⊆ U .

Theorem 2.7. The family U = {U|U ∈ U} is a base for a quasi-uniformity U
on X.

Proof. By definitions 2.2 and Proposition 2.3, it follows that the pair (ξ, ξ)
belongs to every element of U and by the previous Lemma U is a filter.

Let now U, W ∈ U be such that W ◦ W ◦ W ⊆ U and x, y ∈ X with
(x, y) ∈ W ◦W . Then there exists a z in X such that (x, z) ∈ W and (z, y) ∈ W .
If (xxa )a∈A , (z

z
γ
)

γ∈Γ
and (yy

β
)

β∈B
are co-generators of x, z and y respectively,

then Definition 2.2 and Proposition 2.3 imply that τ.((xxa )a, (z
z
γ
)

γ
) ∈ W and

τ.((zzγ )γ , (y
y
β
)

β
) ∈ W . We note that, for each (t

δ
)

δ∈∆
∈ A

y
, it holds that

τ.(yy
β
, t

δ
) −→ 0. Hence, τ.((xxa )a, (tδ )δ ) ∈ W ◦ W ◦ W ⊆ U which implies that

(x, y) ∈ U . �

Proposition 2.8. If ξ ∈ X and (xa)a∈A is a U
⋆-Cauchy net which belong to

A
ξ
, then φ(xa) −→ ξ. Dually, if (yβ )β∈B is a U

⋆-Cauchy net which belong to
B

ξ
, then lim

β
(φ(y

β
), ξ) = 0.

Proof. Let V , U ∈ U such that V ◦ V ⊆ U . If (z
γ
)

γ∈Γ
is a co-generator of

ξ, then (z
γ
, x

a
) −→ 0. Thus there are a

V
and γ

V
such that (z

γ
, x

a
) ∈ V for

γ ≥ γ
V

and a ≥ a
V
. Fix an a ≥ a

V
and pick a net (x

δ
)

δ∈∆
of A

φ(xa )
. Then,

x
δ
−→ xa and so (xa, xδ ) ∈ V , whenever δ ≥ δV for some δV ∈ ∆. Hence,

(z
γ
, x

δ
) ∈ U for γ ≥ γ

V
and δ ≥ δ

V
. Hence (ξ, φ(xa)) ∈ U , whenever a ≥ aV .



∗
-half completeness in quasi-uniform spaces 33

The proof of the dual is similar. �

Theorem 2.9. The quasi-uniform space (X, U) is a ∗-half completion of (X, U).

Proof. We firstly prove that (X, U) is half-complete, and secondly that the

space (X, U) has a τ (U
⋆
)-dense subspace quasi-isomorphic to (X, U). Indeed,

let (ξa)a∈A be a U
⋆
-Cauchy net of X. In the following, for each a ∈ A, (ya

β
)

β∈Ba

denotes a co-generator of ξ
a
. Suppose that W ∈ U. Then, there exists a

W
∈ A

such that (ξγ , ξa) ∈ W whenever γ, a ≥ a
W

. Fix an a ≥ a
W

and suppose that
β(a, W ) is the extreme index of (ya

β
)

β∈Ba
for W .

We consider the set
A⋆ = {(a, W )|a ∈ A, W ∈ U}

ordered by (a′, W ′) ≤ (a′′, W ′′) if a′ ≤ a′′ and W ′′ ⊆ W ′.
We put y(a, W ) = ya

β(a,W )
and we prove that the net

{y(a, W )|(a, W ) ∈ A⋆}

is a U⋆-Cauchy net.
Indeed, let U ∈ U. Pick V ∈ U such that V ◦ V ◦ V ⊆ U . Suppose

that (a′, W ′), (a′′, W ′′) ≥ (a
V
, V ). Then, (y(a′, W ′), ya

′

β′
) ∈ (W ′)⋆ ⊆ V ⋆ and

(y(a′′, W ′′), ya
′′

β′′
) ∈ (W ′′)⋆ ⊆ V ⋆ whenever β′ ≥ β′(a′, W ′) and β′′ ≥ β′′(a′′, W ′′).

Since (ξa)a∈A is a U
⋆
-Cauchy net of X, Proposition 2.4 implies that

τ.((ya
′

β′
)

β′
, (ya

′′

β′′
)

β′′
) ∈ V ⋆ whenever a′, a′′ ≥ a

V
. Hence, (y(a′, W ′), y(a′′, W ′′)) ∈

V ⋆ ◦ V ⋆ ◦ V ⋆ ⊆ U ⋆.

We now prove that (ξa)a∈A is τ (U)-convergent. We have two cases.

Case 1. (y(a, W ))
(a,W )∈A⋆

τ (U)-converges to a point x ∈ X.

In this case, we have that (φ(y(a, W )))
(a,W )∈A⋆

τ (U)-converges to φ(x). Since

(ya
β
)

β∈Ba
belongs to B

ξa
, Proposition 2.8 implies that (φ(y(a, W )), ξ

a
) −→ 0.

Hence, from (φ(x), φ(y(a, W ))) −→ 0 we conclude that (ξa)a∈A τ (U)-converges
to φ(x).

Case 2. (y(a, W ))
(a,W )∈A⋆

is a non τ (U)-convergent net.

Let ξ be the h∗-cut in X which is generated by (y(a, W ))
(a,W )∈A⋆

. It follows,

by Proposition 2.8, that (ξ, φ(y(a, W ))) → 0. Since (ya
β
)

β∈Ba
belongs to B

ξa
,

Proposition 2.8 implies that (φ(y(a, W )), ξ
a
) −→ 0. The rest is obvious.

It remains to prove that (φ(X), U/φ(X) × φ(X)) is a τ (U
⋆
)-dense subspace

of (X, U). Indeed, let ξ ∈ X and let (y
β
)

β∈B
be a co-generator of it. Then,

since the co-generator belongs to both of classes of ξ, Proposition 2.8 implies

that φ(y
β
) τ (U

⋆
)-converges to ξ. �

In the sequel the ∗-half completion (X, U) constructed above will be called
standard ∗-half completion of the space (X, U).

The following example shows that the standard ∗-half completion and the
bicompletion of a quasi-uniform space are in general different.



34 A. Andrikopoulos

Example 2.10. Let X be the set consisting of all nonzero real numbers and
let d be the quasi-metric on X given by:

d(x, y) =

{
y − x if x < y

0 otherwise

Suppose that U is the quasi-uniformity generated by d. Let F be the U⋆-Cauchy
filter generated by {(0, a)|a > 0} and G be the U⋆-Cauchy filter generated by
{(b, 0)|b < 0}. Then a new point is defined by the h∗-cut ξ = (A

ξ
, B

ξ
), where

A
ξ

= {G, F} and B
ξ

= {F}. Hence, X = φ(X) ∪ {ξ}. Clearly, ξ defines the

point 0 in (X, U). On the other hand, there is exactly one minimal U⋆-Cauchy
filter coarser than F and G respectively. More precisely, if F

0
and G

0
are any

bases for F and G respectively, then {U (F
0
) |F

0
∈ F

0
and U is a symmetric

member of U⋆} and {U (G
0
) |G

0
∈ G

0
and U is a symmetric member of U⋆}

are equivalent bases for the minimal U⋆-Cauchy filter H̃ coarser than F and G

respectively. Hence, we have X̃ = i(X) ∪ {H}. The filter H defines the point

0 in (X̃, Ũ) as well. We conclude the following:

(i) The bicompletion of (X, U) differs from the standard ∗-half completion.
Indeed, by the definition of ξ and from the Propositions 2.3 and 2.8,
we conclude that φ(G) and φ(F) converge to 0 with respect to τ (U)

and τ (U
⋆

) respectively. On the other hand, i(G) and i(F) converge to

0 with respect to τ (Ũ
⋆

).
(ii) The standard ∗-half completion is not quasi-uniformly isomorphic to its

bicompletion. This is true by (i) and the fact that the bicompletion of
(X, U) coincides up to a quasi-isomorphism with the bicompletion of
(X, U).

Theorem 2.11. Let (X, U) be a uniform space. Then, the standard ∗-half
completion (X, U) coincides with the usual uniform completion.

Proof. Let (X, U) be a uniform space and let ξ be an h∗-cut in X. Suppose
that (xa)a∈A ∈ Aξ and (yβ )β∈B ∈ Bξ . Then (yβ , xa) −→ 0 and (xa, yβ ) −→ 0.
Hence the nets and the conets of ξ coincide. Thus, the class of equivalent
Cauchy nets, of the uniform case, is identified with an h∗-cut and vice versa.
Hence the “ground set”of the two completions is the X. The rest is obvious. �

Next, we give an equivalent definition for nets for the Definition 5 in [11].

Definition 2.12. Let (X, U) be a quasi-uniform space. A U⋆-Cauchy net
(x

a
)

a∈A
on X is said to be symmetrizable if whenever (y

β
)

β∈B
is a U⋆-Cauchy

net on X such that (y
β
, x

a
) −→ 0, then (x

a
, y

β
) −→ 0.

Definition 2.13. A quasi-uniform space (X, U) is called symmetrizable if each
U⋆-Cauchy net on X, including for each x ∈ X the constant net (x), is sym-
metrizable.

It easy to check that a quasi-uniform space is symmetrizable if and only if the
bicompletion is T

1
. In this case, the space has only one T

0
∗-half completion,



∗
-half completeness in quasi-uniform spaces 35

the bicompletion. From Theorem 2.9 and [11, Theorem 1] we immediate deduce
the following result.

Corollary 2.14. If a T1 quasi-uniform space is symmetrizable, then it has a
T1

∗-half completion which is unique up to a quasi-isomorphism.

3. Standard ∗-half completion and ∗-Compactification

We recall some well known notions from [6].
A net (x

a
)

a∈A
is said to be frequently in S, for some subset S of X, if and

only if for all a ∈ A there is some a′ ≥ a such that x
a′

∈ S. A net is said to
be eventually in S if and only if there is an a

0
in A such that for all a ≥ a

0
,

x
a

is in S. A point x in X is a cluster point of the net (x
a
)

a∈A
if and only if

the net is frequently in all neighborhoods of x. The net (x
a
)

a∈A
converges to

x if and only if (x
a
)

a∈A
is eventually in all neighborhoods of x. The tail sets

of (x
a
)

a∈A
are the sets Ta (a in A) where Ta = {xa′ |a

′ ≥ a}. Note that the Ta
have the finite intersection property, by the directedness of the index set A, so
they generate a filter, the filter of tails of (x

a
)

a∈A
or the filter associated with

the net (x
a
)

a∈A
. Then a point x is a cluster point of (x

a
)

a∈A
if and only if x is

in cl(Ta) for all a (if and only if it is a cluster point of the filter of tails). And
x

a
−→ x if and only if the filter of tails converges to x. This already shows that

there is a close relationship between convergence of filters and convergence of
nets.

Definition 3.1 (see [6, page 81]). A universal net in X is one such that for
each S ⊂ X, either the net is eventually in S, or it is eventually in X \ S.

From the classical theory we have the following statements.

(a) A net is a universal net if and only if its associated filter is an ultrafilter.
(b) Let F be the filter associated with the net (x

a
)

a∈A
and G be a filter

with F ⊂ G. Then (x
a
)

a∈A
has a subnet whose associated filter is G.

(a) and (b) implies that:

(c) Every net has a universal subnet.
(d) A universal net converges to each of its cluster points.
(e) A space is compact if and only if every universal net is convergent.

Definition 3.2 (see [11, Definition 6]). A quasi-uniform space (X, U) is called
⋆-Cauchy bounded if for each ultrafilter F on X there is a U⋆-Cauchy filter G
on X such that (G, F) −→ 0.

Definition 3.2 admits an equivalent definition for nets.

Definition 3.3. A quasi-uniform space (X, U) is called ⋆-Cauchy bounded if
for each universal net (x

a
)

a∈A
on X there is a U⋆-Cauchy net (y

β
)

β∈B
on X

such that (y
β
, x

a
) −→ 0.

Theorem 3.4. Let (X, U) be a ⋆-Cauchy bounded quasi-uniform space. Then
the standard ⋆-half completion (X, U) is a ⋆-compactification of the space (X, U).



36 A. Andrikopoulos

Proof. Let (ξ
a
)

a∈A
be a universal net in (X, U). Suppose that for any a ∈

A, ξ
a

= (A
ξa

, B
ξa

). Let (ya
β
)

β∈Ba
and {y(a, W )|(a, W ) ∈ A⋆} be as in the

proof of Theorem 2.9. Then, {y(a, W )|(a, W ) ∈ A⋆} is a net in X. By the
above statement (c), we have that (y(a, W ))

(a,W )∈A⋆
has a universal subnet,

let {y(a
k
, W

k
)|(a

k
, W

k
) ∈ A⋆, k ∈ K}. Since (X, U) is ⋆-Cauchy bounded,

there is a U⋆-Cauchy net (x
γ
)

γ∈Γ
of X such that (x

γ
, y(a

k
, W

k
)) −→ 0. Hence

(φ(x
γ
), φ(y(a

k
, W

k
))) −→ 0 in (X, U) (1). On the other hand, since the space

(X, U) is half-complete, there exists ξ ∈ X such that (φ(x
γ
))

γ∈Γ
τ (U)-converges

to ξ (2). Hence by (1) and (2) we conclude that {φ(y(a
k
, W

k
))|(a

k
, W

k
) ∈

A⋆, k ∈ K} τ (U)-converges to ξ. Since {φ(y(a
k
, W

k
))|(a

k
, W

k
) ∈ A⋆, k ∈ K} is

a subnet of φ(y(a, W ))
(a,W )∈A⋆

we conclude that ξ is a cluster point of the latter.

Since (ya
β
)

β∈Ba
belongs to B

ξa
, Proposition 8 implies that (φ(y(a, W )), ξ

a
) −→

0. Hence, ξ is a cluster point of (ξ
a
)

a∈A
. There also holds that (ξ

a
)

a∈A
is a

universal net, thus the above statement (d) implies that it τ (U)-converges to
ξ. Finally, by the above statement (e) we conclude that the space (X, U) is

compact. By Theorem 9, the space (X, U) has a τ (U
⋆
)-dense subspace quasi-

isomorphic to (X, U). Hence (X, U) is a ⋆-compactification of (X, U). �

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∗
-half completeness in quasi-uniform spaces 37

Received January 2008

Accepted August 2008

Athanasios Andrikopoulos (aandriko@cc.uoi.gr)
Department of Economics, University of Ioannina, Greece