1. Introduction
Throughout this paper, 
will denote the collection of all finite subsets of the set
. For the other notations and the terminologies in general topology which are not explicitly defined in this paper, the readers will be referred to the reference [1].
Let 
be the set of bounded real continuous functions on a topological space Y. For any subset 
of
, we will show in Section 2 that there exists a unique rf in 
for each f in 
so that for any
Let K be the set
and let V be the set
K and V are called a closed C*D-filter base and an open C*D-filter base on Y, respectively. A closed filter (or an open filter) on Y generated by a K (or a V) is called a basic closed C*D-filter (or a basic open C*D- filter), denoted by ℰ (or Å). If 
for all f in 
at some x in Y, then K, V, ℰ and Å are denoted by Kx, Vx, ℰx and Åx, respectively. Let Y be a topological space, of which, there is a subset 
of 
containing a non-constant function. A compactification 
of Y is obtained by using closed Ãx- and basic closed C*D-filters in a process similar to the Wallman method, where
, 
is the set {Nx|Nx is a closed 
-filter, x is in Y}, 
is the set of all basic closed C*D-filter that does not converge in Y, 
is the topology induced by the base τ = {F*|F is a nonempty closed set in Y} for the closed sets of 
and F* is the set of all ℭ in 
such that 
for all 
in ℭ. Similarly, an arbitrary Hausdorff compactification 
of a Tychonoff space X can be obtained by using the basic closed C*D-filters on X from
, where 
is the set
.
2. Open and Closed C*D-Filter Bases, Basic Open and Closed C*D-Filters
For an arbitrary topological space Y, let 
be a subset of
.
Theorem 2.1 Let ℱ be a filter on Y. For each f in 
there exists a rf in 
such that
for any 
in ℱ and any 
(See Thm. 2.1 in [2, p.1164]).
Proof. If the conclusion is not true, then there is an f in 
such that for each 
in 
there exist an 
in ℱ and an
such that
Since 
is compact and 
is contained in
there exist r1,···,rn in 
such that Y is contained in
Let 
then 
is in ℱ and
contradicting that f is not in ℱ.
Corollary 2.2 Let ℱ (or Q) be a closed (or an open) ultrafilter on Y. For each f in
, there exists a unique 
in 
such that (1) for any 
any 
ℱ
and (2) for any 
any
.
(See Cor. 2.2 & 2.3 in [2, p.1164].)
Therefore, for a given closed ultrafilter ℱ (or open ultrafilter Q), there exists a unique rf in 
for each f in 
such that for any 
Let K be the set
and let V be the set
K and V are called a closed and an open C*D-filter bases, respectively. If for all f in
, 
for some x in Y, then K and V are called the closed and open C*D-filter bases at x, denoted by Kx and Vx, respectively. Let ℰ and ℰx (or Å and Åx) be the closed (or open) filters generated by K and Kx (or V and Vx), respectively, then ℰ and ℰx (or Å and Åx) are called a basic closed C*D-filter and the basic closed C*D-filter at x (or a basic open C*D-filter and the basic open C*D-filter at x), respectively.
Corollary 2.3 Let ℱ and Q be a closed and an open ultrafilters on a topological space Y, respectively. Then there exist a unique basic closed C*D-filter ℰ and a unique basic open C*D-filter Å on Y such that ℰ is contained in ℱ and Å is contained in Q.
3. A Closed (x-Filter and a Modified Wallman Method of Compactification
Let Y be a topological space, of which, there is a subset 
of 
containing a non-constant function. For each x in Y, let Nx be the union of 
and ℰx, if Vx is an open nhood filter base at x; let Nx be the union of 
and
, if Vx is not an open nhood filter base at x. For each x in Y, Nx is a ℘-filter with à being Nx. (See 12E. in [1, p.82] for definition and convergence). Nx is called a closed ℘x-filter. It is clear that Kx is contained in ℰx and ℰx is contained in Nx, Nx converges to x for each x in Y. Let 
be the set of all Nx, x in Y. Let 
be the set of all basic closed C*D-filter ℰ that does not converge in Y and let
.
Definition 3.4 For each nonempty closed set F in Y, let F* be the set of ℭ in 
such that the intersection of F and T is not an empty set for all T in ℭ.
From the Def. 3.4, the following Cor. 3.5 can be readily proved. We omit its proofs.
Corollary 3.5 For a closed set F in Y, (i) x is in F if Nx is in F*; (ii) F is equal to Y if F* is equal to
; (iii) if F is in ℭ, then ℭ is in F*; (iv) ℭ is in 
if there is a T in ℭ such that T is contained in Y – F.
Lemma 3.6 For any two nonempty closed sets E and F in Y,
(i)
,
(ii)
,
(iii)
.
Proof. (i) For [Ü]: If
, pick an x in
, by Cor. 3.5 (i), Nx is in 
and Nx is not in
; i.e.,
. For (Þ) is obvious. (ii) is clear from (i). (iii) For [Í]: If ℭ belongs to 
and does not belong
, then pick 
in ℭ such that
.
Since 
is in ℭ and
.
Thus, ℭ does not belong to
, contradicting the assumption. For [Ê] is obvious from (i).
Proposition 3.7 τ = {F*|F is a nonempty closed set in Y} is a base for the closed sets of
.
Proof. Let ℬ be the set 
We show that ℬ is a base for
. For (a) of Thm. 5.3 in [1, p.38], if ℭ
, then there exist an f in
, a 
such that
ℰ
and
otherwise, if for all f in
, all d > 0, 
then for all f in
, 
, contradicting that 
contains a non-constant function. Thus
, 
is closed, 
is in ℭ and 
imply that ℭ is in
. So,
.
For (b) of Thm. 5.3, if ℭ belongs to
then 
is closed, 
and
is in ℬ. Thus, ℭ is in
.
Equip 
with the topology Á induced by t. For each f Î
, define f*: 
by
, if
ℰ
for all e > 0. Since (i) if ℭ is equal to Nx for some Nx in
, then
is in Nx for all
, (ii) if ℭ is ℰ which is in
, then
is in ℰ for all 
(iii) by Cor. 2.2, the rf is unique for each f in 
and (iv) the K that is contained in ℭ is unique. Thus, f* is well-defined for each f in
. For all f in
, all x in Y,
is in Nx for all 
thus f*(Nx) is equal to f(x) for all f in 
and all x in Y.
Lemma 3.8 For each f in
, let r be in
, then
(i) 
and
Proof. (i): If ℭ is in 
and 
is
, then
for all
, where 
for all
. Thus,
for all
; i.e., 
is
so ℭ is in
. For (ii): If ℭ is in
and 
is
, then
Pick a d > 0 such that
then
Since
thus
. By Cor. 3.5 (iii), ℭ is in
.
Proposition 3.9 For each f in
, f* is a bounded real continuous function on
.
Proof. For each f in 
and each ℭ in
, 
is in
. Thus 
is contained in
; i.e., f* is bounded on
. For the continuity of f*: If ℭ is in 
and 
is tf. We show that for any
there is a 
in t such that ℭ is in
Let
and 
Since
and 
by Cor. 3.5 (iv), ℭ
. Next, for any ℭs in
, if 
for all x in Y, by Cor. 3.5 (iv), pick a 
in ℭs such that
then 
is in ℭs. By Cor. 3.5 (iii) and Lemma 3.8 (i), ℭs
is in
. If ℭs is Nx for some x in Y, by Cor. 3.5 (i), Nx in 
if
, thus
;
i.e., ℭs is Nx which is in
.
Lemma 3.10 Let k: 
be defined by
. Then, (i) k is an embedding from Y into
; (ii) for all f in
,
and (iii) 
is dense in
.
Proof. (i) By the setting, Nx = Ny if x = y. Thus 
is well-defined and one-one. Let 
be a function from
into Y defined by 
To show the continuity of 
and
, for any 
in t, (a): x is in
iff (b): 
is in
. By Cor. 3.5 (i), (b) iff (c): x is not in
. So,
;
i.e.,
.
So, 
and 
are continuous. (ii) is obvious. (iii) For any 
in t such that 
pick a ℭ in 
By Cor. 3.5 (iv), there is a 
in ℭ such that 
Pick an x in
, by Cor. 3.5 (i), 
which is not in
, so 
is in both 
and
; i.e.,
. Thus, 
is dense in
.
Let
. Then 
Let
be a closed C*D*-filter base on 
and let ℰ* be the basic closed C*D*-filter on 
generated by K*. Since 
and 
are one-one, 
for all 
in 
and 
is dense in
, so
for any
, 
(or any
, 
and all 
Thus,
iff
and
iff
for any
, 
(or any
, 
and all e > 0. Therefore, if the K* or ℰ* defined as above is well-defined, so is K or ℰ defined as in Section 2 well-defined and vice versa. If K* or ℰ* is given, then K or ℰ is called the closed C*D-filter base or the basic closed C*D-filter on Y induced by K* or ℰ* and vice versa.
Lemma 3.11 Let ℰ be a basic closed C*D-filter on Y defined as in Section 2. If ℰ converges to a point x in Y, then (i) rf = f(x) for all f in
; i.e. ℰ = ℰx, (ii) Vx is an open nhood base at x in Y and (iii)
is an open nhood base at k(x) in
.
Proof. If ℰ converges to
in Y, (i): for each
,
for all 
thus
; i.e., ℰ = ℰx. (ii): Since ℰ converges to x in Y, for any open nhood 
of
, there is
which is contained in ℰx = ℰ for some 
such that 
Since x is in
and S is in Vx, thus Vx is an open nhood base at x; (iii): For any 
in t such that Nx is not in
, by Cor. 3.5 (i), 
is not in
, and by (ii) of Lemma 3.11 above, 
is in
for some 
Since
Cor. 3.5 (i), Lemmas 3.6 (ii) and 3.8 (i) imply that
where 
We claim that 
For any ℭs in
, if 
for all f in
, then sf
is in 
for all f in
. Pick a
such that 
for all f in
then
and
; i.e. 
So
Thus 
is an open nhood base at
.
Lemma 3.12 Let ℰ be a basic C*D-filter on Y defined as in Section 2. If ℰ does not converge in Y,
is an open nhood base at ℰ in
.
Proof. If ℰ does not converge in Y, then ℰ is in
. Since f*(ℰ) = rf for all f* Î D*ℰ
for any 
For any 
such that ℰ
by Cor. 3.5 (iv) there exists a
ℰ
for some 
such that E Ì Y – F. For 
let
then ℰ
V*. We claim that 
For any ℰt in
, let f*(ℰt) = tf for each f* in
. Then for each f in
,
is in
and 
ℰt
for all 
Pick a 
such that
for each f in
, then
Since 
ℰt, so ℰt
Hence ℰ is in 
Thus, V*ℰ is an open nhood base at ℰ.
Proposition 3.13 For any basic closed C*D*-filter ℰ* on
, ℰ* converges in
.
Proof. For given ℰ*, let K and ℰ be the closed C*D-filter base and the basic closed C*D-filter on Y induced by ℰ*. Case 1: If ℰ converges to an x in Y, then 
is 
for all f in
. For any
in V*k(x), let
where
. Then 
K*
ℰ* and 
Thus, ℰ* converges to 
in
. Case 2: If ℰ does not converge in Y, then ℰ is in
. For any
in V*ℰ, let
then 
ℰ* and 
Thus, ℰ* converges to ℰ in
.
Theorem 3.14 
is a compactification of Y.
Proof. First, we show that 
is compact. Let
be a sub-collection of t with the finite intersection property. Let
then L is a filter base on
. Let ℱ be a closed ultrafilter on 
such that L is contained in ℱ. By Cor. 2.3, there is a unique basic closed C*D*-filter ℰ* on 
such that ℰ* is contained in ℱ. By Prop. 3.13, ℰ* converges to an ℰo in
. This implies that ℱ converges to ℰo too. Hence, ℰo is in F for all F in ℱ; i.e., ℰo
Thm. 17.4 in [1, p.118], 
is compact. Thus, by Lemma 3.10 (i) and (iii), 
is a compactification of Y.
4. The Hausdorff Compactification (Xw,k) of X Induced by a Subset D of C*(X)
Let X be a Tychonoff space and let 
be a subset of 
such that 
separates points of X and the topology on X is the weak topology induced by
. It is clear that 
contains a non-constant function. For each x in X, since Vx is an open nhood base at x, it is clear that ℰx converges to x. Let 
where XE = {ℰx |x
X} and XE = {ℰ|ℰ is a basic closed C*D-filter that does not converge in X}. Similar to what we have done in Section 3, we can get the similar definitions, lemmas, propositions and a theorem in the following:
(4.15.4) (See Def. 3.4) For a nonempty closed set
in X, 
{ℰ
|
for all 
in ℰ}.
(4.15.5) (See Cor. 3.5) For a nonempty closed set F in X, (i) x is in F if ℰx is in F*; (ii) F is X if
; (iii) for each ℰ in
, F is in ℰ implying ℰ is in F*; (iv) ℰ 
there is a 
in ℰ such that 
Proof. (i) (Ü) If ℰx is in
, then
for all f in
, 
Since Vx is a nhood base at
, thus 
is a cluster point of F, so
is in F. (i) implying (ii), (iii) and (iv) are obvious.
(4.15.6) (See Lemma 3.6) For any two nonempty sets 
and 
in X,
(i)
;
(ii) 
(iii) 
(4.15.7) (See Prop. 3.7) t = {F*|F is a nonempty closed set in X} is a base for the closed sets of
.
(4.15.7.1) (See the definitions for the topology Á on 
and f* for each f in 
in Section 3.)
Equip 
with the topology Á induced by t. For each f in
, define 
by f*(ℰ) = rf if
ℰ for all
. Then f* is welldefined and f*(ℰx) is f(x) for all f in 
and all x in X.
(4.15.8) (See Lemma 3.8) For each f in
, let r be in
, then
(i) 
and
(ii) 
for any 
(4.15.9) (See Prop. 3.9) For each f in
, f* is a bounded real continuous function on
.
(4.15.10) (See Lemma 3.10) Let 
be defined by 
ℰx. Then, (i) 
is an embedding from X into
; (ii) 
for all f in
; and (iii) 
is dense in
.
(4.15.11) (See Lemmas 3.11 and 3.12) For each ℰ in
, let
1) If ℰ converges to x, then ℰ is ℰx and V*k(x) is =
V*ℰx =
is an open nhood base at ℰx. 2) If ℰ does not converge in X, then ℰ is in 
and V*ℰ =
is an open nhood base at ℰ in
.
(4.15.13) (See Prop. 3.13) Each basic closed C*D*- filter ℰ* on 
converges to ℰ in
.
(4.15.14) (See Theorem 3.14) 
is a compactification of X.
Lemma 4.16 
separates points of
.
Proof. For ℰs, ℰt in
, let
and similarly for Kt. Since ℰs is not equal to ℰt, Ks is not equal to Kt and that 
has a g such that 
are equivalent, where 
which is contained in ℰs and 
which is contained in ℰt for all 
thus by the definition of g*, g*(ℰs)
g*(ℰt).
Theorem 4.17 
is a Hausdorff compactification of X.
Proof. By 4.15.10 (i) and (iii), 4.15.14 and Lemma 4.16, 
is a Hausdorff compactification of X.
5. The Homeomorphism between (Xw,k) and (Z,h)
Let 
be an arbitrary Hausdorff compactification of X, then X is a Tychonoff space. Let 
denote 
which is the family of real continuous functions on Z, and let
. Then 
is a subset of 
such that 
separates points of X, the topology on X is the weak topology induced by 
and 
contains a non-constant function.
Let 
be the Hausdorff compactification of X obtained by the process in Section 4 and 
is defined as above. For each basic closed C*D-filter ℰ in
, let ℰ be generated by
let °ℰ be the basic closed C*°D-filter on Z generated by
and let h−1 be the function from h(X) to X defined by h−1(h(x)) = x. Since h and h−1 are one-one, f = °f o h and h(X) is dense in Z, similar to the arguments in the paragraphs prior to Lemma 3.11, we have that
iff
for any
(or any
),
(or
)
and all
. Thus, if K or ℰ is well-defined, so is °K or °ℰ and vice versa. If K or ℰ is given, °K or °ℰ is called the closed C*°D-filter base or the basic closed C*°D-filter on Z induced by K or ℰ and vice versa. For any z in Z,
is the closed C*°D-filter base at z. The closed filter °ℰz generated by °Kz is the basic closed C*°D-filter at z. Since Z is compact Hausdorff, each °ℰ on Z converges to a unique point z in Z. So, we define 
by 
(ℰ) = z, where ℰ is in 
and z is the unique point in Z such that the basic closed C*°D-filter °ℰ on Z induced by ℰ converges to it. For ℰs, ℰt in
, let
and similarly for Kt such that ℰs and ℰt are generated by Ks and Kt, respectively. Assume that °ℰs and °ℰt converge to zs and zt in Z, respectively. Then ℰs is not equal to ℰt, °ℰs is not equal to °ℰt and zs is not equal to zt are equivalent. Hence
is well-defined and one-one. For each z in Z, let °ℰz be the basic closed C*°D-filter at z, since Z is compact Hausdorff and
is an open nhood base at z, thus °ℰz converges to z. Let ℰz be the element in 
induced by °ℰz, then, 
(ℰz) = z. Hence, 
is one-one and onto.
Theorem 5.18 (
is homeomorphic to 
under the mapping 
such that
.
Proof. We show that 
is continuous. For each ℰ in F* which is in t, let °ℰ be the basic closed C*°D-filter on Z induced by ℰ. If °ℰ converges to z in Z, 
for each f in
and
Then (a): ℰ is in F* iff (b):
for any 
where
ℰ.
Since 
is one-one, 
for all f in
, so (b) iff (c):
for any
(or
),
(or
)
and any e > 0. Since
for any °f in
, 
(c) iff (d):
for any 
Since
is an arbitrary basic open nhood of z in Z. So, (d) iff z is in
; i.e., ℰ is in F* if 
(ℰ) is equal to z which belongs to
. Hence, T(F*) = ClZ(h(F)) is closed in Z for all F* in t. Thus, 
is continuous. Since 
is one-one, onto and both Z and 
are compact Hausdorff, by Theorem 17.14 in [1, p.123], 
is a homeomorphism. Finally, from the definitions of 
and
, it is clear that 
for all x in X.
Corollary 5.19 Let (bX,
) be the Stone-Čech compactification of a Tychonoff space X,
and
: 
is defined similarly to
as above. Then (bX,
) is homeomorphic to 
such that
Corollary 5.20 Let (gX,
) be the Wallman compactification of a normal T1-space X,
and 
is defined similarly to 
as above. Then (gX,
) is homeomorphic to 
such that
.