Advances in Pure Mathematics, 2012, 2, 296-300
http://dx.doi.org/10.4236/apm.2012.24039 Published Online July 2012 (http://www.SciRP.org/journal/apm)
A ℘x- and Open
D
C
D
C
-Filters Process of Compactifications
and Any Hausdorff Compactification
Hueytzen J. Wu1, Wan-Hong Wu2
1Department of Mathematics, Texas A & M University-Kingsville, Kingsville, USA
2University of Texas at San Antonio, One UTSA Circle, San Antonio, USA
Received February 23, 2012; revised March 15, 2012; accepted March 22, 2012
ABSTRACT
By means of a characterization of compact spaces in terms of open

-filters induced by a D  C*(Y), a ℘x- and open
-filters process of compactifications of an arbitrary topological space Y is obtained in Sec. 3 by embedding Y as a
dense subspace of (, ℑℬ) or (Y, ℑℬ), where = YE  YS, = YE  YT, YE = {Nx|Nx is a ℘x-filter, x  Y}, YS
= {ℰ|ℰ is an open -filter that does not converge in Y}, YT = {Å|Å is a basic open
D
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YW
T
Y
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S
Y
D
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T
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S
D
C

-filter that does not converge in
Y}, ℑℬ is the topology induced by the base ℬ = {U*|U is open in Y, U   } and U* = {ℱ  (or Y)|U  ℱ}.
Furthermore, an arbitrary Hausdorff compactification (Z, h) of a Tychonoff space X can be obtained from a D  C*(X)
by the similar process in Sec. 3.
W
S
YW
T
D
CD C

-Filter; Open Keywords: Net; Open Filter; Open -Filter Base; Basic Open D C

-Filter; ℘-Filter; ℘x-Filter;
Tychonoff Space; Normal T1-Space; Compact Space; Compactifications; Stone-Čech Compactification;
Wallman Compactification
1. Introduction
Throughout this paper, [T]<  denotes the collection of all
finite subsets of the set T. For the other notations and ter-
minologies in General Topology which are not explicitly
defined in this paper, the readers will be referred to the
Ref. [1].
For an arbitrary topological space Y, let C*(Y) be the
set of bounded real-valued continuous functions on Y, D
 C*(Y). It is shown in Sec. 2 that there exists a unique
rf  Cl(f(Y)) for each f in D such that for any H  [D]< ,
 > 0, fHf−1((rf − , rf + )) 
 . Let Vr ={fHf−1((rf − ,
rf + ))|fHf−1((rf − , rf + )) 
 for any H  [D]< ,  >
0}. Vr is called an open -filter base. An open filter ℰr
on Y containing an open D-filter base Vr is called an
open -filter. An open filter År on Y generated by an
open D-filter base Vr is called a basic open D
D
C
C
D
C
CC

-filter.
By a characterization of compact spaces in Sec. 2 and the
℘x- and open D-filters process of compactifications in
Sec. 3, Y can be embedded as a dense subspace of (S
Y,
ℑℬ) or (T
Y, ℑℬ), where S = YE  YS, T = YE 
YT, YE = {Nx|Nx is a ℘x-filter, x  Y}, YS = {ℰ|ℰ is an
open D-filter that does not converge in Y}, YT = {Å|Å
is a basic open -filter that does not converge in Y},
ℑℬ is the topology induced by the base ℬ = {U*|U   ,
U is open in Y} and U* = { ℱ  S (or T
Y)|U  ℱ}.
Furthmore an arbitrary Hausdorff compactification (Z, h)
of a Tychonoff space X can be obtained from a D 
C*(X) by the similar process in Sec. 3.
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2. Open D

-Filters and a Characterization
of Compact Spaces
Let Y, C*(Y) and D be the sets that are defined in Sec. 1.
Theorem 2.1 Let ℱ be a filter on a topological space Y.
For each f  D, there exists a rf  Cl(f(Y)) such that f−1((rf
− , rf + ))  F   for any F  ℱ and any  > 0. (See
Thm 2.1 in the Ref. [2, p. 1164].)
Proof. If the conclusion is not true, then there is an f 
D such that for each rt  Cl(f(Y)), there exist an Ft  ℱ
and an t > 0 such that Ft  f−1((rt − t, rt + t)) =
 . Since
Cl(f(Y)) is compact and Cl(f(Y))  {(rt − t, rt + t)|rt 
Cl(f(Y))}, there exist r1, ···, rn in Cl(f(Y)) such that Y =
f−1(Cl(f(Y))) = {f−1((ri − i, ri + i))|i = 1, ···, n}. Let Fo =
{Fi| i = 1, ···, n}, then Fo  ℱ and Fo = Fo  Y  {[ Fi
 f−1((ri − i, ri + i))]|i = 1, ···, n} = , contradicting that 
 ℱ.

Corollary 2.2 Let Q be an open ultrafilter on Y. For
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H. J. WU, W.-H. WU 297
each f  D, there exists a unique rf  Cl(f(Y)) such that (1)
for any H  [D] , any  > 0, fHf−1((rf − , rf + ))  Q
and (2) for any H  [D] , any  > 0, fHf−1((rf − , rf + ))
  . (See Cor. 2.2 in the Ref. [2, p. 1164].)
Therefore, for a given open ultrafilter Q, Q contains a
unique open filter base Vr = {fHf−1((rf − , rf + ))|
fHf−1((rf − , rf + )) 
 for any H in [D]< ,  > 0}. Vr is
called an open D-filter base. An open filter ℰr on Y
containing an open D-filter base Vr is called an open
-filter. An open filter År on Y generated by an open
D-filter base Vr is called a basic open D-filter. For
each f  D, if rf = f(x) for an x in Y, then Vr and År are
called the open D-filter base and the basic open
-filter at x, denoted by Vx and Åx, respectively.
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D
Definition 2.3 Let L be a family of continuous functions
on Y. A net {xi} in Y is called a L-net, iff {f(xi)} converges
for each f  L.
Theorem 2.4 Let L be a set of continuous functions on
Y. Then Y is compact iff (1) f(Y) is contained in a com-
pact set Cf for each f in L, and (2) every L-net has a cluster
point in Y.
Proof. Let {xi} be an ultranet in Y. For each f in L,
{f(xi)} is an ultranet in Cf, hence converges in Cf; i.e., {xi}
is a L-net. (2) implies that {xi} has a cluster point x in Y.
Since {xi} is an ultranet, {xi} converges to x. Thus, Y is
compact. The converse is obvious.
Corollary 2.5 Let D  C*(Y). If every D-net converges
in Y, then Y is compact.
Definition 2.6 If ℱ is a filter on Y, let ℱ = {(x, F)|x 
F  ℱ}. Then ℱ is directed by the relation (x1, F1)  (x2,
F2) iff F2  F1, so the map P: ℱ  Y defined by P(x, F)
= x is a net in Y. It is called the net based on ℱ. (See
Def.12.16 in the Ref. [1, p. 81].)
Corollary 2.7 If ℱ is a filter on Y, {P(x, F)} is the net
based on ℱ, then ℱ = {S  Y|P(x, F)} is eventually in S}.
(See L2) in the Ref. [3, p. 83].)
Lemma 2.8 Let D  C*(Y). 1) For each open
D-filter ℰ, let Vr, as the Vr defined in Section 1, be an
open D-filter base such that Vr  ℰ. Then the net {xF}
based on ℰ is a D-net such that lim{f(xF)} = rf for each f 
D. 2) For each D-net {xi} in Y, {xi} induces a unique
open -filer base V{xi} on Y.
D
Proof. 1) By Cor. 2.7, {xF} is eventually in f−1((rf − , rf
+ ))  Vr  ℰ for each f  D and any  > 0. Thus lim{f(xF)}
= rf for each f  D; i.e., {xF} is a D-net. 2) Let {xi} be a D-
net. For each f  D, let tf = lim{f(xi)}. Then fHf−1((tf − ,
tf + )) 
 for any H  [D]< , any  > 0. Let V{xi} =
{fHf−1((tf − , tf + ))|fHf−1((tf − , tf + )) 
 for any
H  [D]< ,  > 0}, then V{xi} is an open D-filter base
on Y. Since tf is unique for each f  D, thus V{xi} is
uniquely induced by {xi}.
Theorem 2.9 Let D  C*(Y). Then, 1) and 2) in the
following are equivalent: 1) Every D-net converges in Y.
2) Every open -filter ℰ converges in Y.
Proof. 1)  2) is obvious by Lemma 2.8 1) above and
Thm. 12.17 (a) in the Ref. [1, p. 81]. For 2)  1): Let {xi}
be a D-net in Y, let ℱ = {|O is open and {xi} is eventu-
ally in O}. Clearly, ℱ is an open filter. For each f in D, let
tf = lim{f(xi)}, then {xi}is eventually in f−1((tf − , tf + ))
for any  > 0; i.e., for each f in D, any  > 0, f−1((tf − , tf +
))  ℱ, so ℱ is an open D-filter. 2) implies that ℱ
converges to a point x. Thus, for any open nhood Ux of x,
Ux  ℱ; i.e., {xi} is eventually in Ux. So {xi} converges to
x.
Corollary 2.10 If every open -filter ℰ on Y con-
verges in Y, then Y is compact.
3. An Open D

-Filter Process of
Compactification
For each x  Y, let Nx = {{x}}  {O|O is open, x  O}.
Nx is a ℘-filter (See 12E. in the Ref. [1, p. 83] for its
definition and convergence.) with  = Nx. For each x  Y,
Nx is called a x-filter. Let YE = {Nx|Nx is a ℘x-filter, x 
Y}, YS = {ℰ|ℰ is an open D
C

-filter that does not converge
in Y}, YT = {Å|Å is a basic open D-filter that does not
converge in Y}, Y = YE  YS and = YE  YT.
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S T
Lemma 3.11 For each ℱ  S (or T Y), there is a
unique rf  Cl(f(Y)) for each f  D such that f−1(rf − , rf +
)  Vr  ℱ for all  > 0.
Proof. If ℱ = Nx for an x  Y, then for each f  D,
f−1((rf − , rf + ))  Vx  Nx for all  > 0, where rf = f(x). If
ℱ = ℰ (or Å), then there is an open D-filter base Vr, as
the Vr defined in Sec. 1, such that for each f  D, f−1((rf − ,
rf + ))  Vr  ℰ (or Å) for all  > 0. The uniqueness of rf
for each f  D follows from Cor. 2.2.
Definition 3.12 For each open set U   in Y, define
U* = {ℱ  (or Y)|U  ℱ}.
S T
Lemma 3.13 1) For any open set U in Y, U    U* 
 ; 2) U = Y  U* = S Y (or T); and 3) for any ℱ in
(or ), any open set U   in Y, ℱ  U*  U  ℱ.
T
Proof. 1) If U   , pick an x  U, then U  Nx  Nx 
U*; i.e., U*   . If U*   , pick a ℱ  U*, then U  ℱ 
U   . 2) and 3) are obvious from Def. 3.12.
Lemma 3.14 For any two nonempty open sets S and T
in Y, 1) S  T iff S*  T*, and 2) (S  T)* = S*  T*, if
S  T   .
Proof. 1): (): ℱ  S*  S  ℱ  T  ℱ  ℱ  T*.
(): S ⊈ T  there is a y  (S – T)  Ny  (S* − T*) 
S* ⊈ T*. 2): By 1) above, (S  T)*  S*  T*. If ℱ  S*
 T*, then S  ℱ, T  ℱ and S  T  ℱ; i.e., ℱ  (S 
T)*.
Proposition 3.15 ℬ = {U*|U   is an open set in Y} is
a base for (or ).
S T
Proof. For (a) in Thm. 5.3 in the Ref. [1, p. 38]: For
each ℱ  S (or T Y), pick a O  ℱ. Then O   , ℱ 
O* and O*  ℬ. Thus S
Y (or T Y) = {U*|U*  ℬ}.
For (b): If ℱ  S*  T* for S*, T*  ℬ, then S, T  ℱ,


H. J. WU, W.-H. WU
298
 S  T  ℱ, (S  T)*  ℬ and ℱ  (S  T)*  S*  T*
 ℬ.
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Equip S
Y (or T Y) with the topology induced by ℬ.
For each f in D, define f*:S Y (or T Y)   by f*(ℱ) = rf,
if f−1((rf − , rf + ))  Vr  ℱ for all  > 0. By Lemma 3.11,
for each f  D, f* is well-defined and f*(S
Y) (or f*(T Y)
 Cl(f(Y)), thus f* is a bounded real-valued function on
(or Y) such that f*(Nx) = f(x) for all x  Y.
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T
Proposition 3.16 For each f in D, let t  Cl(f(Y)). For
any ,  with 0 <  < , 1) [f−1((t − , t + ))]*  f*−1((t − ,
t + )), 2) f*−1((t − , t + ))  [f−1((t-, t+))]*.
Proof. 1): If ℱ  [f−1((t − , t + ))]*, then f−1((t − , t +
))  ℱ. If f*(ℱ) = e, then f−1((e − , e + ))  ℱ for all  >
0. Since f−1((t − , t + )  (e − , e + )) = f−1((t − , t + ))
 f−1((e − , e + ))  ℱ for all  > 0, so (t − , t + )  (e
− , e + )   for all  > 0. Thus f*(ℱ) = e  [t − , t+] 
(t − , t + ); i.e., ℱ  f*−1((t − , t + )). 2): If ℱ  f*−1((t
− , t + )), then f*(ℱ) = s  (t − , t + ) and f−1((s − , s +
))  ℱ for all  > 0. Pick  > 0 such that (s − , s + )  (t
− , t + ). Then S = f−1((s − , s + ))  f−1((t − , t + ))
and S  ℱ. Thus f−1((t − , t + ))  ℱ; i.e., ℱ  [f−1((t − ,
t + ))]*.
Proposition 3.17 For each f  D, f* is a bounded real-
valued continuous function on (or ).
T
Proof. For any ℱ  S (or T
Y), let f*(ℱ) = t. We
show that for any  > 0, there is a U*  ℬ such that ℱ 
U*  f*−1((t − , t + )). Let U = f−1((t − /2, t + /2)).
Since f−1((t − , t + ))  ℱ for all  > 0. Thus, U = f−1((t −
/2, t + /2))  ℱ; i.e., ℱ  U*. By Prop. 3.16 1), ℱ  U*
 f*−1((t − , t + )). Thus f* is continuous on Y (or
).
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Y
Lemma 3.18 Let k: Y  S (or T) be defined by
k(x) = Nx. Then, 1) k is well-defined, one-one and k−1(U*)
= U for all nonempty open set U in Y and all U*  ℬ; i.e.,
k is continuous; 2) f* o k = f for all f  D; 3) k(Y) is dense
in (or Y).
Y
S T
Proof. 1) For any x, y in Y, x = y  Nx = Ny, thus x  y
 Nx  Ny, so k is well-defined and one-one. For any U*
 ℬ, by Def. 3.12 and Lemma 3.13 1), U* 
 , U is open,
U   . So (a): x  k−1(U*)  (b): Nx = k(x)  U*. By
Lemma 3.13 3), (b)  (c): U  Nx. By the setting of Nx, (c)
 (d): x  U. Thus k−1(U*) = U for all U*  ℬ, U   and
U is open in Y; i.e., k is continuous. 2) is obvious from (f*
o k)(x) = f*(Nx) = f(x) for all x in Y and all f in D. 3) For
any U*  ℬ, pick a ℱ  U*, then U  ℱ and U   . Pick
an x  U, by 1) above, x  U  k(x)  U*; i.e., k(x)  U*
 k(Y)   . Hence k(Y) is dense in (or Y).
S T
Let D* = {f*|f  D}. Then, D*  C*(S
Y) (or C*(Y)).
For each open C*D*-filter ℰt* on S
Y (or T Y), let t =
{f*H*f*−1((tf* − , tf* + ))|f*H*f*−1((tf* − , tf* + )) 

for any H*  [D*]< ,  > 0} be the open D-filter base
on S (or ) such that  ℰ*
t. Since f* o k = f, k is
one-one and k(Y) is dense in S Y (or ), so k(fHf−1
((tf* − , tf* + ))) = [f*H*f*−1((tf* − , tf* + ))]  k(Y) 
 for any H*  [D*]< , H = {f  D|f*  H*} and any  >
0. Thus Vt = {fHf−1((tf* − , tf* + ))|fHf−1((tf* − , tf* +
))   for any H  [D]< ,  > 0} is a well-defined open
D

-filter base on Y. Let ℒS = {U  Y | U is open, U  
and U*  ℰ*
t} and ℒT = Åt, the basic open D
C

-filter
generated by Vt. Since ℰ*
t is a filter, clearly, by Lemma
3.14, ℒS is an open filter on Y.
Y
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T
Y*
t
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T
Y
Lemma 3.19 ℒS is an open -filter on Y.
D
Proof. For any H  [D]< ,  > 0, let H* ={f*|f  H}, O
= fHf−1((tf* − , tf* + )) and P = f*H*f*−1((tf* − , tf* +
)). Then
  P  t  ℰ*
t. By Lemmas 3.13, 14 and
Prop. 3.16 2), P  O*,
  O*  ℰ*
t, O   and O  ℒS.
This implies that Vt  ℒS.
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Theorem 3.20 (Y, k) is a compactification of Y.
S
Proof. Case 1: If ℒS converges to a point p in Y. Let U
be any open set in Y such that k(p)  U*  ℬ. By Lemma
3.18 1), p  U = k−1(U*), thus U  ℒS; i.e., U*  ℰ*
t. This
implies that ℰt converges to k(p) in S
Y. Case 2: If ℒS
does not converge in Y, then ℒS  S. For any U* in ℬ
such that ℒS  U*, U  ℒS and therefore U*  ℰt. This
shows that ℰ*
t converges to ℒS in S. By Cor. 2.10, S
Y
is compact and by Lemma 3.18 3), (Y, k) is a compac-
tification of Y.
Lemma 3.21 For each open set U  ℒT = Åt, U*  ℰ*
t.
Proof. If U  Åt, then there exist a H  [D]< , an  > 0
such that E = fHf−1((tf* − , tf* + ))  Vt and E  U.
Lemma 3.14 and Prop. 3.16 2) imply that F = f*H*f*−1
((tf* − , tf* + ))  E*  U* and F  ℰ*
t. Thus, U*  ℰ* t.
Theorem 3.22 (Y, k) is a compactification of Y.
T
Proof. Case 1: If ℒT = Åt converges to a point p in Y,
let U be any open set in Y such that k(p)  U*, Lemma
3.18 1) implies that p  U, thus U  ℒT = Åt. So by
Lemma 3.21, U*  ℰ*
t. This implies that ℰ* t converges to
k(p) in T. Case 2: If ℒT = Åt does not converge in Y,
then ℒT = Åt  T. For any U*  ℬ such that Åt  U*, U
 Åt and by Lemmas 3.21, U*  ℰ*
t. Thus ℰ* t converges
to ℒT = Åt in T
Y. Cor.2.10 implies that T Y is compact
and by Lemma 3.18 3), (Y, k) is a compactification of Y.
4. An Arbitrary Hausdorff Compactification
of a Tychonoff Space
For an arbitrary Hausdorff compactification (Z, h) of a
Tychonoff space X, let D = {f|f = ˚f o h, ˚f  ˚D = C(Z)}.
Then D  C*(X), D separates points of X and the
topology on X is the weak topology induced by D. For
each x  X, let Vx, as the Vx defined in Section 2, be the
open D

-filter base at x induced by D. Obviously, we
can easily get Lemma 4.21 as follows:
Lemma 4.21 GD = {Vx|x  X} is a base for the
topology on X and for each x  X, Vx is an open nhood
base at x.

H. J. WU, W.-H. WU 299
Let XW = {Å|Å is a basic open D-filter on X}. For
each År  XW, let Vr, as the Vr defined in Sec. 1, be the
open D-filter base that generates År. If År converges to
an x  X, then for each f  D, x  Cl(f−1((rf − /2, rf + /2))
 f−1([rf − /2, rf + /2])  f−1((rf − , rf + )) for all  > 0;
i.e., rf = f(x) for all f  D, so Vr = Vx and År = Åx. Thus XW
= XE  XF and XE  XF =
 , where XE = {Åx|x  X} and
XF = {Å|Å is a basic open D-filter that does not
converge in X}. Similar to what we have done in Section 3,
we can get the similar definitions and results for XW in the
following:
C
C
C
C

C

D
C
*
t
V
C
C
4.22-1. For each open set U   in X, define U* = {Å 
XW|U  Å}.
4.22-2. 1) for any open set U in X, U    U*   ; 2)
U = X  U* = XW; and (c) for any Å in XW, any open set
U   , Å  U*  U  Å.
4.22-3. For any two nonempty open sets S and T in X, 1)
S  T iff S*  T*, and 2) (S  T)* = S*  T*, if S  T 
 .
4.22-4. ℬ ={U*|U   , U is an open set in X} is a base
for a topology on X.
4.22-5. For each f  D, f*: XW   is defined by f*(År)
= rf, if f−1((rf − , rf + ))  Vr  År for all  > 0. Then
f*(Åx) = f(x) for all x  X.
4.22-6. For each f in D, let t  Cl(f(X)). For any , 
with 0 <  < , 1) [f−1((t − , t + ))]*  f*−1((t − , t + )),
2) f*−1((t − , t + ))  [f−1((t − , t + ))]*.
4.22-7. For each f in D, f* is a bounded real-valued
continuous function on XW.
4.22-8. Define k: X  XW by k(x) = Åx, then 1) k is
well-defined, one-one, and U = k−1(U*) for all open set U
  in X and all U*  ℬ; i.e., k is continuous, 2) f* o k = f
for all f in D and 3) k(X) is dense in XW.
4.22-9. Let D* = {f*|f  D}. Then D*  C*(XW).
Lemma 4.23 D* separates points of XW.
Proof. For Ås, Åt  XW, let Vs = {fHf−1((sf − , sf +
))|fHf−1((sf − , sf + )) 
 for any H  [D]< ,  > 0}
be the open D-filter base that generates Ås and similarly
for Vt. Since Ås = Åt, Vs = Vt and that sf = tf for all f in D
are equivalent, thus Ås  Åt, Vs  Vt and that there is a g in
D such that sg  tg are equivalent. So, if Ås  Åt, then g*(Ås)
= sg  tg = g*(Åt) for some g*  D*.
Lemma 4.24 The topology on XW is the weak topology
induced by D*.
Proof. For each År  XW, let Vr, as the Vr defined in
Sec. 1, be the open D-filter base that generates År and
let U*  ℬ such that År  U*, then U  År. So there exist
a H  [D] , an  > 0 such that fHf−1((rf − , rf + ))  U,
where fHf−1((rf − , rf + ))  Vr  År for all  > 0. By
4.22-2 (c), 4.22-3 and 4.22-6 2), År  [fHf−1((rf − /2, rf
+ /2))]*  f*H*f*−1((rf − , rf + ))  [fHf−1((rf − , rf
+ ))]*  U*; i.e., År  f*H*f*−1((rf − , rf + ))  U*.
For any open -filter ℰ on XW, let = {f*H*
f*−1((tf* − , tf* + ))|f*H*f*−1((tf* − , tf* + ))   for
any H*  [D*]< ,  > 0} be the open D-filter base that
is contained in ℰ*
t. Since f* o k = f for all f  D, k is one-
one and k(X) is dense in XW, so k(fHf−1((tf* − , tf* + )))
= f*H*f*−1((tf* − , tf* + ))  k(X)   for any H* 
[D*]< , H = {f  D|f*  H*}) and any  > 0. Thus Vt =
{fHf−1((tf* − , tf* + ))|fHf−1((tf* − , tf* + )) 
 for
any H  [D]< ,  > 0} is a well-defined open D
*
t

-filter
base on X. Let Åt be the basic open C-filter on X
generated by Vt.
D

*
V
C

Lemma 4.25 For any open set U  Åt, U*  ℰ*
t.
Proof. For any U  Åt, there exist a H  [D] , an  > 0
such that fHf−1((tf* − , tf* + )) = S  Vt and S  U. By
4.22-3 and 4.22-6, T = f*H*f*−1((tf* − , tf* + ))  S* 
U* and T  . Thus U*  ℰ*
t.
t
Theorem 4.26 (XW, k) is a Hausdorff compactification
of X.
Proof. We show that the open D-filter ℰ*
t converges
to Åt in XW. For any open set U in X such that Åt  U*, by
4.22-2 (c), U  Åt, by Lemma 4.25, U*  ℰ*
t. This implies
that ℰ*
t converges to Åt in XW. By Cor. 2.10, XW is
compact. Thus, by 4.22-8 3) and Lemma 4.23, (XW, k) is a
Hausdorff compactification of X.
5. The Homeomorphism between (XW, k) and
(Z, h)
For each basic open D C

-filter År  XW, let Vr, as the Vr
defined in Sec. 1, be the open D-filter base that gener-
ates År. Since h−1: h(X)  X is one-one, f = ˚f o h and h(X)
is dense in Z, so h−1(˚f˚H˚f−1((rf − , rf + ))) = fHf−1
((rf − , rf + ))   for any ˚H  [˚D]< , H = {f|˚f  ˚H}
and any  > 0. Thus, ˚Vr = {°f°H˚f−1((rf − , rf + ))|
˚f˚H˚f−1((rf − , rf + ))   for any ˚H  [˚D] ,  > 0} is
a well-defined open D
C
C

-filter base on Z. Let ˚År be the
basic open D C

-filter on Z generated by ˚Vr. Since Z is
compact, ˚År clusters at a zr  Z. For each ˚f  ˚D, zr 
Cl(˚f−1((rf − /2, rf + /2)))  ˚f−1([rf − /2, rf + /2]) 
˚f−1((rf − , rf + ))  ˚Vr for all  > 0; i.e., ˚f(zr) = rf for all
˚f  ˚D. So ˚Vr = ˚Vzr and ˚År = ˚Åzr. The zr is called the
w- point in Z induced by År such that ˚f(zr) = rf = f*(År) for
all ˚f  ˚D and f*  D*. ˚Vzr and ˚Åzr are called the open
D
C

-filter base and the basic open D-filter at zr in Z
induced by Vr or År, If zs  zr in Z, there is a ˚f  ˚D such
that ˚f(zs)  ˚f(zr) = rf = f*(År), so zr is the unique w-point
in Z induced by År. If Åt  År, let zt be the w-point in Z
induced by Åt. By Lemma 4.23, there is a g*  D* such
that ˚g(zt) = g*(Åt)  g*(År) = ˚g(zr); i.e., zt  zr. So, if ℋ:
XW  Z is defined by ℋ(År) = zr, where zr is the w-point
in Z induced by År, then ℋ is well-defined and one-one.
For any z  Z, let ˚Åz be the basic open D-filter at z 
Z generated by ˚Vz = {°f°H˚f−1((˚f(z) − , ˚f(z) + )) |˚H
 [˚D] ,  > 0}. Since h is one-one, f = ˚f o h and h(X) is
dense in Z, so h(fHf−1((˚f(z) − , ˚f(z) + ))) = °f°H˚f−1
C
C

H. J. WU, W.-H. WU
opyrigh2012 SciRes. APM
300




Ct ©
((˚f(z) − , ˚f(z) + )))  h(X)   for any H  [D]< , ˚H =
{˚f|f  H},  > 0. Thus Vz = {fHf−1((˚f(z) − , ˚f(z) + ))|
fHf−1((˚f(z) − , ˚f(z) + ))   for any H  [D]< ,  > 0}
is a well-defined open D C-filter base on X. Let Åz be the
basic open D C-filter on X generated by Vz. If zo is the
w-point in Z induced by Åz. Then ˚f(zo) = ˚f(z) = f*(Åz) for
all ˚f  ˚D and f*  D*. This implies that z = zo in Z. So,
for any z  Z, there is a unique Åz in XW such that ℋ(Åz)
= z. Hence, ℋ is well-defined, one-one and onto.
Theorem 5.27 (XW, k) is homeomorphic to (Z, h)
under the mapping ℋ such that ℋ(k(x)) = h(x).
Proof. Since the topologies on Z and XW are the weak
topologies induced by ˚D and D*, respectively, to show
the continuity of ℋ, it is enough to show that for any ˚f 
˚D (or f*  D*), any  > 0, ℋ−1(˚f−1((tf − , tf + ))) =
f*−1((tf − , tf + )). For each Ås in XW, let Vs =
{fHf−1((sf − , sf + ))|fHf−1((sf − , sf + ))   for any
H  [D]< ,  > 0} be the open D C-filter base on X that
generates Ås. Let zs be the w-point in Z induced by Ås,
then ˚f(zs) = sf = f*(Ås). Thus (a): [Ås  f*−1((tf − , tf + ))]
iff (b): [˚f(zs) = f*(Ås) = sf  (tf − , tf + )]. Since ℋ(Ås) =
zs, so (b) iff (c): [ℋ(Ås) = zs  ˚f−1((tf − , tf + ))] and (c)
iff (d): [Ås  ℋ−1(˚f−1((tf − , tf + )))]; i.e., f*−1((tf − , tf
+ )) = ℋ−1(˚f−1((tf − , tf + ))). So, ℋ is continuous.
Since ℋ is one-one, onto and Z, XW are compact Haus-
dorff, by Theorem 17.14 in the Ref. [1, p. 123], ℋ is a
homeomorphism. For that ℋ(k(x)) = h(x) is obvious from
the definitions of k and h.
Corollary 5.28 Let (X, h) be the Stone-Čech
compactification of a Tychonoff space X, D = {f|f = ˚f o h,
˚f  C(X)} and ℋ:XW  X is defined similarly to ℋ
as above. Then (X, h) is homeomorphic to (XW, k) such
that ℋ(k(x)) = h(x).
Corollary 5.29 Let (X, h) be the Wallman compactifi-
cation of a normal T1-space X, D = {f|f = ˚f o h, ˚f 
C(X)} and ℋ:XW  X is defined similarly to ℋ as
above. Then (X, h) is homeomorphic to (XW, k) such that
ℋ(k(x)) = h(x).
REFERENCES
[1] S. Willard, “General Topology,” Addison-Wesley, Read-
ing, 1970.
[2] H. J. Wu and W. H. Wu, “An Arbitrary Hausdorff Com-
pactification of a Tychonoff Space X Obtained from
D
C

-Base by a Modified Wallman Method,” Topology
and its Applications, Vol. 155, 2008, pp. 1163-1168.
doi:10.1016/j.topol.2007.05.021
[3] J. L. Kelly, “General Topology,” Van Nostrand, Princeton,
1955.