2 fsd ws2">S T T*.
(): S T there is a y (ST) Ny (S* T*)
S* T*. 2): By 1) above, (S T)* S* T*. If S*
T*, then S , T and S T ; i.e., (S
T)*.
Proposition 3.15 = {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 S (or T
Y), pick a O . Then O
,
O* and O* . Thus S
Y (or T
Y) = {U*|U* }.
For (b): If S* T* for S*, T* , then S, T ,
Copyright © 2012 SciRes. APM
H. J. WU, W.-H. WU
298
S T , (S T)* and (S T)* S* T*
.
W W
W W
W W
W
S
Y
W
S
YW
Y
W
W
S
W
T
Y
W
YW
W
YW
WW
W W
T
W W*
V
C
W
Y
W
C
Equip S
Y (or T
Y) with the topology induced by .
For each f in D, define f*:S
Y (or T
Y) by f*() = rf,
if f1((rf , rf + )) Vr 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.
W
T
Proposition 3.16 For each f in D, let t Cl(f(Y)). For
any , with 0 < < , 1) [f1((t , t + ))]* f*1((t ,
t + )), 2) f*1((t , t + )) [f1((t-, t+))]*.
Proof. 1): If [f1((t , t + ))]*, then f1((t , t +
)) . If f*() = e, then f1((e , e + )) for all >
0. Since f1((t , t + ) (e , e + )) = f1((t , t + ))
f1((e , e + )) for all > 0, so (t , t + ) (e
, e + ) for all > 0. Thus f*() = e [t , t+]
(t , t + ); i.e., f*1((t , t + )). 2): If f*1((t
, t + )), then f*() = s (t , t + ) and f1((s , s +
)) for all > 0. Pick > 0 such that (s , s + ) (t
, t + ). Then S = f1((s , s + )) f1((t , t + ))
and S . Thus f1((t , t + )) ; i.e., [f1((t ,
t + ))]*.
Proposition 3.17 For each f D, f* is a bounded real-
valued continuous function on (or ).
T
Proof. For any S (or T
Y), let f*() = t. We
show that for any > 0, there is a U* such that
U* f*1((t , t + )). Let U = f1((t /2, t + /2)).
Since f1((t , t + )) for all > 0. Thus, U = f1((t
/2, t + /2)) ; i.e., U*. By Prop. 3.16 1), U*
f*1((t , t + )). Thus f* is continuous on Y (or
).
W
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 k1(U*)
= U for all nonempty open set U in Y and all U* ; 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*
, by Def. 3.12 and Lemma 3.13 1), U*
, U is open,
U
. So (a): x k1(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 k1(U*) = U for all U* , 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* , pick a U*, then U 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 t* 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 *
t. Since f* o k = f, k is
one-one and k(Y) is dense in S
Y (or ), so k(fHf1
((tf* , tf* + ))) = [f*H*f*1((tf* , tf* + ))] k(Y)
for any H* [D*]< , H = {f D|f* H*} and any >
0. Thus Vt = {fHf1((tf* , tf* + ))|fHf1((tf* , tf* +
))
for any H [D]< , > 0} is a well-defined open
D
-filter base on Y. Let S = {U Y | U is open, U
and U* *
t} and T = Åt, the basic open D
C
-filter
generated by Vt. Since *
t is a filter, clearly, by Lemma
3.14, S is an open filter on Y.
Y
W
T
Y*
t
V
W
T
Y
Lemma 3.19 S is an open -filter on Y.
D
Proof. For any H [D]< , > 0, let H* ={f*|f H}, O
= fHf1((tf* , tf* + )) and P = f*H*f*1((tf* , tf* +
)). Then
P t *
t. By Lemmas 3.13, 14 and
Prop. 3.16 2), P O*,
O* *
t, O and O S.
This implies that Vt S.
C
*
V
W
*W
W
Y
*
W
YW
W
S
W
W
Y
W
Y
W W
W
T
C
Theorem 3.20 (Y, k) is a compactification of Y.
S
Proof. Case 1: If S converges to a point p in Y. Let U
be any open set in Y such that k(p) U* . By Lemma
3.18 1), p U = k1(U*), thus U S; i.e., U* *
t. This
implies that t converges to k(p) in S
Y. Case 2: If S
does not converge in Y, then S S. For any U* in
such that S U*, U S and therefore U* t. This
shows that *
t converges to S 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 T = Åt, U* *
t.
Proof. If U Åt, then there exist a H [D]< , an > 0
such that E = fHf1((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 *
t. Thus, U* *
t.
Theorem 3.22 (Y, k) is a compactification of Y.
T
Proof. Case 1: If T = 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 T = Åt. So by
Lemma 3.21, U* *
t. This implies that *
t converges to
k(p) in T. Case 2: If T = Åt does not converge in Y,
then T = Åt T. For any U* such that Åt U*, U
Åt and by Lemmas 3.21, U* *
t. Thus *
t converges
to T = 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.
Copyright © 2012 SciRes. APM
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(f1((rf /2, rf + /2))
f1([rf /2, rf + /2]) f1((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. ={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 f1((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) [f1((t , t + ))]* f*1((t , t + )),
2) f*1((t , t + )) [f1((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 = k1(U*) for all open set U
in X and all U* ; 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 = {fHf1((sf , sf +
))|fHf1((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* such that År U*, then U År. So there exist
a H [D] , an > 0 such that fHf1((rf , rf + )) U,
where fHf1((rf , rf + )) Vr År for all > 0. By
4.22-2 (c), 4.22-3 and 4.22-6 2), År [fHf1((rf /2, rf
+ /2))]* f*H*f*1((rf , rf + )) [fHf1((rf , rf
+ ))]* U*; i.e., År f*H*f*1((rf , rf + )) U*.
For any open -filter 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 *
t. Since f* o k = f for all f D, k is one-
one and k(X) is dense in XW, so k(fHf1((tf* , tf* + )))
= f*H*f*1((tf* , tf* + )) k(X)
for any H*
[D*]< , H = {f D|f* H*}) and any > 0. Thus Vt =
{fHf1((tf* , tf* + ))|fHf1((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* *
t.
Proof. For any U Åt, there exist a H [D] , an > 0
such that fHf1((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* *
t.
t
Theorem 4.26 (XW, k) is a Hausdorff compactification
of X.
Proof. We show that the open D-filter *
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* *
t. This implies
that *
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 h1: h(X) X is one-one, f = ˚f o h and h(X)
is dense in Z, so h1(˚f˚H˚f1((rf , rf + ))) = fHf1
((rf , rf + ))
for any ˚H [˚D]< , H = {f|˚f ˚H}
and any > 0. Thus, ˚Vr = {°f°H˚f1((rf , rf + ))|
˚f˚H˚f1((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(˚f1((rf /2, rf + /2))) ˚f1([rf /2, rf + /2])
˚f1((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 :
XW Z is defined by (År) = zr, where zr is the w-point
in Z induced by År, then 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˚f1((˚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(fHf1((˚f(z) , ˚f(z) + ))) = °f°H˚f1
C
C
Copyright © 2012 SciRes. APM
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 = {fHf1((˚f(z) , ˚f(z) + ))|
fHf1((˚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*(z) 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 (Åz)
= z. Hence, is well-defined, one-one and onto.
Theorem 5.27 (XW, k) is homeomorphic to (Z, h)
under the mapping such that (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 , it is enough to show that for any ˚f
˚D (or f* D*), any > 0, 1(˚f1((tf , tf + ))) =
f*1((tf , tf + )). For each Ås in XW, let Vs =
{fHf1((sf , sf + ))|fHf1((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 (s) =
zs, so (b) iff (c): [(Ås) = zs ˚f1((tf , tf + ))] and (c)
iff (d): [Ås 1(˚f1((tf , tf + )))]; i.e., f*1((tf , tf
+ )) = 1(˚f1((tf , tf + ))). So, is continuous.
Since is one-one, onto and Z, XW are compact Haus-
dorff, by Theorem 17.14 in the Ref. [1, p. 123], is a
homeomorphism. For that (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 :XW X is defined similarly to
as above. Then (X, h) is homeomorphic to (XW, k) such
that (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 :XW X is defined similarly to as
above. Then (X, h) is homeomorphic to (XW, k) such that
(k(x)) = h(x).
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ing, 1970.
[2] H. J. Wu and W. H. Wu, “An Arbitrary Hausdorff Com-
pactification of a Tychonoff Space X Obtained from
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,
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