(): 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,

Copyright © 2012 SciRes. APM

H. J. WU, W.-H. WU

298

S T ℱ, (S T)* ℬ and ℱ (S T)* S* T*

ℬ.

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 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.

W

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

).

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 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(fHf−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 = {fHf−1((tf* − , tf* + ))|fHf−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

W

T

Y*

t

V

W

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

= fHf−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.

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 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 = fHf−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.

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(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 = {fHf−1((sf − , sf +

))|fHf−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 fHf−1((rf − , rf + )) U,

where fHf−1((rf − , rf + )) Vr År for all > 0. By

4.22-2 (c), 4.22-3 and 4.22-6 2), År [fHf−1((rf − /2, rf

+ /2))]* f*H*f*−1((rf − , rf + )) [fHf−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(fHf−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 =

{fHf−1((tf* − , tf* + ))|fHf−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 fHf−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 + ))) = fHf−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(fHf−1((˚f(z) − , ˚f(z) + ))) = °f°H˚f−1

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 = {fHf−1((˚f(z) − , ˚f(z) + ))|

fHf−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 =

{fHf−1((sf − , sf + ))|fHf−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).

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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

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.