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
![](https://www.scirp.org/html/5-5300379\de25639d-19d4-4091-8b94-ab1b87e6ae2f.jpg)
Let K be the set
![](https://www.scirp.org/html/5-5300379\003f0328-c594-4180-b1ec-53e815b0133f.jpg)
and let V be the set
![](https://www.scirp.org/html/5-5300379\d0f1e8c2-7d13-4217-a92b-3eaaff67dc6b.jpg)
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
![](https://www.scirp.org/html/5-5300379\2d031559-8ef1-430c-80d6-55ed1849748d.jpg)
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
![](https://www.scirp.org/html/5-5300379\7c71567f-8f57-454d-8988-a8b5ce051a27.jpg)
Since
is compact and
is contained in
there exist r1,···,rn in
such that Y is contained in
![](https://www.scirp.org/html/5-5300379\b407dc74-0369-41c9-ba53-5071a714356b.jpg)
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 ![](https://www.scirp.org/html/5-5300379\d1dc628b-b466-429f-94a4-32ac8cedc433.jpg)
ℱ
![](https://www.scirp.org/html/5-5300379\b27923d7-dd80-45b8-9daa-461052f7e91c.jpg)
and (2) for any
any![](https://www.scirp.org/html/5-5300379\c256efe4-c36c-4245-bcd2-11a485f253f2.jpg)
![](https://www.scirp.org/html/5-5300379\2b964b74-bc3a-4f61-9aba-3b39e04a45a5.jpg)
.
(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 ![](https://www.scirp.org/html/5-5300379\2ed604d8-6e70-465a-b02a-cd33c4188db5.jpg)
![](https://www.scirp.org/html/5-5300379\9743e4f2-2e92-48d0-ac74-327a72074c3a.jpg)
![](https://www.scirp.org/html/5-5300379\e79eab4d-6905-41f4-a444-845bf9d0b76d.jpg)
Let K be the set
![](https://www.scirp.org/html/5-5300379\be7f6f5b-a4ae-49ae-9a87-a3af7c928949.jpg)
and let V be the set
![](https://www.scirp.org/html/5-5300379\c18ee0c9-0909-4cd9-8da1-e999f85c7718.jpg)
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![](https://www.scirp.org/html/5-5300379\238ecbb2-8ac2-4805-a035-d68183df976c.jpg)
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
![](https://www.scirp.org/html/5-5300379\1748eefb-2fbb-46da-9b1c-6cea10f90371.jpg)
is in ℬ. Thus, ℭ is in
.
Equip
with the topology Á induced by t. For each f Î
, define f*:
by
, if
ℰ![](https://www.scirp.org/html/5-5300379\08bc35e9-b96d-4285-b16e-3a3c435617c1.jpg)
for all e > 0. Since (i) if ℭ is equal to Nx for some Nx in
, then
![](https://www.scirp.org/html/5-5300379\372f85a8-2fbc-4ac0-aa3a-e944af3897f7.jpg)
is in Nx for all
, (ii) if ℭ is ℰ which is in
, then
![](https://www.scirp.org/html/5-5300379\6122513f-d76a-42aa-847d-fa3795bb477e.jpg)
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,
![](https://www.scirp.org/html/5-5300379\1dd8640b-efa0-4cbf-83f6-e9db164265b3.jpg)
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) ![](https://www.scirp.org/html/5-5300379\658716a3-8e6d-4167-96f2-22fe3b8bdc73.jpg)
and
![](https://www.scirp.org/html/5-5300379\cd584647-40af-47d3-af3a-c32183774ba7.jpg)
Proof. (i): If ℭ is in
and
is
, then
![](https://www.scirp.org/html/5-5300379\0f8d68ad-6567-4b23-803b-36d5e03703b7.jpg)
for all
, where
for all
. Thus,
![](https://www.scirp.org/html/5-5300379\b685b2f0-0bac-4cb1-9266-1dbb622fc6eb.jpg)
for all
; i.e.,
is
![](https://www.scirp.org/html/5-5300379\bbb69ea0-f0db-4a44-b706-c16550c66d6c.jpg)
so ℭ is in
. For (ii): If ℭ is in
and
is
, then
![](https://www.scirp.org/html/5-5300379\bcc5d9c8-82a4-4ba5-aa02-d02d1c64a910.jpg)
Pick a d > 0 such that
![](https://www.scirp.org/html/5-5300379\7ccc6ad9-abee-47a3-a672-18b11432887b.jpg)
then
![](https://www.scirp.org/html/5-5300379\9177a38e-61eb-4d02-9e71-340c8f8b796e.jpg)
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
![](https://www.scirp.org/html/5-5300379\45531fe9-d717-456f-ab4b-a2bca5b22b85.jpg)
Let
![](https://www.scirp.org/html/5-5300379\0aa0b5ea-bdd7-48e5-8c77-e2e5f149f653.jpg)
and
Since
![](https://www.scirp.org/html/5-5300379\8921f059-41b0-43b0-b07b-95a95aad0f2a.jpg)
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
![](https://www.scirp.org/html/5-5300379\df085703-e3b4-4311-a27f-60c4aaf43351.jpg)
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
![](https://www.scirp.org/html/5-5300379\4b686fe9-4aeb-44f1-824f-f066c51d067e.jpg)
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
![](https://www.scirp.org/html/5-5300379\14088eb7-0442-4b09-8380-a478386eed86.jpg)
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
![](https://www.scirp.org/html/5-5300379\e625bfad-8786-4fb4-a8a6-f6d1ee343c9b.jpg)
for any
,
(or any
,
and all
Thus,
![](https://www.scirp.org/html/5-5300379\c1a99b0c-2eea-435d-a105-05d340084124.jpg)
iff
![](https://www.scirp.org/html/5-5300379\e140730c-c533-46a4-9500-20260f79b229.jpg)
and
![](https://www.scirp.org/html/5-5300379\4ad387dd-4bb6-4b2a-89d1-5bf1e84ca16a.jpg)
iff
![](https://www.scirp.org/html/5-5300379\66fc3c6d-7740-4244-9a0a-dab70a899395.jpg)
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)
![](https://www.scirp.org/html/5-5300379\fe247403-a7ac-4ea3-98a2-ce2710cd54d6.jpg)
is an open nhood base at k(x) in
.
Proof. If ℰ converges to
in Y, (i): for each
,
![](https://www.scirp.org/html/5-5300379\ae8face1-5f14-4617-a351-8f26d64586a5.jpg)
for all
thus
; i.e., ℰ = ℰx. (ii): Since ℰ converges to x in Y, for any open nhood
of
, there is
![](https://www.scirp.org/html/5-5300379\e12bce17-5c60-4850-a4f3-aa55f6d36500.jpg)
which is contained in ℰx = ℰ for some
such that
Since x is in
![](https://www.scirp.org/html/5-5300379\04b9ad7a-12f5-41fd-8c9e-24649ead7323.jpg)
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
![](https://www.scirp.org/html/5-5300379\f95647d9-dd5c-4392-866a-7246fc114966.jpg)
for some
Since
![](https://www.scirp.org/html/5-5300379\842514c2-036c-447b-b897-827d76271edf.jpg)
Cor. 3.5 (i), Lemmas 3.6 (ii) and 3.8 (i) imply that
![](https://www.scirp.org/html/5-5300379\5d178c31-2956-4582-b9dd-b5e7ee159567.jpg)
where
We claim that ![](https://www.scirp.org/html/5-5300379\ebf5eecb-556a-487f-9f76-94acc614fb9c.jpg)
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
![](https://www.scirp.org/html/5-5300379\9514537e-90c2-40a3-8166-9d72e48d1b58.jpg)
and
; i.e.
So
![](https://www.scirp.org/html/5-5300379\f85a9f09-8457-423b-9a61-234a6c9af974.jpg)
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,
![](https://www.scirp.org/html/5-5300379\0196f69d-3791-444d-9a08-790ab0f7890c.jpg)
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![](https://www.scirp.org/html/5-5300379\f87f7b13-3c1b-4f9a-934c-9658822f4621.jpg)
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![](https://www.scirp.org/html/5-5300379\2bc3816f-7bfe-4b3b-a065-5593a2601697.jpg)
![](https://www.scirp.org/html/5-5300379\807ba959-e4cc-4802-b6da-5df69decef31.jpg)
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
![](https://www.scirp.org/html/5-5300379\4f5cb90b-aedc-43b5-aace-b63fa4dcd30a.jpg)
for each f in
, then
![](https://www.scirp.org/html/5-5300379\6b2c3ab1-931a-4ca9-85b9-b7b1dd88d4c5.jpg)
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
![](https://www.scirp.org/html/5-5300379\ee7778ed-7e01-4687-9e81-646f989aeb78.jpg)
in V*k(x), let
where
. Then ![](https://www.scirp.org/html/5-5300379\560ad606-4752-4195-8777-ce9579ca889b.jpg)
K*
ℰ* and ![](https://www.scirp.org/html/5-5300379\413096df-50b3-4e81-8c05-d7cded7388ff.jpg)
Thus, ℰ* converges to
in
. Case 2: If ℰ does not converge in Y, then ℰ is in
. For any
![](https://www.scirp.org/html/5-5300379\5e913ca0-db29-4c74-99dd-64c31a0b55d7.jpg)
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![](https://www.scirp.org/html/5-5300379\c42f35e9-23cb-4fb5-9bd6-85af5167fca5.jpg)
|
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 ![](https://www.scirp.org/html/5-5300379\05bdc3a6-78e9-4903-82fd-220f266db8a6.jpg)
Proof. (i) (Ü) If ℰx is in
, then
![](https://www.scirp.org/html/5-5300379\e9649aa2-6118-47bb-a5f9-85aee739dbc9.jpg)
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) ![](https://www.scirp.org/html/5-5300379\10674d5d-45cd-47a3-b7d3-30f870709e7e.jpg)
(iii) ![](https://www.scirp.org/html/5-5300379\084be282-0df4-4507-9097-99e05468e9de.jpg)
(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) ![](https://www.scirp.org/html/5-5300379\89d938c9-ddd6-455b-8866-790fbef13411.jpg)
and
(ii) ![](https://www.scirp.org/html/5-5300379\147abcd8-e924-4f8d-ada5-9619479e893e.jpg)
for any ![](https://www.scirp.org/html/5-5300379\681d3531-6e44-4ff9-bd3d-12140a576db1.jpg)
(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
![](https://www.scirp.org/html/5-5300379\58240e72-6fd4-4164-9686-bb3b57cd8a7a.jpg)
1) If ℰ converges to x, then ℰ is ℰx and V*k(x) is =
V*ℰx =
![](https://www.scirp.org/html/5-5300379\468f00f6-d4b8-4b8c-bed5-ed8ca9fdfc07.jpg)
is an open nhood base at ℰx. 2) If ℰ does not converge in X, then ℰ is in
and V*ℰ =
![](https://www.scirp.org/html/5-5300379\91c2dd9d-7207-4872-829d-16faf4731af6.jpg)
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
![](https://www.scirp.org/html/5-5300379\b9d98a60-5ea6-4e27-8791-85fcc9de8ac8.jpg)
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
![](https://www.scirp.org/html/5-5300379\c27ac552-133b-430a-b8c5-71568fd38297.jpg)
let °ℰ be the basic closed C*°D-filter on Z generated by
![](https://www.scirp.org/html/5-5300379\2c76f78b-7e9f-41ca-bb10-ac65a484395d.jpg)
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
![](https://www.scirp.org/html/5-5300379\8032bd2f-088a-4c15-ae00-5147cbbe0d05.jpg)
iff
![](https://www.scirp.org/html/5-5300379\342f6dba-4695-4194-87c8-1b4f5ba89855.jpg)
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,
![](https://www.scirp.org/html/5-5300379\bd9139cd-b14d-40ef-a426-453c9fe904c2.jpg)
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
![](https://www.scirp.org/html/5-5300379\2152b328-d62f-4278-95ad-894f51f8f685.jpg)
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
![](https://www.scirp.org/html/5-5300379\3b1683c7-b6e3-415a-a568-af54d5f7ae62.jpg)
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
![](https://www.scirp.org/html/5-5300379\c1d992e7-9091-4afd-a2d7-65b396d600b5.jpg)
Then (a): ℰ is in F* iff (b):
![](https://www.scirp.org/html/5-5300379\78489967-c548-49de-94ff-23b0fe70fdd8.jpg)
for any
where
ℰ.
Since
is one-one,
for all f in
, so (b) iff (c):
![](https://www.scirp.org/html/5-5300379\768cddaa-0dac-4c33-991b-02a692ee2ef4.jpg)
for any
(or
),
(or
)
and any e > 0. Since
![](https://www.scirp.org/html/5-5300379\379cf4fb-9498-4b52-93a1-34e33df8ae93.jpg)
for any °f in
,
(c) iff (d):
![](https://www.scirp.org/html/5-5300379\5bea88ac-4230-4fad-a88d-adf7429e6a84.jpg)
for any
Since
![](https://www.scirp.org/html/5-5300379\967566f3-0cd7-4b57-a9e4-cc415d99fc58.jpg)
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,
![](https://www.scirp.org/html/5-5300379\3c250b2e-e42b-49f2-8268-dfd5acf5fef6.jpg)
and
:
is defined similarly to
as above. Then (bX,
) is homeomorphic to
such that
![](https://www.scirp.org/html/5-5300379\940dc9b6-5bf9-436e-af67-f4903ef78005.jpg)
Corollary 5.20 Let (gX,
) be the Wallman compactification of a normal T1-space X,
![](https://www.scirp.org/html/5-5300379\1e77e971-090b-4785-9eb4-0731697514ae.jpg)
and
is defined similarly to
as above. Then (gX,
) is homeomorphic to
such that
.