**Advances in Pure Mathematics** Vol.3 No.6(2013), Article ID:37446,8 pages DOI:10.4236/apm.2013.36076

A Modified Wallman Method of Compactification

^{1}Department of Mathematics, Texas A&M University, Kingsville, USA

^{2}University of Texas at San Antonio, San Antonio, USA

Email: hueytzen.wu@tamuk.edu, dd1273@yahoo.com

Copyright © 2013 Hueytzen J. Wu, Wan-Hong Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received November 29, 2012; revised December 28, 2012, accepted January 19, 2013

**Keywords:** Closed Ã_{x}-Filter; Open and Closed C*_{D}-Filter Bases; Basic Open and Closed C*_{D}-Filters; Compactification; Stone-Čech and Wallman Compactifications

ABSTRACT

Closed Ã_{x}- and basic closed C*_{D}-filters are used in a process similar to Wallman method for compactifications of the topological spaces Y, of which, there is a subset of containing a non-constant function, where is the set of bounded real continuous functions on Y. An arbitrary Hausdorff compactification of a Tychonoff space X can be obtained by using basic closed C*_{D}-filters from in a similar way, where is the set of real continuous functions on Z.

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 r_{f} 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 K_{x}, V_{x}, ℰ_{x} and Å_{x}, 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 Ã_{x}- and basic closed C*_{D}-filters in a process similar to the Wallman method, where, is the set {N_{x}|N_{x} 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 r_{f} 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 ansuch that

Since is compact and is contained in

there exist r_{1},···,r_{n} 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 r_{f} 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 K_{x} and V_{x}, respectively. Let ℰ and ℰ_{x} (or Å and Å_{x}) be the closed (or open) filters generated by K and K_{x} (or V and V_{x}), respectively, then ℰ and ℰ_{x} (or Å and Å_{x}) 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 N_{x} be the union of and ℰ_{x}, if V_{x} is an open nhood filter base at x; let N_{x} be the union of and, if V_{x} is not an open nhood filter base at x. For each x in Y, N_{x} is a ℘-filter with Ã being N_{x}. (See 12E. in [1, p.82] for definition and convergence). N_{x} is called a closed ℘_{x}-filter. It is clear that K_{x} is contained in ℰ_{x} and ℰ_{x} is contained in N_{x}, N_{x} converges to x for each x in Y. Let be the set of all N_{x}, 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 N_{x} 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), N_{x} is in and N_{x} 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 N_{x} for some N_{x} in, then

is in N_{x} for all, (ii) if ℭ is ℰ which is in, then

is in ℰ for all (iii) by Cor. 2.2, the r_{f} 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 N_{x} for all thus f*(N_{x}) 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 t_{f}. We show that for anythere is a in t such that ℭ is in

Let

and Since

and by Cor. 3.5 (iv), ℭ. Next, for any ℭ_{s} in, if for all x in Y, by Cor. 3.5 (iv), pick a in ℭ_{s} such that

then is in ℭ_{s}. By Cor. 3.5 (iii) and Lemma 3.8 (i), ℭ_{s}_{ }_{}

is in. If ℭ_{s} is N_{x} for some x in Y, by Cor. 3.5 (i), N_{x} in if, thus

;

i.e., ℭ_{s} is N_{x} 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, N_{x} = N_{y} 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) r_{f} = f(x) for all f in; i.e. ℰ = ℰ_{x}, (ii) V_{x} is an open nhood base at x in Y and (iii)

is an open nhood base at k(x) in.

Proof. If ℰ converges toin Y, (i): for each,

for all thus; i.e., ℰ = ℰ_{x}. (ii): Since ℰ converges to x in Y, for any open nhood of, there is

which is contained in ℰ_{x} = ℰ for some such that Since x is in

and S is in V_{x}, thus V_{x} is an open nhood base at x; (iii): For any in t such that N_{x} 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 ℭ_{s} in, if for all f in, then s_{f}

is in for all f in. Pick a

such that for all f inthen

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) = r_{f} for all f* Î D*ℰ

for any For any such that ℰby Cor. 3.5 (iv) there exists a

ℰ

for some such that E Ì Y – F. For let

then ℰV*. We claim that For any ℰ_{t} in, let f*(ℰ_{t}) = t_{f} for each f* in. Then for each f in, is in

and ℰ_{t}

for all Pick a such that

for each f in, then

Since ℰ_{t}, so ℰ_{t} 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. Letbe 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 ℰ_{o} in. This implies that ℱ converges to ℰ_{o} too. Hence, ℰ_{o} is in F for all F in ℱ; i.e., ℰ_{o} 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 (X^{w},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 V_{x} is an open nhood base at x, it is clear that ℰ_{x} converges to x. Let where X_{E} = {ℰ_{x}_{ }|xX} and X_{E} = {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 setin 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 ℰ_{x} 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 ℰ_{x} is in, then

for all f in, Since V_{x} 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) = r_{f} if

ℰ for all. Then f* is welldefined and f*(ℰ_{x}) 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 ℰ_{x}. 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 ℰ_{x} and V*_{k}_{(}_{x}_{)} is =

V*_{ℰ}_{x} =

is an open nhood base at ℰ_{x}. 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 ℰ_{s}, ℰ_{t} in, let

and similarly for K_{t}. Since ℰ_{s} is not equal to ℰ_{t}, K_{s} is not equal to K_{t} and that has a g such that are equivalent, where which is contained in ℰ_{s} and which is contained in ℰ_{t} for all thus by the definition of g*, g*(ℰ_{s}) g*(ℰ_{t}).

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 (X^{w},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 °ℰ_{z} generated by °K_{z} 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 ℰ_{s}, ℰ_{t} in, let

and similarly for K_{t} such that ℰ_{s} and ℰ_{t} are generated by K_{s} and K_{t}, respectively. Assume that °ℰ_{s} and °ℰ_{t} converge to z_{s} and z_{t} in Z, respectively. Then ℰ_{s} is not equal to ℰ_{t}, °ℰ_{s} is not equal to °ℰ_{t} and z_{s} is not equal to z_{t} are equivalent. Henceis well-defined and one-one. For each z in Z, let °ℰ_{z} be the basic closed C*_{°}_{D}-filter at z, since Z is compact Hausdorff and

is an open nhood base at z, thus °ℰ_{z} converges to z. Let ℰ_{z} be the element in induced by °ℰ_{z}, then, (ℰ_{z}) = 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 inand

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*) = Cl_{Z}(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 toas above. Then (bX,) is homeomorphic to such that

Corollary 5.20 Let (gX,) be the Wallman compactification of a normal T_{1}-space X,

and is defined similarly to as above. Then (gX,) is homeomorphic to such that

.

REFERENCES

- S. Willard, “General Topology,” Addison-Wesley, Reading, 1970.
- H. J. Wu and W. H. Wu, “An Arbitrary Hausdorff Compactification of a Tychonoff Space X Obtained from a C*
_{D}-Base by a Modified Wallman Method,” Topology and its Applications, Vol. 155, No. 11, 2008, pp. 1163- 1168. doi:10.1016/j.topol.2007.05.021