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 r₁,···,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 à 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*_ℰx =

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 T₁-space X,

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

.

REFERENCES