A Modified Wallman Method of Compactification

Closed xand 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 D   * C Y containing a non-constant function, where   * C Y is the set of bounded real continuous functions on Y. An arbitrary Hausdorff compactification  ,  Z h   of a Tychonoff space X can be obtained by using basic closed C*D-filters from   C Z | D f h f D       in a similar way, where is the set of real continuous functions on Z   C Z


Introduction
Throughout this paper,   T   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 and let V be the set for any , 0  f x for all f D 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 w  of Y is obtained by using closed  x -and basic closed C* D -filters in a process similar to the Wallman method, where   * C Y Theorem 2.1 Let ℱ be a filter on Y.For each f in there exists a r f in such that for any F in ℱ and any 0   (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 t in there exist an 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

A Closed  x -Filter and a Modified Wallman Method of Compactification
Let Y be a topological space, of which, there is a subset of an open nhood filter base at x; let N x be the union of 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 Thus, ℭ does not belong to , contradicting the assumption.For [] is obvious from (i). * Equip with the topology  induced by .For Proposition 3.9 For each f in , f* is a bounded real continuous function on .
. For the continuity of f*: If ℭ is in and is t f .We show that for any Thus is well-defined and one-one.Let be a function from To show the continuity of and , for any k So, and are continuous.(ii) is obvious.(iii) For any and all  > 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.
and S is in V x , thus V x is an open nhood base at x; (iii): For any * F in  such that N x is not in * F , by Cor.3.5 (i), x is not in F , and by (ii) of Lemma 3.11 above, x is in Cor. 3.5 (i), Lemmas 3.6 (ii) and 3.8 (i) imply that We claim that * : 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, for any For any by Cor.3.5 (iv) there exists a *,   Pick a 0 where

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 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 where X E = {ℰ x |x X} 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 set F in (ii), (iii) and (iv) are obvious.(4.15.6) (See Lemma 3.6) For any two nonempty sets and E F in X, (i) for any 0.

 
  (4.15.9) (See Prop.3.9) For each f in , f* is a bounded real continuous function on D w X .(4.15.10) (See Lemma 3.10) Let be defined by for any * * , 0 is an open nhood base at ℰ in 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

The Homeomorphism between (X w ,k) and (Z,h)
Let   , h Z be an arbitrary Hausdorff compactification of X, then X is a Tychonoff space.Let denote D    C Z which is the family of real continuous functions on Z, and let separates points of X, the topology on X is the weak topology induced by and contains a non-constant function.

D D
Let   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 and all 0   .Thus, if K or ℰ is well-defined, so is °K or and similarly for K t such that ℰ s and ℰ t are generated by K s and K t , respectively.Assume that °ℰs and °ℰt converge to z s and z t in Z, respectively.Then ℰ s is not equal to ℰ t , °ℰs is not equal to °ℰt and z s is not equal to z t 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 is an open nhood base at z, thus °ℰz converges to z.Let ℰ z be the element in w X induced by °ℰz , then,  (ℰ z ) = z.Hence, is one-one and onto. , for any , 0 Since is one-one, h f f h    for all f in , so (b) iff (c): and any  > 0. Since for any °f in , D  0,   (c) iff (d): for any Since   , 0 [2] 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

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* Dfilter), denoted by ℰ (or Å).
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 space X can be obtained by using the basic closed C* D -filters on X from D f V are called a closed and an open C* D -filter bases, respectively.If for all f 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.

0
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, 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) For any basic closed C* D* -filter ℰ* on , ℰ* converges in .w Y 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: 15.7) (See Prop.3.7)  = {F*|F is a nonempty closed set in X} is a base for the closed sets of w X .(4.15.7.1) (See the definitions for the topology  on and f* for each f in in Section 3.) topology  induced by .For each f in , define by f* 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

°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,

X
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 be the basic closed C* °D-filter on Z induced by ℰ.If °ℰ converges to z in Z,

FF
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 .Hence, T(F*) = Cl Z (h(F))is closed in Z for all F* in .Thus, -one, onto and both Z and w X are compact Hausdorff, by Theorem 17.14 in [1, p.123],  is a homeomorphism.Finally, from the definitions of and , it is clear that Let (X, ) be the Stone-Čech compactification of a Tychonoff space X, to  as above.Then (X, ) is homeomorphic to h  , w  X k such that