Int. J. Contemp. Math. Sciences, Vol. 4, 2009, no. 22, 1079 - 1084
Compactifications of Discrete Spaces U. M. Swamy
[email protected] Ch. Santhi Sundar Raj, B. Venkateswarlu and S. Ramesh Department of Mathematics Andhra University Visakhapatnam, India
[email protected] Abstract Compactifications of an infinite discrete space D are discussed. Equivalent conditions are obtained for a compactification of D to be totally disconnected. The existance of infinite compactifications of D which are totally disconnected is proved.
Mathematics Subject Classification: 54D35, 54D60 ˇ Keywords: Compactification, Stone-Cech compactification, Boolean Algebra, Ultra filters and The Stone Space
1
Introduction
For any topological space X, a compact Hausdorff space Y is called a compactification of X, if there is an embedding φ of X into Y such that φ(X) is dense in Y . We shall identify X with the subspace φ(X) of Y . If Y X is a singleton set then Y is called a one-point compactification of X. If Y X is a finite (countable or infinite) set, then Y is called a finite (countable or infinite respectively) compactification of X. It is well known that a space X has a compactification if and only if it is completely regular; for, any compact Hausdorff space is completely regular and any subspace of a completely regular space is completely regular. ˇ A compactification Y of X is called a Stone Cech compactification, if any continuous mapping from X into any compact Hausdorff space Z can be extended (uniquely) to a continuous mapping from Y into Z. The exitance of the
1080
U. M. Swamy, Ch. Santhi Sundar Raj, B. Venkateswarlu and S. Ramesh
ˇ Stone Cech compactification of a completely regular space and its uniqueness (upto homeomorphism) are well known. Further, given any completely regular space X, we know the existence and uniqueness of the one-point compactification of X.
2
Compactifications of Discrete Spaces
In this paper we are mainly interested in the compactifications of a discrete space. If D is a finite discrete space then D itself is a compactification of D and hence we consider infinite discrete spaces only. Throughout this paper D denotes an infinite discrete space. In the following we construct several compactifications of D. First let us recall that the power set P(D) of all subsets of D together with the usual set operations is a Boolean algebra. let B be a Boolean subalgebra of P(D). A non-empty subset U of B is called a filter of B, if it is closed under finite intersections and every superset of a member of U in B is again a member of U. A proper filter which is maximal among the class of proper filters is called an ultrafilter of B. It is well known that a proper filter U of B is maximal if and only if B U is closed under finite unions. Let SpecB denote the set of all ultra filters of B. For any A ∈ B, let Aˆ = {U ∈ SpecB | A ∈ U}. Then {Aˆ | A ∈ B} forms a base for a topology on SpecB which is called the Stone topology. SpecB together with this Stone topology is called the Stone space of B or the Spectrum of B and will be denoted by SpecB. It is well known that the Stone space SpecB is compact Hausdorff and totally disconnected and that any such topological space is homeomorphic to SpecB, for some suitable Boolean algebra B. This result is known as the Stone-Duality [4]. Let B0 = {A ⊆ D | A or D A is finite}. Then B0 is a Boolean subalgebra of P(D) and let us recall the following well known result [1, 5]. Theorem 2.1. Let D be an infinite discrete space. Then Spec(B0 ) is the ˇ one-point compactification of D and Spec(P(D)) is the Stone- Cech compactification of D In the following we shall prove that SpecB is a compactification of D, for any Boolean subalgebra B of P(D) containing B0 . Theorem 2.2. Let D be an infinite discrete space and B a Boolean subalgebra of P(D) containing B0 . Then the Stone space SpecB is a compactification of D in which D is open. Proof. It is well known that SpecB is a compact Hausdorff space. Now define a map α : D → SpecB by α(d) = {A ∈ B | d ∈ A}. Then, since B separates points of D, α is one-one. Since D is discrete, we have α : D → SpecB is a
1081
Compactifications of discrete spaces
continuous map. Now we shall prove that α is an embedding of D into SpecB. and hence each For any d ∈ D, it can be easily seen that {α(d)} = α(D) ∩ {d} singleton set is open in α(D). This proves that α(D) is discrete and hence α is an embedding of D into SpecB. Next, if Aˆ is a non-empty basic open set in SpecB, then A = ∅ and if a ∈ A, then α(a) ∈ α(D) ∩ Aˆ and so α(D) ∩ Aˆ = ∅. Therefore α(D) is dense in SpecB. Thus SpecB is a compactification of D. since any ultra filter of B containing {d} Also, for any d ∈ D, {α(d)} = {d}, must be α(d). This proves that α(D) is open in SpecB. Let us recall that a space X is said to be totally disconnected if clopen sets separate points of X, in the sense that, for any two distinct points of X, there is a clopen set in X containing one of these two points and not containing the other. It is clear that a compact Hausdorff space is totally disconnected if and only if the clopen sets form a base for open sets. Now we have the following, which is a converse of theorem 2.2, in the sense that any totally disconnected compactification of D must necessarily be (homeomorphic to) the Spectrum of a Boolean subalgebra of P(D) containing B0 . Theorem 2.3. A compactification of an infinite discrete space D is totally disconnected if and only if it is homeomorphic to SpecB for some Boolean subalgebra B of P(D) containing B0 . Proof. Let Y be a compactification of D. Suppose that Y is totally disconnected. Consider B = {A ∩ D | A is a clopen subset of Y } First observe that, since D is dense in Y, A∩D = ∅ ⇔ A=∅ A∩D ⊆B∩D ⇔ A⊆B A∩D =B∩D ⇔ A=B
and
for any clopen sets A and B in Y. Now, clearly B is a Boolean subalgebra of P(D). For any d ∈ D, {d} is open in D and hence there is a clopen subset A of Y such that {d} = A ∩ D. This implies that {d} ∈ B for all d ∈ D and hence all finite subsets of D and their complements must be in B, so that B contains B0 . Now we shall prove that Y ∼ = SpecB. Define f : Y → SpecB by f (y) = {A ∩ D | A is clopen in Y and y ∈ A}. It can be easily verified that f (y) is an ultra filter of B and that f is an injection. On the otherhand, if U is any ultra filter of B, then U satisfies the finite intersection property and so is {A | A is clopen in Y and A ∩ D ∈ U}. By the compactness of Y, there exists y ∈ Y such that, for any clopen set A in Y, y ∈ A when ever A ∩ D ∈ U. From this, it follows that U ⊆ f (y) and
1082
U. M. Swamy, Ch. Santhi Sundar Raj, B. Venkateswarlu and S. Ramesh
hence U = f (y). Therefore f is a surjection too. For any A ∩ D ∈ B, where A is clopen in Y. f −1 (A ∩ D) = {y ∈ Y | f (y) ∈ A ∩ D} = {y ∈ Y |A ∩ D ∈ f (y)} = {y ∈ Y |y ∈ A} = A This implies that f is continuous and hence f is a homeomorphism(since both Y and SpecB are compact Hausdorff spaces). The converse follows from Theorem 2.3. In the following, we prove that any finite compactification of a discrete D is totally disconnected and hence,by the above theorem, it must be homeomorphic to the Stone space of a Boolean subalgebra of P(D) containing B0 . Theorem 2.4. Let Y be a compactification of an infinite discrete space D and Y D be finite. Then the following hold. 1. Y is totally disconnected. 2. There exists pair-wise disjoint infinite sets D1 , D2 , · · ·, Dn of D such that n D= Di and Y is the topological union of the one-point compactificai=1
tions of Di ’s. 3. Let Bi = {X ⊆ Di | X or Di X is finite}. Then B1 × B2 × · · · × Bn is isomorphic to the Boolean algebra of all clopen subsets of Y and hence Y is homeomorphic to Spec(B1 × B2 × · · · × Bn ). Proof. (1) Let Y D = {x1 , x2 , · · ·, xn }. Since Y is Hausdorff, we can find open sets A1 , A2 , · · ·, An in Y such that xi ∈ Ai and Ai ∩ Aj = ∅ for all i = j. Since Y D is finite and hence closed in Y , D is open in Y . Also, since D is discrete, {d} is open in D and hence in Y, for all d ∈ D. The class of all singleton sets {d}, d ∈ D, together with the Ai s forms an open cover for Y. Since Y is compact, there exists a finite subset E of D such that E∪A1 ∪A2 ∪, ···, ∪An = Y, each Ai E is open in Y and {E, A1 E, A2 E, · · ·, An E, } is a class of open sets in Y which are pairwise disjoint and cover Y. Therefore, each Ai E is closed also. Since each {d} is clopen in Y, it follows that the points of Y are separated by clopen subsets of Y. Thus Y is totally disconnected. (2) There exists pairwise disjoint clopen sets Y1 , Y2 , · · ·, Yn in Y such that Y = Y1 ∪ Y2 ∪, · · ·, ∪Yn and xi ∈ Yi for each i. (for example, we can take Y1 = E ∪ A1 and Yi = Ai E for i > 1). Put Di = Yi ∩ D. Then Di = ∅ (since D is dense and Yi is a nonempty open set in Y.) Also, each Di is infinite; for, otherwise Di ∪(Y Yi ) is a closed set containing D and hence Y = Di ∪(Y Yi) n which is a contradiction since xi ∈ / Di ∪ (Y Yi ). Further D = Di . Since Yi i=1
Compactifications of discrete spaces
1083
is closed in Y, Yi is compact and hence Yi is the one-point compactification of Di . Thus Y is the topological union of the one-point compactifications of Di s (3) If is well-known that Yi is homeomorphic to the Stone space SpecBi , where Bi = {X ⊆ Di | X or Di X is finite }. Further Y =
n i=1
Yi =
i
SpecBi ∼ = Spec(B1 × · · · × Bn )
and hence B1 × · · · × Bn is isomorphic to the Boolean algebra of all clopen subsets of Y (by the Stone duality). Though we have proved that any finite compactification of D is totally disconnected, we do not know whether finiteness can be dropped in this. However, ˆ there are examples of infinite compactifications, other than the Stone-Cech compactification, which are totally disconnected. Consider the following. Example 2.5. Let D =
∞ n=1
Dn , where each Dn is infinite and
Dn ∩ Dm = ∅ for all n = m. Let B = {A ⊆ D | for each n, either Dn ∩ A or Dn A is finite}. Then B is a Boolean subalgebra of P(D) containing B0 and hence, by theorem 2.3, Spec(B) is a compactification of D which is totally disconnected. It can be easily seen that B is an incomplete Boolean algebra and hence B can not be isomorphic to P(D) so that, by the Stone duality, Spec(B) can not be ˇ homeomorphic to Spec(P(D)). This says that Spec(B) is not the Stone-Cech compactification of D. Further, for each positive integer n, let Un = {A ∈ B | Dn ∩ A is infinite }. Then Un is an ultra filter of B ; that is, Un ∈ SpecB. Let f : D → SpecB be the usual embedding defined by f (d) = {A ∈ B | d ∈ A}. Then Un = f (d) for all d ∈ D and for all n ∈ Z+ . Also, Un = Um for all n = m. Thus SpecB − f (D) contains infinitely many points, so that SpecB is an infinite compactification of D. Observe that each A ∈ B can be uniquely ∞ expressed as A = An , where An is in the Boolean algebra Bn = {E ⊆ n=1
Dn | either E or Dn E is finite } and that B is isomorphic to the product ∞ algebra Bn . By the Stone duality, SpecB is homeomorphic to the direct sum n=1
[1] of the spaces SpecBn , n ∈ Z+ .
1084
U. M. Swamy, Ch. Santhi Sundar Raj, B. Venkateswarlu and S. Ramesh
ACKNOWLEDGEMENTS. The fourth author is thankful to Council of Scientific and Industrial Research for their financial support in the form of CSIR-SRF(NET).
References [1] P.R. Halmos, Lectures on Boolean Algebras, D.Van Nostran Company Inc., Priceton, (1963). ˇ [2] Hindman, Neil. and Dona Strauss, Algebra in Stone-Cech Compactification, Walter de Gruyter, Berlin, NewYork, (1998). [3] J.R.Munkres, Topology, Prentice-Hall of India Private Limited, New Delhi, (1996) [4] M.H.Stone, The theory of Representations of Boolean algebras, Trans. Amer. Math. Soc., 40(1936), 37-111. [5] M.H.Stone, On the Compactification of topological spaces, Ann. Soc. Pol. Math., 21(1948) 153-160. Received: January, 2009
