C-scattered type and minimal compactifications with countable remainder

Topology and its Applications 196 (2015) 120–132

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Topology and its Applications www.elsevier.com/locate/topol

C-scattered type and minimal compactifications with countable remainder Michael G. Charalambous Department of Mathematics, University of the Aegean, 83 200 Karlovassi, Samos, Greece

a r t i c l e

i n f o

Article history: Received 20 December 2014 Received in revised form 2 September 2015 Accepted 5 September 2015 Available online xxxx MSC: 54D35 54D40 54D45 Keywords: Compact Locally compact Rim-compact Čech-complete C-scattered Separable metric spaces

a b s t r a c t R(X) is the closed subset of a space X consisting of the points that do not have a compact neighbourhood. For each ordinal α, R α (X) is defined inductively as follows: 0 X, Rα+1 (X) = R(Rα (X)) and, if α is a non-zero limit ordinal, Rα (X) = R  (X) = β (X). According to R. Telgársky, X is called C-scattered if Rα (X) = ∅ for R β<α some α. The first α for which Rα (X) is locally compact is called here the C-scattered level of X. A space is called a minimal space if it is homeomorphic to its perfect images, and a compactification Y of X is called a minimal compactification of X if for every compactification Z of X smaller than Y , Y \X and Z\X are homeomorphic. For each α < ω1 , we define a C-scattered countable metric space Nα of level α which is contained as a closed subset in every metric C-scattered space of level α. We characterize the spaces Nα topologically and prove that they are minimal spaces. We use the results to refine results of Zippin and Hoshina by showing that a rim-compact separable metrizable space X with Rω0 (X) = ∅ has a minimal metrizable compactification with remainder homeomorphic to Nα or the direct sum of countably many copies of Nα , for a certain α ≤ ω0 . © 2015 Published by Elsevier B.V.

0. Introduction and definitions If Y and Z are compactifications of X, according to the usual order of compactifications, Y ≤ Z if there is a map f : Z → Y that keeps the points of X fixed. As is well known, the Alexandroff or one-point compactification of a locally compact non-compact space is its smallest compactification with respect to ≤, and if a space X has a smallest compactification, then X is locally compact (see [2]). We will say that a compactification Z of X is a minimal compactification of X if for every compactification Y of X with Y ≤ Z, the remainders Z \ X and Y \ X are homeomorphic. We will call a space S a minimal space if S is homeomorphic to every image of S under a perfect map. Evidently, infinite discrete spaces are minimal spaces and a compactification Z of X is a minimal compactification whenever the remainder Z \ X is a minimal space. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.topol.2015.09.004 0166-8641/© 2015 Published by Elsevier B.V.

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L. Zippin in [9] generalized local compactness, introducing semicompact spaces, nowadays also known as rim-compact spaces. He proved that a separable metric, Čech-complete and semicompact space may be compactified by the addition of a countable set. T. Terada [8] partially generalized Zippin’s result by proving that a Čech-complete and rim-compact space X such that R(X) is compact metrizable can be compactified by the addition of a discrete countable space. Here R(X) = X \ L(X), where L(X) denotes the biggest open locally compact subset of X. Finally, T. Hoshina [3] proved that if X is Čech-complete and semicompact and R(X) is separable metrizable, then X has a compactification Y with remainder Y \ X countable. Hoshina then deduced that a metrizable space X can be compactified by adjoining countably many points iff X is Čech-complete and semicompact and R(X) is Lindelöf. The Alexandroff and Terada compactifications are evidently minimal. With regard to Zippin’s or Hoshina’s compactification, however, we have no information other than the countability of the remainder. In this paper, we look for conditions that will produce minimal compactifications with countable remainder. For each ordinal α, Rα (X) is defined inductively as follows: R0 (X) = X, Rα+1 (X) = R(Rα (X)) and, if α  is a non-zero limit ordinal, Rα (X) = β<α Rβ (X). Clearly, for some ordinal α, Rβ (X) = Rγ (X) = R∞ (X) for every β, γ > α. A.H. Stone [6] calls R∞ (X) the non-locally compact kernel of X and R. Telgársky [7] calls a space X C-scattered if Rα (X) = ∅ for some α. We define the C-scattered level of X, denoted by lc(X), to be ∞ if R∞ (X) = ∅ and −1 if X = ∅; otherwise, lc(X) is the first ordinal α for which Rα (X) is locally compact.1 Evidently, for a non-empty C-scattered separable metric space X, lc(X) is a countable ordinal. In the first section of this paper, for each countable ordinal α, we introduce the countable subspace Nα of the real line, in a sense the simplest space of C-scattered level α, and explore some of its properties. An important property in our investigations is that of pointedness: a C-scattered space X is said to be pointed at a point x if Rlc(X) = {x} and lc(F ) < lc(X) for every closed set F of X not containing x. Nα is pointed at 0. The second section of the paper is devoted to the study of the behaviour of C-scattered level and pointed spaces under perfect maps. In the third section, we characterize all locally pointed, separable metric C-scattered spaces X such that every L(Rβ (X)) is discrete for each ordinal β ≤ α = lc(X) and Rα (X) is empty, a singleton or infinite. All such spaces are minimal and are homeomorphic to one of the spaces Nα , Nα \{0} or Nα ⊕(Nα \{0}) (see Theorem 3.8). We apply the results in the fourth section to show that a rim-compact, Čech-complete space X with R(X) separable metric and Rω0 (X) = ∅ has a minimal compactification with countable remainder expressible in terms of the spaces Nα (see Theorem 4.7). This generalizes the Alexandroff and Terada compactifications and refines the results of Zippin and Hoshina quoted above. All spaces in this paper are Tychonoff. I denotes the interval [0, 1], N denotes the natural numbers, Q the rationals, ω0 the first infinite ordinal and ω1 the first uncountable ordinal. A map is a continuous function. If f : X → Y is a function, A ⊂ X, B ⊂ Y and f (A) ⊂ B, we write f : A → B to denote the restriction of f with domain A and range B, instead of the more proper f |A : A → B. We write X ∼ = Y to indicate that X and Y are homeomorphic. The rest of our notation and terminology agrees with [2]. 1. The spaces Nα A space X will be said to have C-type −1 if X = ∅ and ∞ if X is not C-scattered. Otherwise, X will be said to have C-type (α, 0), (α, 1) or (α, 2) if lc(X) = α and, respectively,

1 By analogy with scattered spaces, the first α for which Rα (X) = ∅ should be called the C-scattered height of X: Modify the definition of R(X) to mean the biggest subset of X that contains no isolated points of X, and define Rα (X) as above. Then X is scattered iff Rα (X) = ∅ for some α, and the first α for which this happens is called the scattered height of X.

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(a) Rα (X) = ∅, (b) Rα (X) is compact but not empty or (c) Rα (X) is locally compact but not compact. Clearly, • X has C-type (0, 1) iff X is compact but not empty, • X has C-type (0, 2) iff X is locally compact but not compact, and • X has C-type (α, 0) iff α is a limit ordinal other than 0, Rα (X) = ∅ and Rβ (X) = ∅ for β < α. Let Lα (X) denote the open set X \ Rα (X) of X. Clearly, Lα+1 (X) = Lα (X) ∪ L(Rα (X)) and, for a limit   ordinal α, Lα (X) = β<α Lβ (X). Also, Lα (X) = β<α L(Rβ (X)) and lc(Lα (X)) ≤ α. The following two elementary results are frequently used without explicit mention. Lemma 1.1. Let F be a closed and G an open subset of X. Then for each ordinal α, Lα (X) ∩ F ⊂ Lα (F ), Rα (F ) ⊂ Rα (X) ∩ F , Lα (G) = Lα (X) ∩ G and Rα (G) = Rα (X) ∩ G. Hence, lc(F ), lc(G) ≤ lc(X). Proof. We indicate the proof for G: L(G) ⊂ L(X) and L(X) ∩ G ⊂ L(G) because open sets of locally compact spaces have the same property. Thus, L(G) = L(X) ∩ G. Now, if Lα (G) = Lα (X) ∩ G for every open set G of a space X, then Rα (G) = G \ Lα (G) = G \ (Lα (X) ∩ G) = (X \ Lα (X)) ∩ G = Rα (X) ∩ G and Lα+1 (G) = Lα (G) ∪ L(Rα (X) ∩ G) = Lα (G) ∪ (L(Rα (X)) ∩ (Rα (X) ∩ G)) = Lα (G) ∪ (L(Rα (X)) ∩ G) = (Lα (X) ∪ L(Rα (X))) ∩ G = Lα+1 (X) ∩ G. Hence, an easy inductive argument establishes the result. 2 Lemma 1.2. For a topological sum X = α L (X) =

 s∈S

 s∈S

Xs and each ordinal α,

α α L (Xs ) and R (X) =



α R (Xs ).

2

s∈S

Example 1.3. For each countable ordinal α > 0, we define the subspace Nα of I as follows: N0 = {0} and Nα+1 1 1 1 1 consists of 0 together with a copy of Nα in each of the intervals ( n+1 + m+1 , n+1 +m ), m ≥ n(n + 1), n ∈ N. If α is non-zero limit ordinal, we choose a bijection φ : N → α and let Nα consist of 0 together with a copy 1 of Nφ(n) in the interval ( n+1 , n1 ), for each n ∈ N. 2  It is useful to introduce the notation X = {x} ∨ s∈S Xs for the space obtained by adjoining a new   point x to the sum X together with the s∈S Xs , with topology generated by the open sets of  s∈S s sets X \ Xs , s ∈ S. Clearly, for a countable limit ordinal α, Nα = {0} ∨ β<α Nβ . Moreover, Nα+1 =  {0} ∨ i<ω0 Mi , where each Mi is the direct sum of ℵ0 copies of Nα . Proposition 1.4. For each ordinal α, Rα (Nα ) = {0} and each L(Rβ (Nα )) is discrete. Proof. The proof is by induction on α, the result being evident for α = 0. Suppose the result true for some α. Then from Lemma 1.2, Rα (Nα+1 \ {0}) consists of a single point in each of the pairwise disjoint 1 1 1 1 intervals ( n+1 + m+1 , n+1 +m ), m ≥ n(n + 1). From Lemma 1.1 and the fact that Rα (Nα+1 ) is a closed α subset of Nα+1 , R (Nα+1 ) = Rα (Nα+1 \ {0}) ∪ {0}. Note that any neighbourhood of 0 contains the trace 1 of an interval ( n+1 , n1 ), which contains infinitely many points from Rα (Nα+1 ), from which it follows that Rα (Nα+1 ) is not locally compact at 0. Hence Rα+1 (Nα+1 ) = {0} and Rβ (Nα+1 ) = ∅ for β > α. For β ≤ α, because 0 ∈ / L(Rβ (Nα+1 )), L(Rβ (Nα+1 )) is the sum of copies of L(Rβ (Nα )) and is therefore discrete, by the induction hypothesis. The case of a non-zero limit ordinal is similar. 2

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Thus, Nα is of C-type (α, 1), Nα+1 \ {0} is of C-type (α, 2) and, if α is a non-zero limit ordinal, Nα \ {0} is of C-type (α, 0). Remark 1.5. It is readily seen that if X is C-scattered and each L(Rα (X)) is discrete, then X is scattered and its scattered height and level coincide with the corresponding C-scattered characteristics. Proposition 1.6. For any countable ordinals α, β with α > β and any countable collection of ordinals {βi : i ∈ I} with βi ≤ β, Nα ∼ = Nα ⊕



Nβi .

i∈I

Proof by induction on α. Suppose the result true for an ordinal α. Given a countable collection of ordinals {βi : i ∈ I} with βi ≤ α, let J = {i ∈ I : βi = α}. Observe that subtracting from Nα+1 any number of the 1 1 pairwise disjoint intervals ( 21 + m+1 , 12 + m ), m ≥ 2, leaves a space homeomorphic to Nα+1 . Thus, the sum  of Nα+1 and countably many copies of Nα is homeomorphic to Nα+1 . Hence Nα+1 ∼ = Nα+1 ⊕ i∈I\J Nαi , where αi = α, and Nα+1 ⊕



Nβi ∼ = (Nα+1 ⊕

i∈I



Nβi ) ⊕

i∈J





(Nαi ⊕ Nβi ) ∼ = Nα+1 ⊕

i∈I\J

Nαi ∼ = Nα+1 .

i∈I\J

Now let α be a limit ordinal, β an ordinal < α and {βi : i ∈ I} a countable collection of ordinals with  βi ≤ β. Suppose the result to be proved holds for the ordinal β + 1. Then Nα = {0} ∨ γ<α Nγ and  Nβ+1 ∼ = Nβ+1 ⊕ i∈I Nβi . Hence, Nα ⊕





Nβi ∼ = ({0} ∨

i∈I

Nγ ) ⊕ (Nβ+1 ⊕



Nβi ) ∼ = ({0} ∨

i∈I

γ<α,γ=β+1



Nγ ) ⊕ Nβ+1 ∼ = Nα .

2

γ<α,γ=β+1

Corollary 1.7. For each i ∈ ω0 , let Mi be a copy of Nα and Li a copy of Nα \ {0} or a copy of some Nβ    with β < α. Then i∈ω0 Mi ⊕ i∈ω0 Li ∼ = i∈ω0 Mi . Proof. We can write  i∈ω0

 i∈ω0

Mi ⊕

Li =

 i∈ω0



Li =

i∈ω0

 i∈ω0

Ki where Ki is a copy of some Nβ with β < α. Hence Mi ⊕

 i∈ω0

Ki =

 i∈ω0

(Mi ⊕ Ki ) ∼ =



Mi .

2

i∈ω0

  Proposition 1.8. Let X = {x} ∨ i∈ω0 Xi and Y = {y} ∨ i∈ω0 Yi . Suppose that for each i ∈ ω0 there are infinitely many m and n such that Xi is homeomorphic to a clopen subset of Ym and Yi is homeomorphic   to a clopen subset of Xn . Then i∈ω0 Xi ∼ = i∈ω0 Yi and X ∼ =Y. Proof. We define a homeomorphism f : X → Y by successively defining it on a clopen subset Ai of X onto a clopen subset Bi of Y , where Xi ⊂ Ai ⊂ Ai+1 ⊂ X \ {x} and Yi ⊂ Bi ⊂ Bi+1 ⊂ Y \ {y}. It will then remain to set f (x) = y. We can of course assume that X0 = Y0 = ∅ so that we can set A0 = B0 = ∅. Having defined f from Ai to Bi , we extend it to a homeomorphism from a clopen set Ai+1 of X onto a clopen set Bi+1 of Y , with the restrictions imposed above, as follows. First, we define f from Xi+1 \ Ai onto a clopen set of some Ym , where m ∈ / {k : Yk ∩ Bi = ∅}. Next we define f −1 from Yi+1 \ (Bi ∪ f (Xi+1 )) onto a clopen set of some Xn , where n ∈ / {k : Xk ∩ Ai = ∅} ∪ {i + 1}. It then remains to set Ai+1 = Ai ∪ Xi+1 ∪ f −1 (Yi+1 ) and Bi+1 = Bi ∪ Yi+1 ∪ f (Xi+1 ). 2

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Corollary 1.9. Let (αi )i∈ω0 be a sequence of countable ordinals such that each αi < α = supi∈ω0 αi . Then  Nα = {0} ∨ i∈ω0 Nαi . 2 2. Pointed spaces If Rα (X) consists of a single point x, for some ordinal α, we will say that X is weakly pointed (at x) and will refer to x as the special point of X. Such an α will necessarily equal lc(X). If also lc(F ) < lc(X) for every closed subset F of X that does not contain x, we will call X pointed (at x). Thus, Nα and Nα ⊕ (Nα \ {0}) are weakly pointed at 0, but the second space is not pointed when α is a non-zero limit ordinal. A locally compact space X is weakly pointed iff X is a singleton iff X is pointed. Note that by Lemma 1.1, an open subset G of a weakly pointed space X containing the special point x of X is weakly pointed at x and it will be pointed if X is. Also, by Lemma 1.2, if X is weakly pointed or pointed and lc(Y ) < lc(X), the same is true of X ⊕ Y . A space X will be called locally pointed (resp. locally weakly pointed) if every x ∈ X has an open neighbourhood pointed (resp. weakly pointed) at x. Clearly, open subsets of locally pointed spaces are locally pointed. As can be seen from the following result, Nα \ {0} is locally pointed but not weakly pointed. Proposition 2.1. Nα is pointed and locally pointed for each ordinal α. Proof. We can assume α > 0. By Proposition 1.4, Nα is weakly pointed at 0. Consider a closed subset F of Nα with 0 ∈ / F . Then F is contained in the sum of countably many spaces homeomorphic to Nβ , for some fixed β < α. By Lemmas 1.1 and 1.2 and Proposition 1.4, lc(F ) ≤ β < α = lc(Nα ). Thus, Nα is pointed at 0. Consider a point x = 0. Then x will be contained in some copy N of Nβ that is embedded as a clopen subset of Nα , for some fixed β < α. Hence, by an obvious induction hypothesis, there is an open subset G of N which is pointed at x. Thus, Nα is locally pointed. 2 As we have already observed, X = Nω0 ⊕ (Nω0 \ {0}) is weakly pointed but not pointed. Here Rω0 (X) is a singleton and lc(X) = ω0 . We note, however, the following result. Proposition 2.2. If lc(X) is a successor ordinal and X is weakly pointed, then X is pointed. If Rω0 (X) = ∅ and L(Rn (X)) is discrete for each n < ω0 , then X is locally pointed. Proof. Suppose Rα+1 (X) = {x}. Let F be a closed set of X not containing x. Then Rα+1 (F ) = ∅. Hence Rα (F ) is locally compact, lc(F ) ≤ α < α + 1 = lc(X) and X is pointed. Suppose Rω0 (X) = ∅ and L(Rn (X)) is discrete for each n < ω0 . A point x of X belongs to L(Rn (X)) for some n < ω0 . If n = 0, then {x} is an open set pointed at x. Suppose that n > 0. Then, because L(Rn (X)) is discrete, G = {x} ∪ Ln (X) ⊂ Ln+1 (X) = L(Rn (X)) ∪ Ln (X) is open in X with Rn (G) = {x}. By the first part, G is pointed at x. Hence X is locally pointed. 2 Proposition 2.3. Let f : X → Y be a perfect surjective map and α an arbitrary ordinal. Then (a) (b) (c) (d) (e) (f) (g)

f −1 (L(Y )) ⊂ L(X), f : R(X) → R(Y ) is perfect and surjective, f : Rα (X) → Rα (Y ) is perfect and surjective, X and Y have the same C-type, f −1 (Lα (Y )) ⊂ Lα (X), if L(X) is discrete, so is L(Y ), if L(Rα (X)) is discrete, so is L(Rα (Y )).

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Proof. (a) Recall that compactness and local compactness are invariant and inverse invariant under perfect maps. Thus, the locally compact f −1 (L(Y )) is an open subset of L(X). (b) Let y ∈ Y and suppose the compact space f −1 (y) is contained in L(X). Because L(X) is locally compact, there is an open set U of X with f −1 (y) ⊂ U and cl(U ) is compact. Because f : X → Y is closed, there is an open neighbourhood V of y with f −1 (y) ⊂ f −1 (V ) ⊂ U . But then cl(V ) ⊂ f (cl(U )), cl(V ) is compact and y ∈ L(Y ). Thus, if y ∈ R(Y ), y = f (x) for some x ∈ R(X) and the claim is evident. (c) Suppose that the result holds for all ordinals β < α and α > 1. If α is a successor ordinal, by (b), the result  holds for α. Let α be a limit ordinal and consider y ∈ Rα (Y ). Suppose f −1 (y) ∩ Rα (X) = β<α (f −1 (y) ∩ Rβ (X)) = ∅. Then, since f −1 (y) is compact and Rγ (X) ⊃ Rδ (X) for γ < δ, f −1 (y) ∩ Rβ (X) = ∅ for some β < α. This contradicts the induction hypothesis that f : Rβ (X) → Rβ (Y ) is onto. We conclude that for some x ∈ Rα (X), f (x) = y and the result holds for limit ordinals as well. (d) This follows from (c) and the fact that emptiness, compactness and local compactness are invariant and inverse invariant under perfect maps. (e) Suppose the result holds for an ordinal α. Note that Lα+1 (Y ) = L(Rα (Y )) ∪ Lα (Y ) and, by (a) and (b), f −1 (L(Rα (Y ))) ∩ Rα (X) ⊂ L(Rα (X)). Hence f −1 (Lα+1 (Y )) ⊂ L(Rα (X)) ∪ Lα (X) = Lα+1 (X).  The case of a limit ordinal α is trivial on account of the fact that Lα (Y ) = β<α Lβ (Y ). (f) This follows from (a) and the fact that f : f −1 (L(Y )) → L(Y ) is perfect and surjective. (g) This follows from (c) and (f). 2 Proposition 2.3 (d) refines Theorem 1.3 of [7] which states that the class of C-scattered spaces is perfect. Corollary 2.4. Let Y and Z be compactifications of a space X. Then the remainders Y \ X and Z \ X are of the same C-type. Proof. Let π : βX → Y be the canonical map from the Čech–Stone compactification of X onto Y . Then the restriction π : βX \ X → Y \ X is perfect. Hence βX \ X and Y \ X (and for the same reason Z \ X) have the same C-type. 2 We will need the following two trivial lemmas. Lemma 2.5. If X is locally weakly pointed, then lc(X) < ∞. Proof. It readily follows from Lemma 1.1 that if {Gs : s ∈ S} is a cover of X by weakly pointed open sets, then lc(X) = sup{lc(Gs ) : s ∈ S} < ∞. 2 Lemma 2.6. If α is a successor ordinal, then lc(Lα (X)) < α. If α is a non-zero limit ordinal and K is a compact subset of Lα (X), then K ⊂ Lβ (X) for some β < α. Proof. By Lemma 1.1, Rα (Lα (X)) = Rα (X) ∩ Lα (X) = ∅. Hence, if α = β + 1, Rβ (Lα (X)) is locally compact. Therefore lc(Lα (X)) ≤ β < α.  If K is compact, α > 0 is a limit ordinal and K ⊂ Lα (X) = β<α Lβ (X), then, because Lγ (X) ⊂ Lδ (X) for γ < δ, K is contained in one of the open sets Lβ (X). 2

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Proposition 2.7. Let f : X → Y be a perfect surjective map. (a) (b) (c) (d)

If X is weakly pointed at x, then Y is weakly pointed at f (x). If X is pointed, then so is Y . Suppose ind X = 0, X is weakly pointed and Y is pointed. Then X is pointed. Suppose ind X = 0, L(Rα (X)) is discrete for each ordinal α and X is locally pointed. Then Y is locally pointed.

Proof. (a) This follows from Proposition 2.3. / F . By (b) Suppose X is pointed at x. Then lc(f −1 (F )) < lc(X) for any closed subset F of Y with f (x) ∈ Proposition 2.3, lc(F ) = lc(f −1 (F )) < lc(X) = lc(Y ). By (a), Y is pointed. (c) Suppose Rα (X) = {x}. By Proposition 2.3, Rα (Y ) = {f (x)}. By Proposition 2.2, we can suppose that α is a non-zero limit ordinal. Let F be a closed set of X with x ∈ / F . By Lemma 2.6, the compact set f −1 (f (x)) ∩ F is contained in Lβ (X) for some β < α. As ind X = 0, there is a clopen in X neighbourhood V of f −1 (f (x)) ∩ F inside Lβ (X). Because Y is pointed, by Proposition 2.3, γ = lc(F \ V ) ≤ lc(f −1 (f (F \ V ))) = lc(f (F \ V )) < lc(Y ) = α. Hence lc(F ) ≤ lc(V ⊕ (F \ V )) ≤ max{β, γ} < lc(X). Hence X is pointed. (d) By Lemma 2.5, lc(X) < ∞ and, by Proposition 2.3, lc(Y ) = lc(X) < ∞. Consider y ∈ Y . Then y ∈ L(Rα (Y )) for some ordinal α. By Proposition 2.3, we can suppose α > 0 because L(Y ) is discrete and therefore locally pointed. Also, f −1 (y) ⊂ Lα+1 (X) and f −1 (y) ∩ Rα (X) is non-empty. Because L(Rα (X)) is discrete, the compact f −1 (y) ∩ Rα (X) consists of finitely many points x1 , x2 , . . . , xn . As ind X = 0 and X is locally pointed, we can find pairwise disjoint clopen sets U1 , U2 , . . . , Un of X inside Lα+1 (X) with Ui pointed at xi . Note that lc(Ui ) = α. Put U = U1 ∪ U2 ∪ . . . ∪ Un . By Lemma 2.6, the compact subset f −1 (y) \U of Lα (X) has an open neighbourhood U0 ⊂ Lα (X) \U with lc(U0 ) < α. Then U1 ∪ U0 is pointed at x1 and we can replace U1 by U1 ∪ U0 . Thus, we can assume that f −1 (y) ⊂ U . Now let V be an open neighbourhood of y such that f −1 (V ) ⊂ U . Then each f −1 (V ) ∩Ui is pointed at xi . Consider a closed set F of V not containing y. Then, for each i, lc(f −1 (F ) ∩ Ui ) < lc(f −1 (V ) ∩ Ui ) ≤ lc(f −1 (V )). By Lemma 1.2, lc(f −1 (F )) < lc(f −1 (V )). By Proposition 2.3, lc(F ) = lc(f −1 (F )) < lc(f −1 (V )) = lc(V ). Thus, V is pointed at y and Y is locally pointed. 2  Example 2.8. The space A = {1} ∨ i∈ω0 Mi , where Mi is a copy of Nω0 \ {0}, is weakly pointed but not locally pointed at 1. X = A ⊕ Nβ , where β > ω0 , is countable metric space with each L(Rα (X)) discrete. Let Y be obtained from X by identifying the 0 of Nβ with the 1 of A. Then Y is locally pointed but X is not. 2 3. Minimal spaces Recall that we call a space X minimal if whenever there is a perfect surjective map X → Y , then X and Y are homeomorphic. It is readily seen that the empty set, singletons and infinite discrete spaces are minimal spaces. From the fact that Q is the only countable metric space without isolated points ([5, Theorem 1.9.6.]), it follows that Q is also minimal. For a proof of following lemma see [5, Corollary A.11.6.]. Lemma 3.1. Let F be a decomposition of a metric space X into pairwise disjoint closed subsets. If {F ∈ F : diam(F ) > 1/n} is finite for each n ∈ N, then F is upper semicontinuous. 2

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Proposition 3.2. Let X be subspace of a normal space Z such that cl(Z \ X) is separable metrizable and ind(Z \ X) = 0. Then there is a perfect surjective map π : Z → Y such that π −1 (π(x)) = {x} for each x ∈ X and L(Rα (Y \ π(X))) is discrete for each ordinal α. Proof. For some countable ordinal β, Rβ (Z \ X) = R∞ (Z \ X). For each α ≤ β, L(Rα (Z \ X)) is at most zero-dimensional, locally compact and separable metric. Hence it has a countable cover consisting of pairwise disjoint clopen compact subsets. All these compact sets can be enumerated as {An : n ∈ N}, where An ∩ Am = ∅ for m = n. Now partition the compact set An into a finite number of disjoint clopen sets Bn,m , m < i(n) ∈ N, of diameter < 1/n. The decomposition F = {Bn,m : n ∈ N, m < i(n)} ∪ {{x} : x ∈ R∞ (Z \ X) ∪ X} of Z is upper semicontinuous by Lemma 3.1. Let π : Z → Y be the corresponding quotient map. Evidently, π : Z → Y and π : Z \ X → Y \ π(X) are perfect and surjective. Let y ∈ L(Rα (Y \ π(X)). By Proposition 2.3, ∅ = π −1 (y) ∩ Rα (Z \ X) ⊂ L(Rα (Z \ X)). It follows that π −1 (y) is one of the clopen subsets of L(Rα (Z \ X)) that belongs to F. As π : π −1 (L(Rα (Y \ π(X)) → L(Rα (Y \ π(X)) is perfect, {y} is open in L(Rα (Y \ π(X)). Thus, L(Rα (Y \ π(X)) is discrete. 2 Corollary 3.3. Let X be a minimal zero-dimensional separable metric space. Then L(Rα (X)) is discrete for each ordinal α. Proof. Proposition 3.2 provides a perfect map f : X → Y onto a metric space Y with L(Rα (Y )) discrete for each ordinal α. As X and Y have to be homeomorphic, then L(Rα (X)) and L(Rα (Y )) are homeomorphic and therefore discrete for each ordinal α. 2 Note that each L(Rα (X)) in Corollary 3.3 is necessarily countable. Corollary 3.4. A minimal zero-dimensional separable metric C-scattered space is countable.

2

Lemma 3.5. Let X be a separable metric, pointed and locally pointed space such that L(Rβ (X)) is discrete for every ordinal β. Suppose that α = lc(X) is a non-zero limit ordinal. Then there are pairwise disjoint clopen, pointed (and locally pointed) subsets Xi of X, i ∈ ω0 , such that αi = lc(Xi ) < αi+1 , limi∈ω0 αi = α and X = {x} ∨



Xi .

i∈ω0

Proof. Let α = limi∈ω0 βi , where β0 = 0 and βi < βi+1 . Evidently, X is countable and Rα (X) consists of a single point x. Take a base {Gi : i ∈ ω0 } of the neighbourhoods of x consisting of clopen sets with Gi+1 ⊂ Gi and G0 = X. Let X0 be any clopen subset of X not containing x. Because X is pointed at x, we will have lc(F ) < α for any closed set F of X that does not contain x. We construct pairwise disjoint pointed clopen sets Xi of X that do not contain x and satisfy βi < lc(Xi ) = αi < αi+1 (so  that limi∈ω0 αi = α) and X \ Gi ⊂ j≤i Xj . Suppose this has been done for all members of ω0 smaller  than i. Choose an ordinal β such that sup{lc( j sup{αi−1 , βi } and Xi is pointed at xi .   That X = {x} ∨ i∈ω0 Xi follows from the fact that each X \ j≤i Xj is a clopen neighbourhood of x inside Gi . 2

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Lemma 3.6. Suppose x ∈ Rα (X), X is first countable at x and β < α. Then x has a base of open neighbourhoods {Gi : i ∈ ω0 } such that G0 = X, cl(Gi+1 ) ⊂ Gi and (Gi \ cl(Gi+1 )) ∩ Rβ (X) contains a closed non-compact subset of X. If X is zero-dimensional at x, each Gi can be chosen clopen. Proof. Let {Hi : i ∈ ω0 } be a base of open neighbourhoods of x such that H0 = X and cl(Hi+1 ) ⊂ Hi . If x had an open neighbourhood G such that (cl(G) \ H) ∩ Rβ (X) is compact for every neighbourhood H of x, then G ∩ Rβ (X) would be locally compact at x, x ∈ L(Rβ (X)) and x ∈ / Rβ+1 (X) ⊃ Rα (X). Hence, there is an increasing function φ : ω0 → ω0 such that φ(0) = 0 and (cl(Hφ(i) ) \Hφ(i+1) )) ∩ Rβ (X) is non-compact. We can simply set Gi = Hφ(3i) . If X is zero-dimensional at x, each Hi and so each Gi can be chosen clopen. 2 We digress at this point to prove the following interesting result. Proposition 3.7. Suppose X is a metric space, α a countable ordinal and x ∈ Rα (X). Then X contains a closed subset pointed at x and homeomorphic to Nα . Proof. The proof is by induction on α, the result being evident for α = 0. We suppose α > 0 and that the result holds for smaller ordinals. First, let α be a limit ordinal and consider it as the limit of an increasing sequence (αi )i∈ω0 . By Lemma 1.1, an open neighbourhood of x meets Rβ (X) \ {x} for every β < α. Hence there exist a base {Gi : i < ω0 } of open neighbourhoods of x with cl(Gi+1 ) ⊂ Gi and (Gi \ cl(Gi+1 )) ∩ Rαi (X) = ∅. By the induction  hypothesis, cl(G3i+2 ) \ G3i+1 contains a closed set Mi homeomorphic to Nαi . Let M = {x} ∪ i<ω0 Mi .  Clearly, M is closed in X and M = {x} ∨ i<ω0 Mi ∼ = Nα . Next, suppose α = β + 1. By Lemma 3.6, x has a base of open neighbourhoods {Gi : i ∈ ω0 } such that G0 = X, cl(Gi+1 ) ⊂ Gi and (Gi \ cl(Gi+1 )) ∩ Rβ (X) contains non-compact closed subset of X. Then (Gi \ cl(Gi+1 )) ∩ Rβ (X) contains an infinite sequence without a limit point in the metric space X. Hence, Gi \ cl(Gi+1 ) contains pairwise disjoint open sets Hi,j , j ∈ ω0 , that intersect Rβ (X). By the induction  hypothesis, each Hi,j contains a closed subset Ni,j of X homeomorphic to Nβ . Let Mi = j∈ω0 Ni,j =    j∈ω0 Ni,j and M = {x} ∪ i∈ω0 Mi = {x} ∨ i∈ω0 Mi . Then M is closed subset of X homeomorphic to Nα . 2 Theorem 3.8. Let X be a non-empty C-scattered separable metric space. Suppose that X is locally pointed and L(Rβ (X)) is discrete for each ordinal β ≤ α = lc(X). If Rα (X) is empty, a singleton or infinite, then X is a minimal space. Moreover: (a) (b) (c) (d)

If If If If

X is pointed, then X ∼ = Nα . = Nα+1 \ {0} . Rα (X) is infinite, then X is homeomorphic to the sum of ℵ0 copies of Nα i.e. X ∼ α ∼ R (X) = ∅, then α is a limit ordinal and X = Nα \ {0}.  X is weakly pointed but not pointed, then α is a limit ordinal and X ∼ = Nα (Nα \ {0}).

Proof. Let X → Y be a perfect surjective map. Then Y is a non-empty metric space and, by Proposition 2.3, lc(X) = lc(Y ) = α and L(Rβ (Y )) is discrete for each ordinal β ≤ α. Moreover, if Rα (X) is empty, a singleton or infinite, the same holds for Rα (Y ). Observe that X and Y are countable and therefore zero-dimensional. By Proposition 2.7, Y is locally pointed and, if X is pointed or weakly pointed but not pointed, the same is true of Y . Thus, the minimality of X will follow once we prove that X is homeomorphic to a concrete space. Each point x ∈ X belongs to the discrete L(Rγ (X)) for a unique ordinal γ = γ(x) and for some clopen subset Ux of X inside Lγ+1 (X), Ux ∩ L(Rγ (X)) = {x}. Evidently, Rγ (Ux ) = Ux ∩ Rγ (X) = Ux ∩ L(Rγ (X) = {x} and hence lc(Ux ) = γ(x). As X is locally pointed, we can further suppose Ux to be pointed at x and to be contained in any given neighbourhood of x.

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(a) Suppose X is pointed and Rα (X) = {x}. The result to be proved being obvious for α = 0, suppose that α > 0 and the result is true for all ordinals smaller than α.  First we consider the case of a limit ordinal α. By Lemma 3.5, X = {x} ∨ i∈ω0 Xi , where αi = lc(Xi ) < α, limi∈ω0 αi = α and each Xi is pointed and locally pointed. By the induction hypothesis, Xi ∼ = Nα . = Nαi . Hence, by Corollary 1.9, X ∼ Suppose next that α = β + 1. By Lemma 3.6, x has a base of clopen neighbourhoods {Gi : i ∈ ω0 } such that G0 = X, Gi+1 ⊂ Gi and (Gi \ Gi+1 ) ∩ Rβ (X) is infinite. It follows from the second paragraph of the proof that each Gi \ Gi+1 has a clopen cover {Ui , Vi : i ∈ ω0 } consisting of pairwise disjoint sets such that each Ui is pointed at a point of L(Rβ (X)) and each Vi has C-scattered level less than β. Then Ui ∪ Vi is pointed and locally pointed and, by the induction hypothesis, is homeomorphic to Nβ . Thus, the clopen set Mi = Gi \ Gi+1 of X is the direct sum of countably infinitely many copies of Nβ . Hence  X = {x} ∨ i∈ω0 Mi = Nβ+1 = Nα . (b) Suppose Rα (X) is infinite. By the preceding argument, the zero-dimensional space X has a cover {Ui , Vi : i ∈ ω0 } consisting of pairwise disjoint clopen sets such that each Ui is pointed at some point {xi } of Rα (X) and each Vi has C-scattered level < α. Then Ui ∪ Vi is pointed and locally pointed at xi . It follows from (a) that Ui ∪ Vi ∼ = Nα , and X is homeomorphic to the sum of ℵ0 copies of Nα . (c) Suppose that Rα (X) = ∅. Then α is a limit ordinal and a point of X is contained in some L(Rγ (X)), where γ < α. Hence we can cover X with pairwise disjoint pointed clopen sets Xi , i ∈ ω0 , where  αi = lc(Xi ) < α = supi∈ω0 αi . By (a), Xi ∼ = Nαi . Finally, by Corollary 1.9, {0} ∨ Xi is homeomorphic  to Nα . Hence X = Xi is homeomorphic to Nα \ {0}. (d) There is a closed set F of X with x ∈ / F and lc(F ) = α, where x is the special point of X. Let U ⊂ X \F be a clopen set of X that is pointed at x. Then Rα (U ) = {x}, Rα (X \ U ) = ∅ and lc(X \ U ) = α. By (a), Rα (U ) ∼ = Nα and, by (c), α is a limit ordinal and Rα (X \ U ) ∼ = Nα \ {0}. It follows that  ∼ X = Nα (Nα \ {0}). 2 Corollary 3.9. For each countable ordinal α, Nα , Nα \ {0} and Nα



(Nα \ {0}) are minimal spaces.

2

Corollary 3.10. Let X be a countable metric space such that L(Rβ (X)) is discrete for each ordinal β ≤ α = lc(X), where α is a limit ordinal. Suppose that Rα (X) = {x} and X \ {x} is locally pointed but X is not.  Then X is minimal and X ∼ = {x} ∨ i∈ω0 Mi , where Mi ∼ = Nα \ {0}. Proof. Because X is not locally pointed at x, there is a base of neighbourhoods {Gi : i ∈ ω0 } of x consisting of clopen sets with Gi+1 ⊂ Gi and lc(Gi \ Gi+1 ) = α. Clearly, Rα (Gi \ Gi+1 ) = ∅ and, by Theorem 3.8,  Gi \ Gi+1 ∼ = {x} ∨ i∈ω0 Mi , where Mi ∼ = Nα \ {0}. Hence, X ∼ = Nα \ {0}. Any perfect image of Y of X, by Propositions 2.3 and 2.7 will have the properties of X listed in the enunciation of the result. Hence,  Y ∼ = {x} ∨ i∈ω0 Mi ∼ = X and X is minimal. 2 4. Compactifications Proposition 4.1. Let X = A ∪ B where X is compact and A ∩ B = ∅. Let α and n be ordinals with n < ω0 . (a) (b) (c) (d) (e) (f) (g)

Rα (A) ∪ Rα (B) is closed in X and Lα (A) ∪ Lα (B) is open. If A has C-type (α, 0), then B has C-type (α, 1). If Rα (A) is compact, then Rα (B) is locally compact. If Rα (A) is locally compact, then Rα+1 (B) is compact. If A is dense in X, then cl Rn (A) = Rn (A) ∪ Rn (B). If A is dense in X, A has C-type (n + 1, 1) iff B has C-type (n, 2). If A is dense in X, A has C-type (n, 2) iff B has C-type (n, 1).

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Proof. (a) Let x ∈ L(A) ∪ L(B). Without loss of generality, x ∈ L(A). In A there is an open set U and a compact set F such that x ∈ U ⊂ F ⊂ L(A). Write U = V ∩ A, where V is an open set of X. Then V \ F is a locally compact open subset of B. Thus, V \ F ⊂ L(B) and x ∈ V ⊂ (V \ F ) ∪ F ⊂ L(A) ∪ L(B). We conclude that L(A) ∪ L(B) is open in X and its complement R(A) ∪ R(B) is closed. By trivial induction, Rα (A) ∪ Rα (B) is closed in X and Lα (A) ∪ Lα (B) is open. (b) Suppose A has C-type (α, 0). Then α is a limit ordinal > 0, Rα (A) = ∅ and, for each β < α,  Rβ (A) ∪ Rβ (B) is non-empty and compact. Hence Rα (B) = Rα (A) ∪ Rα (B) = β<α (Rβ (A) ∪ Rβ (B)) is non-empty and compact. (c) By (a), Y = Rα (A) ∪ Rα (B) is a compact. Hence, if Rα (A) is compact, then Rα (B) = Y \ cl Rα (A) is locally compact. (d) If Rα (A) is locally compact, then Rα+1 (A) = ∅ and, by (a), Rα+1 (B) = Rα+1 (A) ∪ Rα+1 (B) is compact. (e) Suppose now that A is dense in X. Then L(A) is open in X and cl B = B ∪ R(A) = R(A) ∪ R(B) ∪ L(B). As cl B \ cl R(A) ⊂ L(B), R(A) ∪ R(B) ⊂ cl R(A). We conclude from (a) that cl R(A) = R(A) ∪ R(B), and the result now follows by repeated applications of R. (f) Suppose A is dense in X and has C-type (n + 1, 1). Then Rn+1 (A) is compact and non-empty. By (e), Rn+1 (B) = cl(Rn+1 (A)) \ Rn+1 (A) = ∅. Hence Rn (B) is locally compact. By (c), if Rn (B) were compact, Rn (A) would be locally compact and Rn+1 (A) empty. Thus, B is of C-type (n, 2). By (e), if Rn (A) is locally compact, then Rn (B) is compact. Thus, if B is of C-type (n, 2), by (d) and (e), Rn+1 (A) is compact and non-empty, i.e, A has C-type (n + 1, 1). (g) Suppose A is dense in X and has C-type (n, 2). Then Rn (A) is locally compact and non-compact. By (e), Rn (B) = cl(Rn (A)) \ Rn (A) is compact and non-empty. Thus, B is of C-type (n, 1). The converse follows from (c) and (e). 2 Example 4.2. Let X = I2 , A = {1} × [0, 12 ) and B = X \ A. Then R(A) = ∅, R(B) = {(1, 12 )} and cl(R(A)) = R(A) ∪ R(B). Corollary 4.3. (cf. [7, Corollary 1.2]) Let Y be a compactification of X. Then X is C-scattered iff Y \ X is. Proof. This follows from parts (c) and (d) of Proposition 4.1. 2 Corollary 4.4. (cf. [6, Theorem 10] and [7, Theorem 1.7]) A separable metric C-scattered space X is Čechcomplete. Proof. Let Y be a metric compactification of X. By Corollary 4.3, α = lc(Y \ X) is a countable ordinal. Now the locally compact, Lindelöf space L(Rβ (Y \ X)) is σ-compact for each β ≤ α. Hence Y \ X =  β β≤α L(R (Y \ X)) is σ-compact and X is Čech-complete. 2 The above result shows that for separable metric C-scattered space X, the assumption of Čech completeness in Theorems 4.5 and 4.7 is redundant. However, if S is uncountable, A = {0} ∨ s∈S {s} and X = I ×A \ {(0, s) : s ∈ S}, then R(X) = {(0, 0)}, lc(X) = 1 but X is not Čech-complete. At this point, we state Hoshina’s Theorem in the form proved in [1]. Theorem 4.5. Let X be a Čech-complete and rim-compact space such that R(X) is separable metric. Let Z be compactification of X with ind(Z \ X) ≤ 0. Then X has a compactification Y such that Y ≤ Z and Y \ X is countable. 2

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Remark 4.6. (a) The condition we imposed in [1] on R(X) is that of having a countable network, implying that this gives a more general result than Hoshina’s. We apologize for failing to notice at the time that a Čech-complete space that has a countable network is separable metrizable. For the weight wX of a Čech-complete space X equals its network weight nwX (see [2, Exercise 3.9.E]). Note that in Theorem 4.5, the compact space Y \L(X) has a countable network and is therefore separable metric. (b) Without the assumption that R(X) is separable metrizable, there is no guarantee that a Čech-complete, rim-compact space X will have a compactification with countable remainder, even if X is metrizable [3, Example 4.2] or R(X) is compact ([8, Example 1, Example 4], [1, Example 1]) or R(X) is (uncountable) discrete [1, Example 2]. (c) A rim-compact space X has a compactification Z with ind(Z \ X) ≤ 0. One such compactification is that of Freudenthal. For details see [4]. Recall that a compactification Y of X is a minimal compactification when the remainder Y \ X is a minimal space. Thus, if N is a compactification of Nα , then Y = I ×N is a minimal compactification of X = Y \ ({0} × Nα ). Theorem 4.7. Let X be a rim-compact, Čech-complete space with R(X) separable metric and Rω0 (X) = ∅. Let Z be a compactification of X with ind(Z \ X) = 0. Then X has a minimal compactification Y that precedes Z. (a) If X has C-type (n, 2), where n ∈ ω0 , then Y \ X ∼ = Nn . (b) If X has C-type (n + 1, 1), where n ∈ ω0 , then Y \ X is homeomorphic to the sum of ℵ0 copies of Nn . (c) If X has C-type (ω0 , 0), then Y \ X is homeomorphic to Nω0 . Proof. By Theorem 4.5, X has a compactification preceding Z with countable remainder that has metrizable closure. By Proposition 3.2, X has a compactification W that precedes Z with W \ X countable metric and each L(Rα (W \ X)) discrete. (a) Suppose X has C-type (n, 2). Then, by Proposition 4.1, W \ X has C-type (n, 1). This means that Rn (W \ X) is compact but not empty. By Proposition 2.3, shrinking the compact set Rn (W \ X) to a point, we obtain a compactification Y of X with Y \ X weakly pointed and each L(Rα (Y \ X)) discrete. By Proposition 2.2, Y \ X is pointed and locally pointed. Hence, by Theorem 3.8, Y \ X is a minimal space homeomorphic to Nn . Thus, Y is a compactification of X with the required properties. (b) Suppose next that X has C-type (n + 1, 1). Then, by Proposition 4.1, W \ X has C-type (n, 2). This means that Rn (W \ X) is locally compact but not compact. Thus, Y = W is a compactification of X with each L(Rα (Y \ X)) discrete and Rn (Y \ X) infinite. By Proposition 2.2, Y \ X is locally pointed. By Theorem 3.8, Y \ X is a minimal space homeomorphic to the sum of ℵ0 copies of Nn . Thus, Y has the required properties. (c) Suppose X has C-type (ω0 , 0). By Proposition 4.1, W \X has C-type (ω0 , 1). This means that Rω0 (W \X) is compact and non-empty. By Proposition 2.3, shrinking the compact set Rω0 (W \ X) to a point x, we obtain a compactification Y of X with Y \ X weakly pointed and each L(Rα (Y \ X)) discrete. By Proposition 2.2, Y \ (X ∪ {x}) is locally pointed. Suppose Y \ X is not pointed. Then it contains a closed set F such that lc(F ) = ω0 and x ∈ / F . Clearly, Rω0 (F ) = ∅ and the compact set Rω0 (cl(F ) \F ) ∪ Rω0 (F ) is not empty. As cl(F ) \F ⊂ X, this implies that Rω0 (X) = ∅. Thus, Y \X is pointed and locally pointed. We can now deduce from Theorem 3.8 that Y is a minimal compactification of X with Y \X ∼ = Nω0 . 2

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Let X be rim-compact, Čech-complete space with R(X) separable metric. According to Theorem 4.7, if Rω0 (X) = ∅, the C-type of X determines the minimal compactification remainder of X. The following examples show that this is not so if X has C-type (ω0 , 1).  Example 4.8. Let Yn be a compactification of a space Xn with Yn \Xn = Nn , n ∈ ω0 . Let Y = {0}∨ n∈ω0 Yn .  Let A = {0} ∨ n∈ω0 Xn . Then Y is a minimal compactification of A with remainder Nω0 \ {0}. Here A is of C-type (ω0 , 1) and the remainder is of C-type (ω0 , 0).  Let B = n∈ω0 Xn . Then Y is a minimal compactification of B with remainder Nω0 . Here B is of C-type (ω0 , 0) and the remainder is of C-type (ω0 , 1). Let C be the subspace A ⊕ B of Y ⊕ Y . Then Y ⊕ Y is a minimal compactification of C with remainder (Nω0 \ {0}) ⊕ Nω0 . Here C is of C-type (ω0 , 1) and the remainder is of C-type (ω0 , 1). Let K = {0, 12 , 13 . . .} ⊂ I and Z = K × Y . Then Z is a minimal compactification of D = ({0} × Y ) ∪ ({ 12 , 13 . . .} × B) with remainder { 12 , 13 . . .} × Nω0 , which is minimal by Theorem 3.8. Clearly, Rω0 (D) = {(0, 0)}, D has C-type (ω0 , 1) while the remainder has C-type (ω0 , 2). 2 References [1] [2] [3] [4] [5] [6] [7] [8] [9]

M.G. Charalambous, Compactifications with a countable remainder, Proc. Am. Math. Soc. 78 (1980) 127–131. R. Engelking, General Topology, revised and completed edition, Heldermann Verlag, Berlin, 1989. T. Hoshina, Countable-points compactifications of metric spaces, Fundam. Math. 103 (1979) 123–132. J.R. Isbell, Uniform Spaces, Amer. Math. Soc., Providence, 1964. J. van Mill, The Infinite-Dimensional Topology of Function Spaces, Elsevier, Amsterdam, 2001. A.H. Stone, Kernel constructions and Borel sets, Trans. Am. Math. Soc. 107 (1963) 58–70. R. Telgársky, C-scattered and paracompact spaces, Fundam. Math. 73 (1971) 59–74. T. Terada, On countable discrete compactifications, Gen. Topol. Appl. 7 (1977) 321–327. L. Zippin, On semicompact spaces, Am. J. Math. 57 (1935) 327–341.