Dec 31, 2016 - Evidently, every LOTS, and thus any GO-space, is a Hausdorff topological space, but not necessarily separable or LindelÃ¶f. The Sorgenf...

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COUNTABLE SUCCESSOR ORDINALS AS GENERALIZED ORDERED TOPOLOGICAL SPACES ROBERT BONNET 1 AND ARKADY LEIDERMAN Abstract. We prove the following Main Theorem: Assume that any continuous image of a Hausdorff topological space X is a generalized ordered space. Then X is homeomorphic to a countable successor ordinal (with the order topology). The converse trivially holds.

1. Introduction and Main Theorem All topological spaces are assumed to be Hausdorff. Remind that L is a Linearly Ordered Topological Space (LOTS) whenever there is a linear ordering ≤L on the set L such that a basis of the topology λL on L consists of all open convex subsets. A convex set C in a linear ordering M is a subset of M with the property: for every x < z < y in M , if x, y ∈ C then z ∈ C. The above topology, denoted by λL is called an order topology. Since the order ≤L defines the topology λL on L (but not vice-versa), we denote also by hL, ≤L i the structure including the topology λL . A topological space hX, τ X i is called a Generalized Ordered Space (GO-space) whenever hX, τ X i is homeomorphic to a subspace of a LOTS hL, λL i, that is τ X = λL ↾X := {U ∩ X : U ∈ λL } (see [2]). Evidently, every LOTS, and thus any GO-space, is a Hausdorff topological space, but not necessarily separable or Lindel¨ of. The Sorgenfrey line Z is an example of a GO-space, which is not a LOTS, and such that every subspace of Z is separable and Lindel¨ of (see [4]). By definition, every subspace of a GO-space is also a GO-space. In this article, in a less traditional manner, we say that a space X is a hereditarily GO-space if every continuous image of X (in particular, X itself) is a GO-space. Main Theorem 1.1. Every hereditarily GO-space is homeomorphic to a countable successor ordinal, considered as a LOTS. The converse obviously holds. This result is closely related to the following line of research: characterize Hausdorff topological spaces X such that all continuous images of X have the topological property P. All questions listed below for concrete P are still open.

Date: December 26, 2016. 1991 Mathematics Subject Classification. 03E10, 06A05, 54F05, 54F65. Key words and phrases. Interval spaces, linearly ordered topological spaces, generalized ordered spaces, compact spaces. 1 The first listed author gratefully acknowledges the financial support he received from the Center for Advanced Studies in Mathematics of the Ben-Gurion University of the Negev. 1

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Problem 1.2. (1) Characterize Hausdorff spaces such that all continuous images of X are regular. (2) Characterize Hausdorff spaces X such that all continuous images of X are normal. (This was partially solved by W. Fleissner and R. Levy in [5, 6]). (3) Characterize Hausdorff spaces such that all continuous images of X are realcompact. (This question has been formulated in [1]). (4) Characterize Hausdorff spaces such that all continuous images of X are paracompact. (5) Characterize Hausdorff spaces such that all continuous images of X are monotonically normal. (This is related to “Niekel Conjecture” answered positively by M. E. Rudin [12]). Remark 1.3. Let b be the minimal cardinality of unbounded subsets of ω ω . Recently M. Bekkali and S. Todorˇcevi´c proved the following relevant result: Continuous zero-dimensional images of a compact LOTS of weight less than b is itself a LOTS [3, Theorem 4.2]. Note also a recent paper [14], which studies topological properties P that are reflectable in small continuous images. For instance, a GO-space X is hereditarily Lindel¨ of iff all continuous images of X have countable pseudocharacter [14]. As a special case of Main Theorem 1.1, we obtain the following fact. Corollary 1.4. Assume that any continuous image of a Hausdorff topological space X is a LOTS. Then X is homeomorphic to a countable successor ordinal. In order to make this paper widely readable, we have tried to give self-contained and elementary proofs, even when our results could be deduced from more general theorems. By these reasons, and for the readers’ convenience, we include a separate and short proof of Corollary 1.4 in Section 2. In our paper, a GO-structure, formally, is a 4-tuple hX, τ X , L , ≤L i where (1) X ⊆ L, (2) hL, ≤L i is a linear ordering with the order topology λL , and (3) the order ≤X on X is the restriction ≤L ↾ X of ≤L to X, and τ X is the topology λL ↾X := {U ∩ X : U ∈ λ} Any GO-space X can be written under the above structure. In [7], [10] GO-space are denoted by hX, τ X , ≤X i where ≤X is the restriction of ≤L to X. It is easy to see that the following are equivalent. (i) hX, τ X , ≤X i is a GO-space and (ii) λX ⊆ τ X and τ X has a base consisting of convex sets. Many times we denote hX, τ X , L , ≤L i by hX, τ X , L , λL i. The proof of Main Theorem 1.1, is organized as follows. In §3, we present the basic facts on LOTS and GO-spaces. In particular, Proposition 3.4 shows that if hX, τ X , L , ≤L i is a GO-structure, then we may assume that L is a complete ordering and that X is topologically dense in hL, λL i. In §4.1 we show that any hereditarly GO-space satisfies c.c.c. property: every family of pairwise disjoint nonempty open set is countable (Lemma 4.1). In §4.2 we prove that any hereditarily GO-space has no countable closed and relatively discrete subset (Corollary 4.5): we recall that a subset D of a space Y is relatively discrete whenever there is a family UD := {Ud : d ∈ D} of open subsets of Y such that D ∩ Ud = {d} for every d.

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Proposition 4.8 shows that a hereditarily GO-space is a subspace of a scattered linear order (with the order topology). Finally, in §4.4 we conclude the proof of Main Theorem.

2. Elementary proof of Main Theorem for LOTS: Corollary 1.4 We assume that the reader is familiar with the properties of LOTS and give a selfcontained proof of Corollary 1.4. To prove the result, we need some preliminaries. Fact 1. Let hL, ≤L i be a LOTS. If hL, ≤L i is a scattered ordering then hL, λL i is a scattered topological space. P The converse does not hold: consider the lexicographic sum N := r∈Q Zr of copies Zr of the integers Z over the rational chain Q: hN, ≤N i is not a scattered ordering but hN, λN i is a discrete space (this example will be used again in Part (1) of Remark 3.3). The next fact is used implicitly in the arguments that follow. Fact 2. Let hL, ≤L i be a LOTS and F be a closed subspace of hL, λL i. Then the induced topology λL ↾F on F is the order topology on F defined by the restriction ≤L ↾F of ≤L on F . L L Fact 3. Let hL, ≤ i be a scattered linear ordering then hL, λ i is 0-dimensional (i.e. hL, λL i has a base consisting of clopen sets). Proof. The proof uses the fact that if hL, ≤L i is a scattered linear ordering, then c the Dedekind completion hLc , ≤L i of hL, ≤L i is also a scattered linear ordering. Fact 4. Let any continuous image of a Hausdorff topological space Y is a LOTS. Then Y satisfies c.c.c. . In particular, ω1 and ω1∗ are not order-embeddable in Y . Fact 5. Let Y be a 0-dimensional LOTS. If D is a countable closed and discrete subset of Y then D is a continuous image of Y . Proof. Let hUd id∈D be clopen subsets of Y such that Ud ∩ D = {d} for d ∈ D. Fix d0 ∈ D. Let ≈ be the equivalence relation on Y defined by x ≈ y whenever there is S d ∈ D \ {d0 } such that x, y ∈ Ud , or x, y 6∈ U := { Ud : d ∈ D\{d0 } }. Then Y /≈ is a continuous image of Y , Y /≈ is Hausdorff and D is homeomorphic to Y /≈. Fact 6. Let D be a countable and discrete space. Then D has a continuous image which is not homeomorphic to a LOTS. Proof. Consider ω as a discrete space. Let U on ω be a non-principal ultrafilter on ω and let ∗ be a new element with ∗ 6∈ ω. We equip N∗ := ω ∪ {∗} with the ˇ topology induced from the Stone–Cech compactification βN. It is well-know that N∗ is not a LOTS: this is so because N∗ is separable and its topology does not have a countable base [4, §3.6]. Evidently, countable N∗ is a continuous image of ω. As a consequence of Facts 4–6, we have the following result. Fact 7. Let hX, ≤X , λX i be a scattered linear ordering. Assume that any continuous image of X is a LOTS. Then the following holds (1) If x is in the topological closure of a nonempty subset A in X, then there is a countable monotone sequence of elements of A converging to x. (2) Every monotone sequence hxn in∈ω converges. In particular, there exist both minimum and maximum in hX, ≤ i. Now we are in a position to finish the argument.

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Proof of Main Theorem (for LOTS). Let ≡ be the equivalence on hX, ≤X , λX i defined by: x ≤ y and [x, y] is a scattered linear ordering x ≡ y if and only if x ≤ y and [y, x] is a scattered linear ordering Note that each ≡-class is closed for the topology λX and each ≡-class is convex and scattered for the order ≤X . Moreover, X/≡, denoted by X1 , is a LOTS. Note that there are no consecutive classes in X1 , and thus X1 is order-dense. Also the map ϕ : X → X/≡, preserving supremum and infimum, is increasing and continuous. Let N be a linear ordering and N c its Dedekind completion. Recall that a cut is a member of N c \ L ∪ {min(N c ), max(N c )} . For instance, in the chain Q of rationals, the cuts are the irrationals. Case 1. The set Γ ( := X1 c \ X1 ) of cuts of X1 has no consecutive elements and Γ is topologically dense in X1 . So, Γ is order–dense and Γ has no first and no last element. Note that, in that case, X1 is 0-dimensional. Let c ∈ Γ. Let hcα iα<λ be a strictly increasing sequence cofinal in (−∞, c) with λ regular. By Fact 4, λ = ω. Since c ∈ Γ, D := {xα : α ∈ ω} is a countable discrete and closed subset of X1 , which contradicts Fact 6. So, Case 1 does not occur. Case 2. The set Γ ( := X1 c \ X1 ) of cuts of X1 has two consecutive elements, or Γ is not topologically dense in X1 . Then there is a nonempty open interval (u, v) of X such that (u, v)∩Γ = ∅. We set X1′ := [u, v]. So X1′ is connected, infinite and order–dense. Also X1′ is a continuous image of X1 . Let X2 be the quotient space of X1′ , obtained by identification of u and v. Obviously, X2 is connected. Since for every t ∈ X2 the set X2 \ {t} is connected, X2 is not a LOTS. We have proved that |X1 | = 1, that is: X is a scattered linear ordering. By Fact 1, X is a scattered topological space. Moreover, X satisfies c.c.c., and thus ω1 and ω1∗ are not order-embeddable in the scattered linear ordering X. In particular, the space X has only countably many isolated points. Also the space X has no infinite and discrete subset. Hence the linear ordering X is complete and thus the space X is compact. We have proved that X is a countable compact and scattered space, that is, X is homeomorphic to α + 1 for some α < ω1 .

3. Basic facts on LOTS and GO-spaces Let S be a set, U ⊆ P(S) and T ⊆ S. We set U ↾ T = {U ∩ T : U ∈ U }. Recall that a linear ordering hM, ≤M i is complete whenever every subset A of M has the supremum supM (A) and the infimum inf M (A). In particular, there exist both the maximum max(M ) and the minimum min(M ) in M . Let N be a linear ordering. The Dedekind completion of N , denoted by N c , is a complete linear ordering containing N , such that N c is minimal with respect to this property. That is, (D1) N c is a complete chain, c (D2) for every x, y ∈ N : if x

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(D3) for every x ∈ N c there are sets A, B ⊆ N such that x 6∈ A ∪ B and c c supN (A) = x = inf N (B). c

c

For instance, min(N c ) = supN (∅) = inf N (N ). Notice also that the Dedekind completion N c of N is unique up to an order-isomorphism. In a linear ordering N , a cut in N is a member of N c \ N ∪{min(N c ), max(N c )} , [11, 2.22, 2.23]. For example, the cuts for the chain Q of rationals are irrationals. Next we say that ha, b i are consecutive in N whenever a, b ∈ N , a < b and (a, b)N = ∅. So, for any linear ordering N : (1) if ha′ , b′ i are consecutive in N then ha′ , b′ i are consecutive in N c , and (2) if ha′ , b′ i are consecutive in N c then a′ , b′ ∈ N and ha′ , b′ i are consecutive in N . The following fact is well-known. Proposition 3.1. Let N be a LOTS. (1) hN, λN i is a compact space if and only if hN, ≤N i is order complete. (2) hN, λN i is a connected space if and only if hN, ≤N i has no cuts and no consecutive elements. ✷ Next we introduce some basic notions on scattered linear orders and scattered spaces. Let N be a linear ordering. We say that N is order-dense, or dense if between two elements of N there is a member of N . Notice that for any dense linear order N , the rational chain Q is order-embeddable in N . A linear order N is called order-scattered or simply scattered, whenever the rational chain Q is not embeddable in N . For example, ω1 and its converse ordering ω1∗ are scattered linear ordering (by the definition, hω1∗ , ≤ i is the ordering hω1 , ≥ i). A space Y is dense-in-itself if Y is nonempty and has no isolated point. A densein-itself and closed subspace Y of a space Z is called a perfect subspace of Z. A space X is called topologically-scattered, or simply scattered (space), whenever X does not contain a perfect subspace, that is, every nonempty subset A of X with the induced topology has an isolated point in A. We state other well-known facts about LOTS. For completeness we include the proofs. Proposition 3.2. Let N be a LOTS. (1) The following hold. (a) If hN, ≤N i is a scattered linear ordering then hN, λN i is topologically scattered. (b) Assume that hN, ≤N i is a complete chain, i.e., by Proposition 3.1(1), hN, λN i is a compact space. Then the following are equivalent. (i) hN, λN i is a scattered topological space. (ii) hN, ≤N i is a scattered linear ordering. (2) If hN, ≤N i is order-scattered then hN, λN i is 0-dimensional. (3) If N has only countably many isolated points then N is countable. Proof. (1) (a) Assume that hN, λN i is not a scattered space. Let D ⊆ N be a dense-in-itself subset of N . Then hD, ≤N ↾D i contains an order-dense subset, and thus N is not a scattered chain. (b) Suppose that hN, λN i is compact and that hN, ≤N i is not order-scattered. Let S ⊆ N be an order-dense subset of N , and let T be its topological closure in hN, λN i. Then T has no isolated points, i.e. T is dense-in-itself. By compactness, T is compact and thus T is perfect. Hence hN, λN i is not a scattered space.

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We prove a little bit more. We have N = N c . Let M be a linear ordering and let M c be its the Dedekind completion. The following are equivalent: (i) M is a scattered chain, (ii) Q is not order-embeddable in M , (iii) Q is not orderc embeddable in M c , and (iv) M c is a scattered chain. Now since hM c , ≤M i is c a complete chain, hM c , λM i is a compact space. Therefore the previous items are equivalent to each of the following (v) Q is not order-embeddable in M c , and c (vi) hM c , λM i is a scattered space. (2) In the proof of Part (1), we have seen that if N is a scattered chain, then its Dedekind completion N c is a scattered chain and thus N c , considered as a LOTS, is compact and topologically-scattered. Therefore N c is 0-dimensional, and thus N is also 0-dimensional. (3) By the hypothesis, the set S := Iso(N ) of isolated points in N is countable. Since N is a scattered space, S is topologically dense in N . Since S is a chain, by the proof of Part (1), the chain S c is scattered. We claim that S c is countable. This is so, because if S c is an uncountable scattered chain, then ω1 or ω1∗ is order-embeddable in S c and thus the same holds for S, that is, S is uncountable contradicting our assumptions. Next, since S c is countable, N must be countable. Indeed, there exists a continuous (increasing) map f from N into S c such that |f −1 (x)| ≤ 2 for any x ∈ N . (That is, the case if N := ω + 1 + 1 + ω ∗ , and thus S := Iso(N ) = ω + ω ∗ and S c = ω + 1 + ω ∗ .) Remark 3.3. (1) Proposition P3.2(1)(a) is not reversible. As an a example, consider the lexicographic sum N := q∈Q Zq of copies Zq of (the the chain of integers) Z, indexed by the chain of rationals Q. Then N is a non-scattered linear ordering, but N is a topological discrete LOTS and thus N is a scattered topological space. (2) Recall that the Dedekind completion of M := (0, 1) ∩ Q is M c = [0, 1]R . On the other hand, M is topologically dense in the Cantor set 2ω (considered as a subset of R). (3) Let X := {1/n : n > 0} ∪ {−1} ⊂ R := L. Then hX, λX i is compact, but hX, τ X i is infinite and discrete. • Let hX, τ X , L , ≤L i be a GO-space. Then λL ↾X ⊆ τ X . Indeed, let a < b in X. So (a, b)X ∈ λX and thus, by the definition, (a, b)X = (a, b)N ∩ X ∈ τ X . The following result is well-known. For completeness we include its proof. Proposition 3.4. Let hX, τ X i be a GO-space. Let hL, ≤L i be such that hX, τ X , L , ≤L i is a GO-space. Without loss of generality we may assume that L satisfies: (H1) X is topologically dense in hL, λL i; (H2) hL, ≤L i is a complete linear ordering. Proof. The proof follows from the following two facts. c

Fact 1. Let hN, ≤N i be a linear ordering and hN c , ≤N i be its Dedekind completion. c c Then λN = τ N ↾N . That is, the order topology λN is the induced topology of τ N on N .

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Proof. Let c ∈ N c . Then c is a cut if and only if c 6∈ N ∪ {min(N ), max(N )} and c c has no predecessor nor a successor in N c . Obviously, λN ⊆ τ N ↾N . Next let c c c (a, b)N ∈ λN be an open convex set in hN c , ≤N i. So a, b ∈ N c . If a 6∈ N then a = inf({a′ ∈ N : a′ > a}) and if b 6∈ N then b = inf({b′ ∈ N : b′ < b}). So c (a, b)N ∩ N is an union of open convex sets (a′ , b′ )N where a′ , b′ ∈ N . X N N Fact 2. Let hX, τ , N , ≤ i be a GO-structure such that hN, τ i is a complete ordering. Let L be the topological closure of X in the space hN, λN i. Then τ X = λL ↾X. That is, hX, τ X , L , ≤N ↾L i is a GO-structure. Proof. It suffices to show that for every a < b in N there are a′ < b′ in L such that (∗): (a′ , b′ )L ∩X = (a, b)N ∩X. Fix a < b in N . If a ∈ X (b ∈ X) set a′ = a (b′ = b). Next suppose that a ∈ N \ L. Since N is a complete ordering and N \ L is open in L, there is a (maximal) open convex set (α, α′ )N in N such that a ∈ (α, α′ )N , (α, α′ )N ∩ L = ∅ and α, α′ ∈ L; and we set a′ = α. Similarly, suppose that b ∈ N \ L. Again, since N is complete and N \ L is open in L, there is a (maximal) open convex set (β, β ′ )N such that b ∈ (α, β ′ )N , (β, β ′ )N ∩ L = ∅ and β, β ′ ∈ L; and we set b′ = β ′ . Now obviously (a′ , b′ )L is as required in (∗). Now let hX, τ X , N , ≤N i be a GO-structure. By Fact 1 we may assume that N is a complete ordering. Finally, the result follows from Fact 2. Remark 3.5. (1) In general λX $ τ X . For example, consider L = ω1 and let Lim be the set of all countable limit ordinals. We set X = ω1 \ Lim. We have: (i) τ X is the discrete topology, (ii) Y is topologically dense in hω1 , λω1 i, and (iii) X is order-isomorphic to ω1 and thus hY, λω1 i is homeomorphic to the ordinal space ω1 . (2) The one-point compactification of an uncountable discrete space is not a GO-space (by Lemma 4.1), and there is a countable space which is not a GO-space (by Lemma 4.4). For completeness we recall the proof of the following fact. Proposition 3.6. [10, Lemma 6.1] Let hX, τ X , L , ≤L i be a GO-structure satisfying (H1) and (H2). So hX, ≤L ↾X i is a linear ordering. (1) If hX, τ X i is a compact space then X = L and τ X = λX . (2) If hX, τ X i is a connected space then τ X = λX . Proof. (1) Since X is compact then X is closed in hL, λL i. By (H1), X = L and thus τ X = λX . (2) Next suppose that hX, τ X i is connected. Then X, considered as the LOTS hX, λX i, has no consecutive point and no cut because for each final subset of hX, ≤L ↾X i is closed in hX, τ X i. Therefore, X \{min(L), max(L)} = L\{min(L), max(L)} and thus τ X = λX .

4. Proof of Main Theorem As one of the main parts of Main Theorem 1.1 Proposition 4.8 implies that it suffices to assume that hL, ≤ i is a scattered linear order. To prove this result, we use Corollary 4.5 (in §4.2) which says that a GO-space does not contain an infinite countable relatively discrete closed subset.

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Recall that hX, τ i is a GO-space means that hX, τ , L , ≤ i is a GO-structure. So, we have X ⊆ L. Also for simplicity hX, τ i is denoted by X. In the sequel, by Proposition 3.4, we assume that the GO-structure hX, τ , L , ≤ i satisfies properties (H1) and (H2). 4.1. A hereditarily GO-space satisfies c.c.c. property Lemma 4.1. Assume that hX, τ , L , ≤ i is a 0-dimensional hereditarily GO-structure. Then (1) X satisfies c.c.c. property . (2) The linear orderings ω1 and ω1∗ are not order-embeddable in X. Proof. (1) Since hX, τ i is 0-dimensional, any nonempty open subset of hX, τ i contains a clopen convex subset of the form (a, b)X := (a, b)L ∩ X where a, b ∈ L. By contradiction, assume that {Ui : i ∈ I} is an uncountable family of pairwise nonempty clopen convex subsets of X. So each Ui is of the form (ai , bi )X with ai , bi ∈ L. S Fix i0 ∈ I. Let U := Ui : i ∈ I \ {i0 } and ∼ be the equivalence relation on X defined as follows: x ∼ y if and only if x, y ∈ Y \U or there is i ∈ I such that x, y ∈ Ui . Denote by X ′ the set X/∼ and by f : X → X ′ the quotient map. We endow X ′ with the quotient topology τ ′ on X ′ . So V ′ ∈ τ ′ if and only if f −1 [V ′ ] ∈ τ . In particular, ui := f [Ui ] is anSisolated point in X ′ for any i ∈ I \ {i0 }. Setting u = f [X \ U ] we have X ′ = {u} ∪ ui : i ∈ I \ {i0 } and the set {ui } : i ∈ I \ {i0 } ∪ X ′ \ {ui } : i ∈ I \ {i0 } is a subbase of a topology τ ′′ on X ′ satisfying:

(1) τ ′′ ⊆ τ ′ and thus f : hX, τ i → hX ′ , τ ′′ i is continuous, (2) hX ′ , τ ′′ i is compact (this follow from the definition of τ ′′ ), and (3) hX ′ , τ ′′ i is homeomorphic to the one-point compactification of the uncountable discrete set. This is so because u is the unique accumulation point of hX ′ , τ ′′ i. We show that hX ′ , τ ′′ i is not a GO-space. By contradiction, suppose that hX ′ , τ ′′ , L′ , ≤′ i is a GO-structure. Since hX ′ , τ ′′ i is compact, by Proposition 3.6(1), ′ L′ = X ′ and τ ′′ = λ′ := λ≤ . So it suffices to prove that (4) hX ′ , τ ′′ i := hX ′ , λ′ i is not a LOTS. ′

Assume that hL′ , ≤′ i is a chain. Hence, for instance, (−∞, u)L is uncountable. ′ Consider any y ∈ (−∞, u) such that (−∞, y)L is infinite. By the definition, ′ (−∞, y]L is infinite, discrete, closed and thus compact, that contradicts the fact ′ that u 6∈ (−∞, y]L . We have proved that X satisfies c.c.c. property . (2) follows from Part (1).

Now remind the classical result which is due to Mazurkiewicz and Sierpi´ nski (for example, see [8, Theorem 17.11], [13, Ch. 2, Theorem 8.6.10]). Lemma 4.2. Every topologically scattered compact and countable space is homeomorphic to a countable and successor ordinal space.

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4.2. A hereditarily GO-space has no countable closed and relatively discrete subsets Lemma 4.3. Let M be an order-scattered LOTS. If M contains a closed and countable relatively discrete subset D, then D is a continuous image of M . Proof. First we introduce a new definition. Let Y be a topological space. For a family V of pairwise disjoint subsets of Y , we denote by acc(V ) the set of accumulation points of V . By definition, x ∈ acc(V ) if and only if for every neighborhood W of x the set {V ∈ V : V ∩W 6= ∅} is infinite. So if Z ⊆ Y , acc(Z) = acc {z} : z ∈ Z . We say that a subset D of a space Y is strongly discrete whenever UD satisfies acc(D) = acc(UD ). The next result is well-known. For completeness we recall its proof. Fact 1. Let M be a 0-dimensional LOTS and D ⊆ M . If D is relatively discrete then D is strongly discrete. Proof. For d ∈ D let Ud := (ad , bd )L be a clopen convex set such that D ∩Ud = {d}. Note the following property (∗): if d < d′ then x < x′ for every x ∈ Ud and x′ ∈ Ud′ . We set UD = {Ud : d ∈ D}. Obviously, acc(D) ⊆ acc(UD ). Conversely, let x ∈ acc(UD ) and V be a neighborhood of x. We may assume that V is of the form (a, b) with a < b in M . Therefore, by (∗), there are infinitely many di ’s such that di ∈ Udi ⊆ V , and thus x ∈ acc(UD ) ⊆ acc(D). M M Since hM, ≤ i is an order-scattered LOTS, by Proposition 3.2(2), hM, λ i is 0-dimensional. By Fact 1, let UD := {Ud : d ∈ D} be a family of clopen convex subsets of M such that Ud ∩ DS= {d} for d ∈ D and acc(D) = acc(UD ). Since D is closed, acc(D) = ∅ and thus UD is a clopen subset of M . S S Let d0 ∈ D be fixed and UD\{d0 } = {Ud : d ∈ D\{d0 }}. So UD\{d0 } = UD \ Ud0 is a clopen subset of M . Let ≈ be the equivalence relation S on M defined by x ≈ y whenever x, y ∈ UD for some d ∈ D \ {d0 }, or x, y ∈ M \ UD\{d0 } . For each x ∈ M there is an unique f (x) ∈ D such that f (x) ≈ x. It is easy to check that the mapping f : M → D is onto and that f is continuous: this is so, because f −1 (d) is clopen in M for any d ∈ D. Our next result, which is apparently well-known, strengthens Fact 6 from the proof of Corollary 1.4. Consider again ω as a discrete space. Let U on ω be a non-principal ultrafilter on ω and let ∗ be a new element with ∗ 6∈ ω. The ˇ space N∗ := ω ∪ {∗} is equipped with the topology induced from the Stone–Cech compactification βN. Lemma 4.4. The countable space N∗ is not a GO-space. Proof. Recall that the character χ(X) of the infinite topological space X is the supremum of cardinalities of minimal local neighborhood bases of all points in X. Fact 1. Let hX, τ X , L , λL i be a GO-structure satisfying (H1) and (H2). If X is countable then χ(hX, τ X i) = ℵ0 . Proof. Since X is a countable subset of L and X is topologically dense in L, the chain L is order-embeddable in the segment [0, 1] of R. Since Q ∩ [0, 1] ⊆ [0, 1] we have χ([0, 1]) = ℵ0 . So, ℵ0 ≤ χ(X) ≤ χ(L) ≤ χ([0, 1]) = ℵ0 . Now it suffices to remind a well-known fact that χ(N∗ ) > ℵ0 (see [4, 3.6.17]). Therefore, N∗ is not a GO-space. As a consequence of the above results 4.3 and 4.4, we have:

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Corollary 4.5. Let X be a hereditarily GO-space. Then X does not contain an infinite countable relatively discrete closed subset. ✷ 4.3. A hereditarily GO-space comes from a scattered linear ordering Let GO-structure hX, τ X , L , ≤L i satisfies the conditions: (H1) X is topologically dense in hL, λL i. (H2) hL, ≤L i is complete, meaning that hL, λL i is a compact LOTS. Hence, by Proposition 3.2(1)(b), (H3) For any u < v in L: [u, v]L is order-scattered if and only if [u, v]L is topologically-scattered. Let hX, τ i be a hereditarily GO-space, meaning that hX, τ X , L , ≤L i is a hereditarily GO-structure satisfying (H1)–(H3). For simplicity denote the space hX, τ X i by X. We shall show that hL, ≤L i is order-scattered (Proposition 4.8), and thus, by Proposition 3.2(1)(b), hL, λL i is topologically–scattered. Therefore X, as subset of L, is also topologically–scattered. Let ≡L be the equivalence relation on L defined as follows. For x, y ∈ L, we set x ≡L y if x ≤ y and [x, y]L is an order-scattered subset of L, or y ≤ x and [y, x]L is an order-scattered subset of L. Note that, by (H3), in the definition of ≡L we have: [u, v]L is order-scattered if and only if [u, v]L is topologically-scattered. Now, each equivalence class is an order-scattered and convex subset of the LOTS hL, ≤ i and each equivalence class is closed in hL, λ i. ( ≡L is standard (see the proof of Theorem 19.26, in [9] )). We denote L/≡L by L1 and by π : L → L1 the projection map. So π is increasing and thus π induces a linear order ≤1 on L1 . Lemma 4.6. The linear ordering hL1 , ≤1 i has the following properties. (1) L1 is a complete linear ordering and the quotient topology on L1 is the order topology λ1 := λ≤L1 . (2) L1 is order-dense, i.e. L1 has no consecutive elements. (3) L1 is a compact and dense-in-itself space. (4) L1 is a connected space. Proof. (1) This part follows from the fact that L is complete and π is increasing and onto. (2)–(3) Notice first that hL1 , ≤1 i is a complete chain, and thus hL1 , λ1 i is compact. Secondly, there are no consecutive ≡L -classes in L, this is so because the union of two consecutive ≡L -classes is an ≡L -class. Hence L1 has no consecutive elements. Therefore, hL1 , ≤1 i is a dense chain and thus L1 is dense-in-itself. (4) By Part (2), L1 has no consecutive elements. Also since L1 is a complete chain, L1 has no cuts. So, by Proposition 3.6, L1 is a connected. Now we recall that X ⊆ L and that for x < y in L: (1) [x, y]L is a scattered subspace of hL, λ i if and only if [x, y]L is a scattered subchain of hL, ≤ i, and (2) if [x, y]L is a scattered subspace of L then [x, y]X := [x, y]L ∩X is a scattered subspace of X (but not vice-versa).

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Now the relation ≡L induces an equivalence relation ≡X on X, setting for x, y ∈ X: x ≡X y if and only if x ≡L y . We denote by X1 the space X/≡X . Lemma 4.7. The following hold for hX1 , τ X1 , L1 , ≤L1 i. (1) The continuous inclusion embedding X ⊆ L induces a continuous inclusion embedding X1 ⊆ L1 . (2) X1 is a topologically dense subset of L1 for the topology λ1 (on L1 ). (3) X1 considered as subordering of L1 has no consecutive points. (4) hX1 , τ1 , L1 , ≤1 i is a GO-structure. Proof. (1) Let π : L → L1 be the projection map. Then π[X] := X1 ⊆ L1 and the embedding X1 ⊆ L1 is continuous. (2) Since X is a topologically dense subset of L, X1 is topologically dense in L1 . (3) Since X1 is topologically dense in L1 , if a1 < b1 are consecutive elements in X1 then a1 < b1 are also consecutive in L1 . This contradicts Lemma 4.6(2). (4) follows from the definitions. We have seen that hX1 , τ1 , L1 , ≤1 i is a GO-structure with the properties (H1), (H2), and the properties (1)–(4) of Lemma 4.6 and (1)–(3) of Lemma 4.7. Proposition 4.8. Let hX, τ , L , ≤ i be a hereditarily GO-space satisfying (H1) and (H2). Then hL, ≤ i is a scattered chain. Proof. Now, with the above notations of §4.3, we consider the GO-space hX1 , τ X1 , L1 , ≤L1 i instead of the GO-space hX, τ X , L , ≤L i. Let Γ1 = L1 \ X1 ∪ {min(L1 ), max(L1 )} .

That is, “Γ1 is the set of cuts of the chain hX1 , ≤X1 i considered as linear subordering order of hL1 , ≤L1 i”. An obvious characterization of the elements of Γ1 is stated in the following fact. Fact 1. We have that γ ∈ Γ1 if and only if γ defines two nonempty clopen sets, namely (−∞, γ)X1 and (γ, +∞)X1 , for the induced topology τ X1 . Therefore hX1 , τ X1 i is a connected space if and only if Γ1 is empty and X1 has no consecutive elements. To prove Proposition 4.8, we distinguish two cases, and in fact we prove that |X1 | = 1. This implies that hL, ≤L i is order-scattered. Case 1. Γ1 has no consecutive elements and Γ1 is topologically dense in L1 for the order topology λ1 . So the set Γ1 is a dense linear order and every nonempty open convex subset of L1 contains a cut. Since X1 is topologically dense in L1 , any nonempty open convex subset of L1 contains a cut and thus, in that case, • hX1 , τ1 i is 0-dimensional. Let c ∈ Γ1 . Since Γ1 has no first element, let hcα : α < λi be a cofinal strictly increasing sequence in (−∞, c)X1 where λ is an infinite regular cardinal, that is, for every x ∈ (−∞, c)X1 there is α such that x ≤ cα . Fact 2. λ = ω.

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Proof. If not then ω1 is order-embeddable in Γ1 . Choose xα ∈ (cα , cα+1 ) ∩ X1 for any α, then hxα : α < λi is a sequence in X1 , order-isomorphic to ω1 . Since hX1 , τ1 i is 0-dimensional (but not necessarily an interval space), this contradicts Lemma 4.1(2). We have proved that λ = ω. Keeping the same notations, we have (∗): the set D := {xα : α ∈ ω} is countable and relatively discrete. Also since c is a cut, (∗∗): D is closed in X1 . (∗) together with (∗∗) contradicts Corollary 4.5. Therefore, Case 1 does not occur. Case 2. Not Case 1. This implies that either Γ1 := L1 \ X1 ∪ {min(L1 ), max(L1 )} has two consecutive elements in L1 or Γ1 is not topologically dense in L1 . Anycase, there is an infinite open interval (u, v)L1 of L1 (with u < v in L1 ) that does not contain a member of Γ1 . Recall that we have the following properties. (P1) The elements of L1 are exactly the ≡L -classes of L, and (P2) hL1 , ≤1 i is a dense linear order. Since (u, v)L1 is infinite, we may assume that u, v 6∈ Γ1 . So, we have the additional properties. (P3) [u, v]L1 ∩ Γ1 = ∅, and thus [u, v]L1 = [u, v]X1 . (P4) [u, v]L1 is infinite. (P5) X1 , and thus [u, v]X1 , has no consecutive elements (Lemma 4.7(3)). (P6) Hence, by (P3), (P5) and Proposition 3.1(2), [u, v]X1 is a connected space. Next we prove that |X1 | = 1. By contradiction, assume that |X1 | > 1. We consider the equivalence relation ≅ on X1 which identifies all elements of (−∞, u]∪[v, +∞)X. So X1 /≅, denoted by X2 , is a continuous image of X1 . Also X2 is a connected space and, by (P4), X2 is an infinite continuous image of X. Fact 3. Let hY, τ Y , M , ≤M i be a GO-structure such that hY, τ Y i is connected. Then, for every y ∈ Y : if y is not the minimum nor the maximum of Y (if they exist) then the subspace Y \ {y} is not a connected space. Proof. By Proposition 3.6(2), τ Y = λY . Let y ∈ Y be such that y is not the minimum nor the maximum of Y . Set U = (∞, y)M and V = (y, ∞)M . By the definition, U and V are open subsets of M . Hence U ∩ Y and V ∩ Y define a partition of Y into two nonempty open sets of Y . We have proved Fact 3. We claim that Fact 4. For every x ∈ X2 , the subspace X2 \ {x} is connected. Proof. First recall that X2 is a connected space. Also we can define X2 as follows: X2 is the quotient of [u, v]X1 identifying u and v. Fact 4 follows from the claim that [u, v]X1 is a connected interval subspace of X1 . Now, from (P6) it follows that: X2 is connected, and by Fact 4: for any x ∈ X2 the space X2 \ {x} is connected. Hence, by Fact 3, the space X2 is not a GO-space. So X1 and thus X is not a GO-space. In other words, by (P1) and (P2), if X (or equivalently L) has more than one ≡L class, then X2 is a continuous image X and X2 is not a GO-space. This contradicts the fact that X is a hereditarily GO-space. We have proved that |X1 | = 1. Further, |X1 | = 1 means that X1 consists of exactly one ≡X -class, or, equivalently, there is the unique ≡L -class in L. Since for x < y in X: x ≡L y if and only if [x, y]L is a scattered chain, the chain L is scattered.

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4.4. End of the proof of Main Theorem Let hX, τ , L , ≤ i be a hereditarily GO-space. We prove first that X is countable. By Proposition 4.8, hL, ≤ i is a scattered chain. Since hL, λ i is compact and topologically-scattered, by Lemma 4.1, hX, τ i satisfies c.c.c. property, so X has only countably many isolated points. Denote by Iso(Y ) the set of isolated points of Y . Since X is topologically dense in L, we have Iso(L) = Iso(X) and thus Iso(L) = ℵ0 . Therefore, by Proposition 3.2(3), L, and thus X is also countable. Next, by Lemma 4.3(3), the space X does not contain a countable relatively discrete set. Since X is countable, X is closed under supremum and infimum in L, and thus X, as a linear order, is complete. We have seen that L and X are compact and countable. Finally, since X is topologically-scattered, by Lemma 4.2, X is homeomorphic to the LOTS α + 1 where α is a countable ordinal. We have proved Main Theorem 1.1.

References [1] A. V. Arkhangel’skii and O. V. Okunev, Characterization of properties of spaces by properties of their continuous images, Moscow Univ. Bull. 40(5) 1985, 32–35. [2] H. R. Bennett, D. J. Lutzer, Topology and order structures, Mathematical Centre Tracts, (142) 1981 and (169) 1983. [3] M. Bekkali and S. Todorˇ cevi´ c, Algebras that are hereditarily interval, Algebra Universalis 73(1) 2015, 87–95. [4] R. Engelking, General Topology, Heldermann Verlag, Berlin 1989. [5] W. Fleissner and R. Levy, Ordered spaces all of whose continuous images are normal, Proc. Amer. Math. Soc., 105(1) 1989, 231–235. ˇ [6] W. Fleissner and R. Levy, Stone-Cech remainders which make continuous images normal, Proc. Amer. Math. Soc., 106(3) 1989, 839–842. [7] K. P. Hart, J. Nagata and J. E. Vaughan, Encyclopedia of General Topology, Elsevier Science Publishers, B.V., Amsterdam 2004. [8] S. Koppelberg, Handbook on Boolean Algebras, Vol. 1, Edited by J. D. Monk and R. Bonnet. North-Holland Publishing Co., Amsterdam 1989. [9] J. K¸ akol, W. Kubi´s, M. L´ opez-Pellicer, Descriptive Topology in Selected Topics of Functional Analysis, Developments in Mathematics, 24. Springer, New York, 2011. [10] D. J. Lutzer, On generalized ordered spaces, Dissertationes Mathematicae 89 1971, 32 pp. [11] J. G. Rosenstein, Linear orderings, Pure and Applied Mathematics, 98. Academic Press, Inc., 1982 . [12] M. E. Rudin, Nikiel’s conjecture, Topology Appl. 116(3) 2001, 305–331. [13] Z. Semadeni, Banach spaces of continuous functions, Vol. 1, Monografie Matematyczne, Tome 55, PWN–Polish Scientific Publishers, Warsaw, 1971. [14] V. V. Tkachuk and R. G. Wilson, Reflections in small continuous images of ordered spaces, European J. Math. 2 2016, 508–517. R. Bonnet: Laboratoire de Math´ ematiques, Universit´ e de Savoie, Le Bourget-du-Lac, France E-mail address: [email protected] and [email protected] A. Leiderman: Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, Israel E-mail address: [email protected]