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A topological characterization of ordinals: van Dalen and Wattel revisited Chris Good and Kyriakos Papadopoulos School of Mathematics, University of Birmingham, Birmingham B15 2TT, UK Abstract We revisit van Dalen and Wattelโs characterization of linearly ordered topological spaces in terms of nests of open sets and use this to give a topological characterization of ordinals. In particular we characterize ๐1 . Key words: Nests, LOTS, linearly ordered topological spaces, GO spaces, generalized ordered spaces, ordinals, ๐1 AMS subject classi๏ฌcation: 54F05, 06B30, 06F30 1 Nests and ordered spaces If < is a linear on a space then the order topology on ๐ is the topology generated by the collection of <-intervals. If ๐ is given the order topology, then the resulting space is called a linearly ordered topological space, or LOTS. A subspace of a LOTS is known as a generalized order, or GO, space (equivalently ๐ has a topology ๏ฌner than the order topology and each point has a local base of order convex sets). LOTS, GO spaces and ordinals with their order topology are naturally occurring topological objects and are canonical building blocks for topological examples. The problem of characterizing arbitrary LOTS and GO spaces topologically was solved by van Dalen and Wattel [12]. Previously a number of characterization of particular LOTS had been given (there are, for example, characterizations of โ, [0, 1], โโโ, compact LOTS). For a survey of such characterizations see [9]. For a general survey of LOTS and GO spaces see for example [7]. Email addresses: [email protected] (Chris Good), (Kyriakos Papadopoulos). Preprint submitted to Topology and its Applications 3 December 2010 In this paper we look again at van Dalen and Wattelโs characterization from a more order-theoretic point of view. Motivated in particular by Reedโs โmisnomed intersection topologyโ (see [10] and also [13], [4] and [6]), we ask whether it it possible to characterize ordinal spaces in purely topological terms. There are other essentially internal characterizations of certain ordinals and subspaces of ordinals due to Baker [1], van Douwen [14], Purisch [8], for example. However, these tend not to be as general or so simply stated as our own. There are also external characterizations in terms of selections, see for example [5], [3], [2]. De๏ฌnition 1 Let ๐ be a set and โ โ ๐ซ(๐). โ is said to be ๐0 -separating if and only if for each ๐ฅ โ= ๐ฆ in ๐ there is some ๐ โ โ such that either ๐ฅ โ ๐ฟ โโ ๐ฆ or ๐ฆ โ ๐ฟ โโ ๐ฅ. โ is said to be ๐1 -separating if and only if for each ๐ฅ โ= ๐ฆ in ๐ there are ๐ and ๐ in โ such that ๐ฅ โ ๐ โโ ๐ฆ and ๐ฆ โ ๐ โโ ๐ฅ. De๏ฌnition 2 Let ๐ be a set and let โ โ ๐ซ(๐). The order โฒโ is de๏ฌned by declaring ๐ฅ โฒโ ๐ฆ if and only if ๐ฅ โ ๐ฟ โโ ๐ฆ, for some ๐ฟ โ โ. De๏ฌnition 3 Let ๐ be a set and let โ โ ๐ซ(๐). โ is said to be a nest if and only if โ is linearly ordered by inclusion. There is obviously a close link between nests and linear orders. Theorem 4 Let ๐ be a set and โ โ ๐ซ(๐). (1) If โ is a nest, then โฒโ is a transitive order. (2) โ is a nest if and only if for every ๐ฅ, ๐ฆ โ ๐, either ๐ฅ = ๐ฆ, ๐ฅ โโฒโ ๐ฆ or ๐ฆ โโฒโ ๐ฅ. (3) โ is ๐0 -separating if and only if for every ๐ฅ, ๐ฆ โ ๐, either ๐ฅ = ๐ฆ, ๐ฅ โฒโ ๐ฆ or ๐ฆ โฒโ ๐ฅ. (4) โ is a ๐0 -separating nest if and only if โฒโ is a linear order. PROOF. (1) is immediate from the de๏ฌnition of โฒโ . For (2), suppose ๏ฌrst that โ is a nest. If ๐ฅ โ= ๐ฆ and both ๐ฅโฒโ ๐ฆ and ๐ฆโฒโ ๐ฅ, then there are ๐ and ๐ in โ such that ๐ฅ โ ๐ โโ ๐ฆ and ๐ฆ โ ๐ โโ ๐ฅ, so that ๐ is not a subset of ๐ and ๐ is not a subset of ๐ , contradicting the fact that โ is a nest. Conversely, suppose that ๐ and ๐ are elements of โ. If ๐ is not a subset of ๐ and ๐ is not a subset of ๐ , then are ๐ฅ โ ๐ โ ๐ and ๐ฆ โ ๐ โ ๐ , so that both ๐ฅโฒโ ๐ฆ and ๐ฆโฒโ ๐ฅ. For (3), if โ is ๐0 -separating and ๐ฅ โ= ๐ฆ, then there is ๐ โ โ such that either ๐ฅ โ ๐ โโ ๐ฆ, so that ๐ฅโฒโ ๐ฆ, or ๐ฆ โ ๐ โโ ๐ฅ, so that ๐ฆโฒโ ๐ฅ. Conversely, if ๐ฅ โ= ๐ฆ, then without loss of generality ๐ฅโฒโ ๐ฆ, so that there is ๐ โ โ such that ๐ฅ โ ๐ โโ ๐ฆ. 2 (4) now follows from (1), (2) and (3). โก { } Note that if ๐ = {0, 1, 2} and ๐ฉ = {0}, {1}, {2} , then โฒ๐ฉ is transitive but not a linear order. Theorem 5 Let ๐ be a set. Suppose that โ and โ are two nests on ๐. โ โช โ is ๐1 -separating if and only if โ and โ are both ๐0 -separating and โฒโ = โณโ . PROOF. Suppose that โ โช โ is ๐1 -separating. If ๐ฅ โ= ๐ฆ, then there are ๐ and ๐ in โ โช โ such that ๐ฅ โ ๐ โโ ๐ฆ and ๐ฆ โ ๐ โโ ๐ฅ. Without loss of generality ๐ โ โ and, since โ is a nest, ๐ โ / โ so ๐ โ โ. Hence ๐ฅโฒโ ๐ฆ and ๐ฆโฒโ ๐ฅ. Since ๐ฅ and ๐ฆ were arbitrary, it follows that โ and โ are ๐0 -separating and that โฒโ = โณโ . Conversely, suppose that โ and โ are two ๐0 -separating nests such that โฒโ = โณโ . If ๐ฅ โ= ๐ฆ, then there is ๐ฟ โ โ such that, without loss of generality, ๐ฅ โ ๐ฟ โโ ๐ฆ. Hence ๐ฅโฒโ ๐ฆ so that ๐ฆโฒโ ๐ฅ, which implies that there is some ๐ โ โ such that ๐ฆ โ ๐ โโ ๐ฅ. Hence โ โช โ is ๐1 -separating. โก Let โ and โ be two nests whose union is ๐1 -separating. Topologically speaking, if the elements of โ and โ are open sets, it is relatively simple to show that the order-topology generated by โฒโ is coarser than the topology on ๐. As we shall see in Theorem 10, the following notion of interlocking, due to van Dalen and Wattel [12], is the key idea in ensuring that the topology induced by the order โฒโ coincides with the topology generated by the subbase โ โช โ. De๏ฌnition 6 Let ๐ be a set and โ โ ๐ซ(๐). We say that โ is interlocking โฉ โช if and only if, for each ๐ฟ โ โ, ๐ฟ = {๐ โ โ : ๐ฟ โ ๐ } implies ๐ฟ = {๐ โ โ : ๐ โ ๐ฟ}. The next two propositions clarify the relationship between an interlocking nest and the properties of its induced order. Lemma 7 Let ๐ be a set and โ be a ๐0 -separating nest on ๐. โฉ (1) ๐ฟ = {๐ โ โ : ๐ฟ โ ๐ } if and only if ๐ โ ๐ฟ has no โฒโ -minimal element. โช (2) ๐ฟ = {๐ โ โ : ๐ โ ๐ฟ} if and only if ๐ฟ has no โฒโ -maximal element. PROOF. By Theorem 4, โฒโ is a linear order on ๐. For (1), if ๐ฅ is the โฒโ minimal element of ๐ โ ๐ฟ, then for all ๐ โ โ such that ๐ฟ โ ๐ , ๐ฅ โ ๐ , so โฉ that ๐ฟ โ= {๐ โ โ : ๐ฟ โ ๐ }. Conversely, if ๐ โ ๐ฟ has no minimal element, 3 then for all ๐ฅ โ / ๐ฟ, there is some ๐ฆโฒโ ๐ฅ such that ๐ฆ โ / ๐ฟ. Since โ is a ๐0 separating nest, there is some ๐ โ โ such that ๐ฆ โ ๐ โโ ๐ฅ. Since ๐ฆ โ ๐ โ ๐ฟ, โฉ โฉ ๐ฟ โ ๐ , so that ๐ฅ โ / {๐ โ โ : ๐ฟ โ ๐ } and ๐ฟ = {๐ โ โ : ๐ฟ โ ๐ }. (1) follows. The proof of (2) is exactly similar. โก It is immediate from De๏ฌnition 6, that a collection โ is interlocking if and only โช โฉ if, for all ๐ฟ โ โ, either ๐ฟ = {๐ โ โ : ๐ โ ๐ฟ} or ๐ฟ โ= {๐ โ โ : ๐ฟ โ ๐ }. Theorem 4 and Lemma 7 therefore imply the following. Theorem 8 Let ๐ be a set and let โ be a ๐0 -separating nest on ๐. The following are equivalent: (1) โ is interlocking; (2) for each ๐ฟ โ โ, if ๐ฟ has a โฒโ -maximal element, then ๐ โ ๐ฟ has a โฒโ -minimal element; (3) for all ๐ฟ โ โ, either ๐ฟ has no โฒโ -maximum element or ๐ โ ๐ฟ has a โฒโ -minimal element. โก Lemma 9 Let < be a linear order on ๐. Let { โ< = (โโ, ๐) : ๐ โ ๐ } and { } โ< = (๐, โ) : ๐ โ ๐ . Then โ< and โ< are ๐0 separating, interlocking nests such that โ โช โ is ๐1 separating and โฒโ = โณโ =<. Moreover โ โช โ forms a subbase of order open sets for the order topology on ๐. PROOF. Clearly โ and โ are ๐0 -separating nests whose union is ๐1 -separating. By Theorem 5, โฒโ = โณโ . If ๐ฅ < ๐ฆ, then ๐ฟ = (โโ, ๐ฆ) โ โ and ๐ฅ โ ๐ฟ โโ ๐ฆ, so that ๐ฅโฒโ ๐ฆ. On the other hand if ๐ฅโฒโ ๐ฆ, then for some ๐ง โ ๐, ๐ฅ โ (โโ, ๐ง) โโ ๐ฆ, so that ๐ฅ < ๐ง and ๐ง < ๐ฆ, which implies that ๐ฅ < ๐ฆ. It remains to show that โ and โ are interlocking. Suppose that ๐ฟ = (โโ, ๐) โ โ has a <-maximal element ๐. Then ๐ < ๐ and if ๐ < ๐ฅ โค ๐, ๐ฅ = ๐, so that ๐ is the <-minimal element of ๐ โ ๐ฟ. By Theorem 8, then, โ is interlocking. That โ is interlocking follows by the same argument. By de๏ฌnition of the order topology induced by < on ๐, โ โช โ forms a subbase of order open sets. โก We are now in a position to give a slightly di๏ฌerent, more direct, proof of van Dalen and Wattelโs characterization of GO and LOTS. Theorem 10 (van Dalen & Wattel) Let ๐ be a space with topology ๐ฏ . Suppose that (1) If โ and โ are two nests of open sets whose union is ๐1 -separating, then every โฒโ -order open set is open in ๐. 4 (2) ๐ is a GO space if and only if there are two nests โ and โ of open sets whose union is ๐1 -separating and forms a subbase for ๐ฏ . (3) ๐ is a LOTS if and only if there are two interlocking nests โ and โ of open sets whose union is ๐1 -separating and forms a subbase for ๐ฏ . โช PROOF. Clearly, for any ๐ โ ๐, the โฒโ -interval (โโ, ๐) = {๐ฟ โ โ : ๐ โ / โช ๐ฟ} and the โฒโ -interval (๐, โ) = {๐ โ โ : ๐ โ / ๐ }. It follows immediately that, if the sets in โ and โ, are open in ๐, then every order-open set is open in ๐, so that (1) holds. For (3), if ๐ is a LOTS with linear order <, then the existence of two such nests follows by Lemma 9. Conversely, suppose that there are two interlocking nests โ and โ of open sets whose union is ๐1 -separating and forms a subbase for the topology on ๐. By Theorem 4, โฒโ is a linear order on ๐ and by (1), every order open set is open. It remains to show that every open set is orderopen. Since โ โช โ forms a subbase for the topology ๐ฏ , and โ and โ are both nests, every ๐ โ ๐ฏ can be written as a union of sets of the form ๐ฟ โฉ ๐ , where ๐ฟ โ โ and ๐ โ โ. It su๏ฌces, then, to show that each ๐ฟ โ โ and each ๐ โ โ is order-open. So suppose that ๐ฟ โ โ. If โ has no โฒโ -maximal element, then there is some ๐ด โ ๐ฟ that is co๏ฌnal in ๐ฟ with respect to the order โฒโ . But โช then ๐ฟ = ๐โ๐ด (โโ, ๐), so that ๐ฟ is order open. On the other hand, if ๐ฟ does have a maximal element ๐, then, since โ is interlocking, ๐ โ ๐ฟ has a minimal element ๐โฒ and ๐ฟ = (โโ, ๐] = (โโ, ๐โฒ ) is also order open. That each ๐ โ โ is order open follows in exactly the same way. To see that (2) holds, if ๐ is a GO space then ๐ โ ๐ for some LOTS ๐ . Since ๐ is a LOTS, it has two interlocking nests of open sets, โ and โ, whose union forms a ๐1 -separating subbase for the topology on ๐ . Setting โโฒ = {๐ฟ โฉ ๐ : ๐ฟ โ โ} and โโฒ = {๐ โฉ ๐ : ๐ โ โ}, we obtain two nest of sets open in ๐ whose union is forms a ๐1 -separating subbase for the topology on ๐. For the converse, suppose that the space ๐ has two nests โ and โ whose union forms a ๐1 -separating subbase for ๐. We will construct a LOTS ๐ such that ๐ is a subspace of ๐ . Let โโ be the set of all ๐ฟ in โ such that ๐ฟ has a โฒโ -maximal element but ๐ โ ๐ฟ has no โฒโ -minimal element. Let โโ be the set of all ๐ in โ such that ๐ has a โฒโ -maximal element (i.e. a โฒโ -minimal element) but ๐ โ ๐ has no โฒโ -minimal element (i.e. a โฒโ maximal element). For each ๐ฟ โ โโ , let ๐ฅ๐ฟ denote the โฒโ -maximal element of ๐ฟ and, for each ๐ in โโ , let ๐ฆ๐ denote the โฒโ -minimal element of ๐ . For โ each ๐ฟ โ โโ and ๐ โ โโ , choose distinct points ๐ฅ+ ๐ฟ and ๐ฆ๐ not in ๐. Let โ โ โ ๐ = ๐ โช {๐ฅ+ ๐ฟ : ๐ฟ โ โ } โช {๐ฆ๐ : ๐ โ โ }, de๏ฌne ๐ : ๐ โ ๐ by โง ๏ฃด ๏ฃด โจ๐ฅ ๐(๐ฅ) = ๏ฃด๐ฅ๐ฟ ๏ฃด โฉ ๐ฆ๐ 5 if ๐ฅ โ ๐, if ๐ฅ = ๐ฅ+ ๐ฟ, โ if ๐ฅ = ๐ฆ๐ . and de๏ฌne the linear order < on ๐ by declaring ๐ฅ < ๐ฆ if and only if either โ ๐(๐ฅ) โ= ๐(๐ฆ) and ๐(๐ฅ)โฒโ ๐(๐ฆ), or ๐ฅ = ๐ฅ๐ฟ and ๐ฆ = ๐ฅ+ ๐ฟ , or ๐ฅ = ๐ฆ๐ and ๐ฆ = ๐ฆ๐ . Clearly ๐ โ ๐ and the restriction of < to ๐ is equal to โฒโ . It remains to show that the topology ๐ฏ on ๐ coincides with the subspace topology on ๐ inherited from the order topology on ๐ . As in the argument for (3), since โโชโ is a subbase for ๐ฏ consisting of two nests, every ๐ in ๐ฏ can be written as a union of sets of the form ๐ฟ โฉ ๐ , where ๐ฟ โ โ and ๐ โ โ. It su๏ฌces therefore to show that every ๐ฟ โ โ and ๐ โ โ can be written as the intersection of an order-open set with ๐. If ๐ฟ โ / โโฒ , then ๐ฟ = ๐ โฉ ๐ โ1 (๐ฟ) and ๐ โ1 (๐ฟ) is order-open. On the other hand, if ๐ฟ is in โโฒ with โฒโ -maximal element ๐ฅ๐ฟ , then ๐ฟ = ๐ โฉ (โโ, ๐ฅ+ ๐ฟ ). The argument for ๐ โ โ is the same. โก As van Dalen and Wattel point out [12], if ๐ is a compact space and the two nests โ and โ form a ๐1 -separating subbase for ๐, then both โ and โ are interlocking, corresponding to the fact that a compact GO space is LOTS. In fact more is true. Theorem 11 Let ๐ be a space and let โ and โ be two nests of open sets whose union forms a ๐1 -separating subbase for ๐. Suppose that โ has the property that for all ๐ฟ โ โ there is a compact set ๐ถ such that ๐ฟ โ ๐ถ, (1) โ is interlocking. (2) If โ is not interlocking, then this is only because there is a singleton โฉ ๐ 0 โ โ such that ๐ 0 = {๐ โ โ : ๐ 0 โ ๐ }. PROOF. Suppose that โ is not interlocking. By Theorem 8, there is some โฉ ๐ฟ โ โ such that ๐ฟ has a โฒโ -maximal element ๐ฅ๐ฟ , but ๐ฟ = {๐ โ โ : ๐ฟ โ ๐ }. Choose ๐ โ โ and a compact set ๐ถ such that ๐ฟ โ ๐ โ ๐ถ. There is an in๏ฌnite decreasing subset โณ of {๐ โ โ : ๐ฟ โ ๐ โ ๐ โ ๐ถ} such that โฉ โณ = ๐ฟ. Since โ โช โ is ๐1 -separating, for each ๐ and ๐ โฒ in โณ such that ๐ โ ๐ โฒ , there is ๐ฅ๐ โ ๐ , ๐ฆ๐ โ ๐ โฒ and ๐ โ โ such that ๐ฅ๐ โ / ๐ โฉ๐ โฒ and ๐ฆ๐ โ ๐ โฉ ๐ . But then there is an in๏ฌnite increasing subset ๐ฎ of โ that covers ๐ โ ๐ฟ. It follows that {๐ฟ} โช ๐ฎ is an open cover of ๐ถ with no ๏ฌnite subcover. This contradiction proves (1). Suppose now that โ is not interlocking, so that for some ๐ โ โ, ๐ has a โฉ โฒโ -minimal element ๐ฅ๐ but ๐ = {๐ โ โ : ๐ โ ๐}. If ๐ is not a singleton (and thus the least element of โ), then there is some ๐ฆ โ ๐ such that ๐ฅ๐ โฒโ ๐ฆ. Let ๐ โ โ be such that ๐ฅ๐ โ ๐ โโ ๐ฆ and let ๐ถ be a compact set such that ๐ โ ๐ถ. Then as for (1), {๐ } โช {๐ฟ โ โ : ๐ฟ โฉ ๐ = โ } is a cover of ๐ถ by sets open in ๐ that has no ๏ฌnite subcover. โก The next corollary now follows easily from Theorems 10 and 11. In fact it is 6 not much harder to argue locally to see that a locally compact GO space is a LOTS. Corollary 12 If ๐ is compact GO space, then ๐ is a LOTS. โก Suppose that โ an ๐ฉ are two nests and โฒโ =โฒ๐ฉ . How do โ and ๐ฉ relate? Proposition 13 Let ๐ be a set and let โ โ ๐ซ(๐). (1) If โ โ โโช and each element of โโช is a union of elements from โ, in particular if โโช is the closure of โ under arbitrary unions, then โฒโ =โฒโโช . (2) If โ โ โโฉ and each element of โโฉ is an intersection of elements from โ, in particular if โโฉ is the closure of โ under arbitrary intersections, then โฒโ =โฒโโฉ . (3) If โ is an interlocking nest, โ โ โโช and each element of โโช is a union of elements from โ, then โโช is an interlocking nest. { } (4) If โ is a ๐0 -separating nest and โโฒ = (โโ, ๐) : ๐ โ ๐ , is nest leftin๏ฌnite open โฒโ -intervals, then ๐๐ฟโฒ is an interlocking nest and โฒโ =โฒ๐๐ฟโฒ PROOF. For (1) and (2) note ๏ฌrst that, if ๐ฟ โ โ, then ๐ฟ is in both โโช and โโฉ , so that ๐ฅ โฒโโช ๐ฆ and ๐ฅ โฒโโฉ ๐ฆ, whenever ๐ฅโฒโ ๐ฆ. If ๐ฅ โฒโโช ๐ฆ, then for some โช โณ โ โ, ๐ฅ โ โณ โโ ๐ฆ, so that for any ๐ โ โณ, ๐ฅ โ ๐ โโ ๐ฆ and ๐ฅโฒโ ๐ฆ. If โฉ ๐ฅ โฒโโฉ ๐ฆ, then for some โณ โ โ, ๐ฅ โ โณ โโ ๐ฆ, so that for some ๐ โ โณ, ๐ฅ โ ๐ โโ ๐ฆ and ๐ฅโฒโ ๐ฆ. (3) is immediate from the De๏ฌnition 6. (4) is routine given the proof of Lemma 9. โก Let < be the usual order on โ. Recall that the Sorgenfrey {Line is โ with } the topology generated by the base of half open <-intervals (๐, ๐] : ๐ < ๐ . { } { } Clearly the two nests โ = (โโ, ๐] : ๐ โ โ and โ = (๐, โ) : ๐ โ โ form a ๐1 -separating subbase for the Sorgenfrey line. โ is interlocking but โ is not interlocking. On the other hand, <= โฒโ . The Michael is formed from the real line by de๏ฌning the topology of the reals by declaring each irrational to be isolated. The nests { } { ๐ฉ = (โโ, ๐) : ๐ โ โ โช (โโ, ๐] : ๐ โ /โ } and { } { } โณ = (๐, โ) : ๐ โ โ โช [๐, โ) : ๐ โ /โ form a ๐1 -separating subbase }for the Michael Line. Notice that the nests โ, { ๐ฉ and ๐ฌ = (โโ, ๐) : ๐ โ โ are all distinct, indeed โ and ๐ฌ are disjoint, 7 yet all three generate the usual order on โ. 2 Nests and well-orders In this section we turn our attention to ordinal spaces. Recall that a topological space ๐ is said to be scattered if and only if every subset ๐ด โ ๐ has an isolated point. ๐ is said to be right-separated if and only if there is a well-order on ๐ for which initial segments are open [11]. De๏ฌnition 14 A collection โ โ ๐ซ(๐) scatters ๐ if and only if for every non-empty subset ๐ด of ๐, there is some ๐ฟ โ โ such that โฃ๐ด โฉ ๐ฟโฃ = 1. The following lemma is obvious. Lemma 15 Let ๐ be a space. The following are equivalent. (1) ๐ is scattered, (2) ๐ is right separated. (3) ๐ is scattered by a nest of open subsets. โก Theorem 16 Let ๐ be a set and โ be a nest. The following are equivalent: (1) (2) (3) (4) โ scatters ๐. โฒโ is a well order. โ is ๐0 -separating and well-ordered by โ. โ is ๐0 -separating and, for every subset ๐ด of ๐, there is an ๐ โ ๐ด such that for any ๐ฅ โ ๐ด and any ๐ฟ โ โ, if ๐ฅ โ ๐ฟ, then ๐ โ ๐ฟ. PROOF. (1) implies (2): If ๐ด is a subset of ๐ and ๐ฟ โฉ ๐ด = {๐}, then ๐ is clearly the โฒโ -least element of ๐ด. (2) implies (3): Since โฒโ is a linear order, โ is ๐0 -separating by Theorem 4. Suppose that ๐ฟ1 โ ๐ฟ2 โ ๐ฟ3 โ . . . is an in๏ฌnite decreasing chain in โ. Then there are points ๐ฅ๐ โ ๐ฟ๐ โ ๐ฟ๐+1 , which form an in๏ฌnite decreasing โฒโ -chain, contradicting (2). (3) implies (4): Suppose that ๐ด is a subset of ๐. Let ๐ฟ be the โ-least element of โ such that ๐ฟ โฉ ๐ด is non-empty. Since โ is ๐0 -separating and well-ordered by โ, ๐ฟ โฉ ๐ด = {๐} for some ๐. Then, if ๐ โ โ and ๐ โ= ๐ฅ โ ๐ด โฉ ๐ , ๐ฟ โ ๐ , so that ๐ โ ๐ . (4) implies (1): Let ๐ด be a subset of ๐. Let ๐ be the point furnished by (4) for 8 ๐ด and let ๐ be the point furnished by (4) for ๐ด โ {๐}. Since โ is ๐0 separating, there is ๐ฟ โ โ which ๐0 -separates ๐ and ๐. By (4), ๐ โ ๐ฟ if ๐ โ ๐ฟ, so ๐ โ ๐ฟ โโ ๐. If ๐ฅ โ= ๐ and ๐ฅ โ ๐ฟ, then ๐ โ ๐ฟ, so ๐ฟ โฉ ๐ด = {๐}. โก Theorem 17 Let ๐ be a set. Let โ and โ be two nests on ๐ that are ๐0 separating. (1) Suppose that for all ๐ด โ ๐, there is some ๐ โ ๐ด such that, if ๐ โ ๐ โ โ, then ๐ด โ ๐ . Then โณโ is a well-order and โ is well-ordered by โ. (2) Suppose that โ โช โ ๐1 -separates ๐. โ is well-ordered by โ if and only if โ is well-ordered by โ. PROOF. Clearly, if ๐ด is a subset of ๐, and ๐ is as in the statement of the theorem, then ๐ is the โฒโ -maximal, hence โณโ -minimal, element of ๐ด. Hence (1) holds. (2) is immediate by Theorem 5 โก Lemma 18 Let ๐ be a set and let โ and โ be subsets of ๐ซ(๐). Suppose that โ scatters ๐. (1) โ is ๐0 -separating. (2) If โ is a nest, then โ is interlocking. (3) If โ and โ are nests of open sets whose union ๐1 -separates ๐, then there is a subset โณ of โ that ๐0 -separates and scatters ๐ consisting of clopen sets. PROOF. For (1), given ๐ฅ โ= ๐ฆ, there is ๐ฟ โ โ such that ๐ฟ โฉ {๐ฅ, ๐ฆ} is a singleton. For (2), pick ๐ฟ โ โ. Since โ scatters ๐, there is some ๐ โ โ such that ๐ โฉ (๐ โ ๐ฟ) = {๐ฅ}. Since โ is a nest, ๐ฟ โ ๐ = ๐ฟ โช {๐ฅ} and, whenever โฉ ๐ฟ โ ๐ โฒ โ โ, ๐ โ ๐ โฒ . Hence ๐ฟ โ= {๐ โ โ : ๐ฟ โ ๐ }, so that โ is vacuously interlocking. To see that (3) holds, note ๏ฌrst that by Lemma 16 that โฒโ is a well-order. Let ๐ฅ โ= ๐ฆ and let ๐ฅ+ denote the immediate โฒโ -successor of ๐ฅ, so that ๐ฅ+ โดโ ๐ฆ. Sinceโ โช โ ๐1 -separates ๐, there are ๐ฟ โ โ and ๐ โ โ such that ๐ฅ โ ๐ฟ โโ ๐ฅ+ and ๐ฅ โ / ๐ โ ๐ฅ+ . Since the interval (๐ฅ, ๐ฅ+ ) is empty, ๐ โ ๐ฟ = ๐ . Since ๐ is open, ๐ฟ is clopen, ๐ฅ โ ๐ฟ โโ ๐ฆ and ๐ฆ โ ๐ โโ ๐ฅ๐ฟ. โก Lemma 19 Let โ be a nest of subsets of the set ๐ and let โ be the nest โ = {๐ โ ๐ฟ : ๐ฟ โ โ}. (1) โ is interlocking if and only if, for all ๐ฟ โ โ, ๐ฟ = โช whenever ๐ฟ = {๐ โ โ : ๐ โ ๐ฟ}. 9 โฉ {๐ โ โ : ๐ฟ โ ๐ } (2) If ๐ is a space and each ๐ฟ โ โ is compact and open, then โ is interlocking. (3) If โ is ๐0 -separating, in particular if โ scatters ๐, then โ โช โ is ๐1 separating. PROOF. That (1) holds is an immediate consequence of de Morganโs Laws. โช For (2), suppose that ๐ฟ = {๐ โ โ : ๐ โ ๐ฟ}. Since ๐ฟ is compact, we have ๐ฟ = ๐1 โช โ โ โ โช ๐๐ , for some ๐๐ โ ๐ฟ, but โ is a nest so we have ๐ฟ = ๐๐ for some ๐๐ โ โ such that ๐๐ โ ๐ฟ, which is impossible. So the condition in (1) holds vacuously. Given Lemma 18, (3) follows immediately. โก Theorem 20 Let ๐ be a space. The following are equivalent. (1) ๐ is homeomorphic to an ordinal. (2) ๐ has two interlocking nests โ and โ of open sets whose union is ๐1 separating subbase such that โ scatters ๐. (3) ๐ has two interlocking nests โ and โ of open sets whose union is ๐1 separating subbase, one of which is well-ordered by โ or โ. (4) ๐ is scattered by a nest โ of clopen subsets such that โช (a) ๐ฟ โ= {๐ : ๐ โ ๐ฟ} for any ๐ฟ โ โ; (b) {๐ฟ โ ๐ : ๐ฟ, ๐ โ โ} is a base for ๐. (5) ๐ is scattered by a nest of compact clopen sets. PROOF. The equivalence of (1), (2) and (3) follows immediately from Theorems 16 and 10 and Lemma 18. { } (1) implies both (4) and (5) since, if ๐ผ is an ordinal, then [0, ๐ฝ] : ๐ฝ < ๐ผ is a nest of compact clopen subsets that scatter ๐ผ and satisfy conditions (4a) and (4b). Lemmas 18 and 19 imply that if either (4) or (5) holds, then both โ and โ = {๐ โ ๐ฟ : ๐ฟ โ โ} are interlocking nests of open sets whose union ๐1 separates ๐. If (4b) holds, then โ โช โ is a subbase for ๐ and we see that (4) implies (2). To see that (5) implies (1), we argue as follows. We have that โฒโ is a well-order on ๐ and that the order topology induced by โฒโ is coarser than the topology on ๐ by Theorem 10. If ๐ is compact, then we note that the order topology is Hausdor๏ฌ and coarser than the compact topology on ๐. Hence the two topologies coincide. If ๐ is not compact, then since the elements of โ are clopen and compact, ๐ is locally compact and we may form the one-point compacti๏ฌcation ๐ โ of ๐. But then โ โช {๐ โ } is a nest of compact clopen sets that scatters ๐ โ , which by the previous sentence is homeomorphic to an ordinal. Clearly ๐ is a โฒโ -initial segment of ๐ โ , so that ๐ is also homeomorphic to an ordinal. โก 10 The following corollary is now immediate. Corollary 21 ๐ is homeomorphic to a cardinal if and only if ๐ is scattered by a nest โ of compact clopen sets such that โฃ๐ฟโฃ < โฃ๐โฃ for each ๐ฟ โ โ. In particular, ๐ is homeomorphic to ๐1 if and only if ๐ is uncountable and scattered by a nest of compact clopen countable sets. โก As in Theorem 10, we also have the following. Proposition 22 Let ๐ be a space. (1) ๐ admits a continuous bijection onto and ordinal if and only if it is scattered by a nest of clopen sets. (2) ๐ is homeomorphic to a subspace of an ordinal if and only if it is scattered by a nest of clopen sets โ and {๐ฟ โ ๐ : ๐ฟ, ๐ โ โ} forms a subbase for ๐. PROOF. One direction is obvious in each case. For (1), we note that the order โฒโ is a well-order and that every order open set is open in ๐ by Theorem 10. For (2), if โ is a nest of clopen sets that scattered ๐, then โฒโ is well-ordered and โ is ๐0 -separating (and interlocking) by Lemma 18. Let โ = {๐ โ ๐ฟ : ๐ฟ โ โ}. Then, as โ is ๐0 -separating, โ โช โ is ๐1 -separating, so that the result follows by the proof of Theorem 10 (2). To see this note that by Lemma 19, โช interlocking fails in โ for elements ๐ โ ๐ฟ where ๐ฟ = {๐ โ โ : ๐ โ ๐ฟ} but โฉ ๐ฟ โ= {๐ โ โ : ๐ฟ โ ๐ }. Let โโฒ be the set of all such ๐ฟ. For each such ๐ฟ we introduce a new point ๐ฅ๐ฟ โโ ๐ and de๏ฌne an order < on ๐ โ = ๐ โช {๐ฅ๐ฟ : ๐ฟ โ โโฒ } by declaring โง ๏ฃด ๐ฅ, ๐ฆ โ ๐ and ๐ฅโฒโ ๐ฆ, ๏ฃด ๏ฃด ๏ฃด ๏ฃด โจ๐ฅ = ๐ฅ , ๐ฆ โ ๐ and ๐ฆ โ / ๐ฟ, ๐ฟ ๐ฅ < ๐ฆ i๏ฌ ๏ฃด ๏ฃด๐ฅ โ ๐, ๐ฆ = ๐ฅ๐ฟ and ๐ฅ โ ๐ฟ, ๏ฃด ๏ฃด ๏ฃด โฉ ๐ฅ = ๐ฅ๐ฟ , ๐ฆ = ๐ฆ๐ and ๐ฟ โ ๐. Then it is not hard to show that < is a well-order on ๐ โ that agrees with โฒโ on ๐ and that ๐ is subspace of ๐ โ . โก Lemma 15 shows that the existence of nest of open sets that scatters a space is equivalent to right-separation. We exploit this in the next theorem. De๏ฌnition 23 Let ๐ be a space and < a well-order on ๐. We say that <weakly left-separates ๐ if and only if {๐ฆ โ ๐ : ๐ฆ โค ๐ฅ} is closed for every ๐ฅ โ ๐. < left-separates ๐ if and only if {๐ฆ : ๐ฆ < ๐ฅ} is closed for all ๐ฅ โ ๐. 11 Theorem 24 Let ๐ be a space. (1) ๐ admits a continuous bijection onto an ordinal if and only if it is rightand weakly left-separated by the same well order. (2) ๐ is homeomorphic to a subspace of an ordinal if and only if it is rightand weakly-left separated by a well-order whose intervals form a subbase for ๐. (3) ๐ is homeomorphic to an ordinal if and only if it is right- and weakly left-separated by the order < so that if ๐ถ = {๐ฅ๐ผ : ๐ผ โ ๐} is a <-increasing sequence indexed by a limit ordinal and ๐ถ is closed, then ๐ถ is <-co๏ฌnal in ๐. PROOF. (1) and (2) follow easily from Proposition 22. For (3), note ๏ฌrst that the order topology induced by < is coarser than the topology on ๐. But then if the topology of ๐ is strictly ๏ฌner than the order topology on ๐, there is some order limit point ๐ฅ that is not a limit point in ๐, which contradicts the condition of the theorem. The space ๐1 +1+๐โ, where ๐ โ denotes ๐ with the reverse order, is a compact scattered LOTS that is not scattered by a nest of clopen sets. Purisch [8] has shown that every scattered GO space is LOTS, hence the isolated points of ๐1 is a locally compact LOTS and has an subbase consisting of two interlocking nest whose union is ๐1 -separating. It also has a nest of clopen countable sets that scatter ๐, but is not scattered by a nest of compact clopen sets. Let ฮจ = ๐ โช {๐ฅ๐ผ : ๐ผ โ ๐ } denote Mrowkaโs ฮจ-space and ฮจโ denote its one { point compacti๏ฌcation. If ๐ฟ๐ผ = ๐ โช {๐ฅ๐ฝ : ๐ฝ < ๐ผ} and โ = [0, ๐] : ๐ โ } ๐ โช {๐ฟ๐ผ : ๐ผ โ ๐ } is a ๐0 -separating nest of open sets that scatters ฮจโ . Note โฉ โช that ๐ฟ0 = {๐ โ ๐ฉ : ๐ฟ0 โ ๐ } but ๐ฟ0 โ= {๐ โ ๐ฉ : ๐ โ ๐ฟ0 }, so that โ is not interlocking. It follows that two nests whose union is ๐1 -separating for the conclusion of Theorem 11 to hold. ฮจโ is both right- and left-separated, but not by the same order. { } Let ๐ = ๐1 โช (๐ผ, ๐) : ๐ โ ๐, ๐ผ โ ๐1 a limit . 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