Given a compact scattered space K, we call the derivative of K (denoted by K ′ ) the subset of K formed by its accumulation points and we inductively define K (α) = (K (β) ) ′ if α = β+1 and K (α) = β<α K (β) if α is a limit ordinal. The height of K, ht(K), is the smallest ordinal α such that K (α) is finite and nonempty, and the width of K, wd(K), is the supremum of the cardinalities |K (α) \K (α+1) | for α < ht(K). We call K = α
The purpose of this work is to show that the existence of compact hereditarily separable scattered spaces of height ω 2 is consistent with the usual axioms of set theory. For a given ordinal θ let us consider the following notation:• A cw(θ) space is a compact scattered space of countable width and height equal to θ. • A hs(θ) space is a compact scattered space which is hereditarily separable and of height equal to θ. cw(ω 1 ) spaces are usually called thin-tall spaces and cw(ω 2 ) spaces are the thinvery tall spaces. First we remark that any hs(θ) space is a cw(θ) space as the Cantor-Bendixson levels form discrete subspaces. Whether there is or not in ZFC a cw(ω 1 ) space was a question posed by Telgársky in 1968 (unpublished) and first (consistently) answered by Ostaszewski [21], using ♦. Rajagopalan constructed the first ZFC example of a cw(ω 1 ) in [23]. Further, Juhász and Weiss generalized these results (and simplified their proofs) in [12] proving in ZFC that for any ordinal θ < ω 2 , there is a cw(θ) space.
For higher θ’s the situation changes: in any model of CH there are no cw(ω 2 ) spaces and Just proved in [13] that neither are there such spaces in the Cohen model (where ¬ CH holds). On the other hand, Baumgartner and Shelah [2] constructed by forcing the first consistent example of a cw(ω 2 ) space. An interesting point of this forcing construction was the use of a new combinatorial device called a function with the property ∆.
The main purpose of this work is to prove the consistency of the existence of a hs(ω 2 ) space. In fact, our space has even stronger properties: each of its finite powers is hereditarily separable. Whether consistently there are hs(ω 3 ) or even cw(ω 3 ) spaces remains a well-known open question. On the other hand, Martínez in [17] adopted the method of [2] to obtain the consistency of the existence of cw(θ) spaces for each θ < ω 3 .
It follows from an old result of Shapirovskiȋ [25] that for any compact space K, hd(K) ≤ hL(K) + . Our construction shows that the dual inequality does not follow from ZFC, since for our compact space K, we have that hL(K) = ℵ 2 ≤ ℵ 1 = hd(K) + . Nevertheless, the dual inequality holds under GCH for regular spaces: since the weight w(K) of a regular space K is less or equal to 2 d(K) (see, for example, [8]), we trivially conclude that hL(K) ≤ w(K) ≤ 2 d(K) = d(K) + ≤ hd(K) + under GCH.
Turning to properties of Banach spaces, let us first recall some definitions and results: a Banach space X is an Asplund space if every continuous and convex real-valued function on X is Fréchet smooth at all points of a G δ dense subset of X. For separable Banach spaces, this is equivalent to admitting a Fréchet smooth renorming (see [4]). Namioka and Phelps proved in [19] that C(K) is Asplund if and only if K is scattered. Thus, our C(K) is an Asplund space.
Haydon constructed in [7] the first nonseparable Asplund space C(K) which does not admit a Fréchet smooth renorming, concluding that the situation changes for nonseparable Asplund spaces. Later, Jiménez Sevilla and Moreno analyzed in [10] the structural properties of the space C(K), where K is the well-known Kunen line constructed under CH (see [20]). They showed, for the Kunen line K, that C(K) is also a nonseparable Asplund space with no Fréchet smooth renorming.
The weight of our space K is ℵ 2 , so that C(K) is an Asplund space of density ℵ 2 . The fact that K is compact scattered and every finite power of K is hereditarily separable implies, in the same way as for the Kunen line, that C(K) does not admit any Fréchet smooth renorming, but as in the case of the Kunen line we do not know if it admits a Gâteaux smooth renorming, or a Fréchet smooth bump function.
A biorthogonal system on a Banach space X is a family [29] (Theorem 9 together with the results of [3]) the existence of uncountable semi-biorthogonal systems in Banach spaces C(K) of density strictly greater than ℵ 1 . On the other hand, the fact that our space K is compact scattered and every finite power of K is hereditarily separable implies, in the same way as for the Kunen line, that C(K) does not admit an uncountable biorthogonal system. It follows that Todorcevic’s result cannot be improved in ZFC by replacing the existence of uncountable semi-biorthogonal systems by the existence of uncountable biorthogonal systems in spaces C(K) of large density. On the other hand it is proved in [29]
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