A first-countable non-remainder of H

The purpose of this note is to confirm a suspicion raised in [55; 5, Question 4.2]: we show that Bell's example, from [33,3], of a first-countable compact space that is not an N * -image can be adapted to produce admits a connected variation that is neither an N * -image nor an H * -image. The interest in this variation stems from the authors' version of Parovičenko's theorem from [99,9]. That theorem states that every compact Hausdorff space of weight ℵ 1 or less is an N * -image; the Continuum Hypothesis then implies that the N * -images are exactly the compact Hausdorff spaces of weight 2 ℵ0 or less. We proved in [44,4] a parallel result for H * and continua (connected compact Hausdorff spaces). Since, by Arkhangel ′ skiȋ's theorem [11,1], first-countable compact spaces have weight at most 2 ℵ0 it follows that under CH first-countable compacta/continua are N * -images/H * -images respectively.

Bell’s graph. A major ingredient in our construction is Bell’s graph, constructed in [22,2]. It is a graph on the ordinal ω 2 , represented by a symmetric subset E of ω 2 2 . The crucial property of this graph is that there is no map ϕ : ω 2 → P(N) that represents this graph in the sense that α, β ∈ E if and only if ϕ(α) ∩ ϕ(β) is infinite.

Bell’s graph exists in any forcing extension in which ℵ 2 Cohen reals are added; for the reader’s convenience we shall describe the construction of E and adapt Bell’s proof so that it applies to continuous maps defined on H * .

Our starting point is a connected version of the Alexandroff double of the unit interval. We topologize the unit square as follows.

(1) a local base at points of the form x, 0 consists of the sets

(2) a local base at points of the form x, y , with y > 0 consists of the sets

We call the resulting space the connected comb and denote it by C. It is straightforward to verify that C is compact, Hausdorff and connected; it is first-countable by definition.

For each x ∈ [0, 1] and positive a we define to be the following cross-shaped closed subset of C 2 :

We note the following two properties of the sets D x,a

(1) if a < b then D x,b is in the interior of D x,a , and (2) if x = y then D x,a ∩D y,a is the union of two squares:

Now take any ℵ 2 -sized subset of [0, 1] and index it (faithfully) as {x α : α < ω 2 }. We use this indexing to identify E with the subset { x α , x β : α, β ∈ E} of the unit square. Next we remove from C 2 the following open set:

The resulting compact space we denote by C E . Observe that the intersections

To begin: the square S of the base line of C is a subset of C E and homeomorphic to the unit square so that it is (arcwise) connected.

Let x, a, y, b be a point of

where

Because the intersections of the sets D E xα,a represent E the intersections of the O α will do this as well: the conditions ‘O α ∩ O β is unbounded’ and ’ α, β ∈ E’ are equivalent.

In the next subsection we show that for (many) α, β this equivalence does not hold and that therefore C E is not a continuous image of H * .

Note also that our continuum is not an N * -image either: if g : N * → C E were continuous and onto we could use clopen subsets of N * and their representing infinite subsets of N to contradict the unrepresentability property of E.

Destroying the equivalence. We follow the argument from [22,2] and we rely on Kunen’s book [88; 8, Chapter VII] for basic facts on forcing. We let L = { α, β ∈ ω 2 2 : α ≤ β} and we force with the partial order Fn(L.2) of finite partial functions with domain in L and range in {0, 1}. If G is a generic filter on Fn(L, 2) then we let

To show that E is as required we take a nice name Ḟ for a function from ω 2 to (Q 2 ) ω that represents a choice of open sets α → O α as in above in that F (α) = a α,n , b α,n : n ∈ ω for all α. As a nice name Ḟ is a subset of ω 2 ×ω×Q 2 ×Fn(L, 2), where for each point α, n, a, b the set {p : α, n, a, b, p ∈ Ḟ } is a maximal antichain in the set of conditions that forces the nth term of Ḟ (α) to be a, b .

For each α we let I α be the set of ordinals that occur in the domains of the conditions that appear as a fifth coordinate in the elements of Ḟ with first coordinate α. The sets I α are countable, by the ccc of Fn(L, 2). We may therefore apply the Free-Set Lemma, see [66; 6, Corollary 44.2], and find a subset A of ω 2 of cardinality ℵ 2 such that α / ∈ I β and β / ∈ I α whenever α, β ∈ A and α = β. Let p ∈ Fn(L, 2) be arbitrary and take α and β in A with α < β and such that α > η whenever η occurs in p. Consider the condition q = p∪ α, β, 1 . If q forces O α ∩ O β to be bounded in [0, ∞) then we are done: q forces that the equivalence fails at α, β .

If q does not force the intersection to be bounded we can extend q to a condition r that forces O α ∩ O β to be unbounded. We define an automorphism h of Fn(L, 2) by changing the value of the conditions only at α, β : from 0 to 1 and vice versa. The condition p as well as the names ẋα and ẋβ are invariant under h. It follows that h

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