Some compactifications of the integers

Extract

Suppose that K = {K₀, K₁} is a partition of some finite combinatorial power [S]^r of a set S. Let X(K) be the compact subspace of the Tychonoff cube [0, 1]^S consisting of those functions ƒ whose supports S_ƒ = {ƒ > 0} are 0-homogeneous i.e., [S_ƒ]^r ⊆ K⁰. It can be said that almost every example of a space witnessing the distinction between various chain conditions is a variation of X(K) (see [2]). This comes from the fact that the subject is closely related to the subject of Partition Calculus. (See [11] for an explanation of this point.) To have examples which are even more topologically interesting one usually tries to make them small, i.e., as closely related to the unit interval as possible. The operation X(K) has a considerable drawback in that respect. For example, if we want X(K) to be ccc this becomes equivalent to the fact that the poset of all finite 0-homogeneous sets is ccc which amounts to the fact that K₁ must be very small. Hence K₀ is big in the sense that there exist large 0-homogeneous sets. This usually results in X(K) having large size and having points of large character. One attempt to solve this problem was given by van Douwen in [5] by going to the subspace X_m(K) of X(K) consisting of those ƒ for which S_ƒ are maximal 0-homogeneous subsets of S. Unfortunately, while X_m(K) usually does have small character it is almost never compact. This might have been the reason for his question ([2], p. 207) whether the Continuum Hypothesis implies that the class of all first countable compacta distinguishes the standard chain conditions that lie between ‘ccc’ and ‘separable’. In this paper we solve this problem completely. Moreover, we shall not go beyond the usual axioms of set theory in constructing the examples. The sequence of examples will start with a compact space of small character whose chain condition is not productive and it will end with a compact space of size c and small character which is ccc in a strong sense but which fails to have calibre θ for some regular uncountable cardinal θ, i.e., it fails to have the property of Shanin. Note that one cannot go further and show that, for example, compact spaces of small character distinguish between ‘the property of Shanin’ and ‘separable’. This follows from an old result of Efimov [3] that, under CH, first-countable spaces of calibre ℵ₁ are separable. The combinatorics behind our examples have been developed in a series of papers that deal with the subject of forcing axioms in general and Martin's axiom in particular ([10, 11, 12, 13, 14]). Martin's axiom was originally invented in connection with the Souslin Problem, i.e., to show that certain compact ccc spaces must be separable (see [8]). A result of the aforementioned study of MA showed that this axiom is nothing more than the statement that all compact ccc spaces of π-weight < c must be separable (see [12]). An analysis of the fact that MA implies SH, due to Hajnal and Juhasz[6] (see also [4], §43 for a definite result in that direction due to Shapirovskii), revealed that MA implies that every compact ccc space X with the property χ(X)⁺ < c must be separable. This result explains why the compact ccc non-separable spaces X that we construct in this paper have the property that χ(x, X) < c for all x m X rather than the stronger property χ(X) < c or even χ(X) = ℵ₀. Note that our examples show, answering a question from [6], that the Hajnal–Juhasz result is sharp in the sense that the assumption χ(X)⁺ < c cannot be weakened to χ(X) < c. The first example to show this was constructed by Bell [1] using a consequence of MA rather than just ZFC for its construction. Another feature of our examples is that they all are remainders of certain compactifications of the integers. This is of independent interest in certain constructions of weak P-points in compact F-spaces. An explanation of this can be found in [1] and [9] where the first examples of ccc non-separable remainders were constructed and used.