Research Interests

General topology, compactness and other covering properties, function spaces, applications to functional analysis, Tukey order.

Publications

PhD Thesis


Summary

Let A be a collection of subsets of a set S. If A is a directed set with respect to set containment ⊆ or reverse set containment ⊇, then A is called a pre-ideal or pre-filter on S, respectively. The broad aim of this thesis is to investigate the relationship between (i) the topological properties of spaces and (ii) the order properties of various pre-ideals and pre-filters associated with those spaces.

Perhaps the most obvious pre-filters of interest in any topological space X are the neighborhood filters. If S is any subset of X, then the collection N(S,X) of all neighborhoods of S in X is called the neighborhood filter of S. Let Δ={(x,x):x∈X} denote the diagonal in the square of a space X. One of the main results in this thesis is the following generalization of Schneider's classic theorem that any compact space with a Gδ diagonal is metrizable:

Theorem 1. If X is any compact (Hausdorff) space, and if N(Δ,X2) has calibre (ω1,ω), then X must be metrizable.

Here, a directed set (D,≤) is said to have calibre (ω1,ω) if each uncountable subset of D contains an infinite subset with an upper bound with respect to ≤. Thus, the neighborhood filter N(Δ,X2), which is directed with respect to ⊇, has calibre (ω1,ω) precisely if each uncountable family of neighborhoods of the diagonal contains an infinite subfamily whose intersection is a neighborhood of the diagonal.

Let (E,τ) be a locally convex topological vector space, and suppose τ' is a weaker LCTVS topology on E that contains the weak topology on E (that is, σ(E,E*) ⊆ τ' ⊆ τ). Let N = N(0,(E,τ')) be the τ'-neighborhood filter of the zero element in E. If N has calibre (ω1,ω), then Theorem 1 can be used to show that every compact subset of (E,τ) is metrizable. This generalizes a result of Cascales and Orihuela (1987) concerning the compact subsets of any LCTVS in a class of spaces they called 𝔊.

If M is any separable metrizable space, then the pre-ideal K(M) of all compact subsets of M, ordered by ⊆, has calibre (ω1,ω). Note that K(M) is also a separable metrizable topological space via the Hausdorff metric. The directed sets with calibre (ω1,ω) play a crucial role in many of the results in this thesis, but some results seem to require the extra structure of the K(M)s. We define a large class C of directed sets with calibre (ω1,ω) that includes every K(M), and many results that use the extra structure of the K(M)s actually extend to every member of the class C.

We show that if (E,τ) is any LCTVS with a topology τ' as above such that N(0,(E,τ')) is in the class C, then every weakly compact subset K of E is Gul'ko compact (meaning the Banach space C(K) is countably determined in its weak topology). This generalizes Cascales and Orihuela's result that any weakly compact subset K of a LCTVS in their class 𝔊 is Talagrand compact (meaning C(K) is k-analytic in its weak topology). We would still like know whether or not this restriction to the class C is necessary:

Question 1. Does the same (or a similar) result hold for the weakly compact subsets of E when the neighborhood filter of 0 in (E,τ') just has calibre (ω1,ω)?

Calibre (ω1,ω) plays a key role not just in Theorem 1, but in many of the results in this thesis. For this reason, a chapter is devoted to studying the behavior of calibre (ω1,ω), especially its preservation under products. Todorcevic showed that calibre (ω1,ω) is generally not even preserved by finite products, and we show that uncountable products rarely have calibre (ω1,ω). Nevertheless, any countable product of K(M)s, where each M is separable metrizable, has calibre (ω1,ω). In fact, any Σ-product of K(M)s has calibre (ω1,ω). Here, the Σ-product of {K(Mα) : α∈A} is the subspace of their Tychonoff product consisting of all points (Kα)α which have at most countably many nonempty coordinates. We also extend this result to Σ-products of directed sets in the class C mentioned above.

Let C(X) denote the set of real-valued continuous functions on a space X, and for any pre-ideal S of compact subsets of X, let CS(X) denote C(X) with the topology of uniform convergence on elements of S. Thus, if S = K(X) is the pre-ideal of all compact subsets of X, then CS(X) = Ck(X) is just C(X) with the compact-open topology, while if S = [X]<ω is the pre-ideal of all finite subsets of X, then CS(X) = Cp(X) is just C(X) with the pointwise-convergence topology.

Gabriyelyan, Kakol, and Leiderman (2014) investigated the so-called strong Pytkeev property for spaces of the form Ck(X). They showed that when the points in Ck(X) have neighborhood filters with a certain nice order structure, then the strong Pytkeev property in Ck(X) is equivalent to X being Lindelof. We provide a complete characterization of when the space CS(X) has the strong Pytkeev property for some pre-ideals S. In particular, we show that Ck(X) has the strong Pytkeev property if and only if X is `Lindelof cofinally Σ', a property which strengthens the notion of a Lindelof Σ-space, and this also occurs if and only if Ck(X) is countably tight and the neighborhood filter N(0,Ck(X)) is in the class C mentioned above.