Reverse Mathematics and parameter-free Transfer

Recently, conservative extensions of Peano and Heyting arithmetic in the spirit of Nelson's axiomatic approach to Nonstandard Analysis, have been proposed. In this paper, we study the Transfer axiom of Nonstandard Analysis restricted to formulas without parameters. Based on this axiom, we formulate a base theory for the Reverse Mathematics of Nonstandard Analysis and prove some natural reversals, and show that most of these equivalences do not hold in the absence of parameter-free Transfer.     

Continuity of posets via Scott topology and sobrification

In this paper, posets which may not be dcpos are considered. The concept of embedded bases for posets is introduced. Characterizations of continuity of posets in terms of embedded bases and Scott topology are given. The main results are:

(1)

A poset is continuous iff it is an embedded basis for a dcpo up to an isomorphism;

(2)

A poset is continuous iff its Scott topology is completely distributive;

(3)

A topological T₀ space is a continuous poset equipped with the Scott topology in the specialization order iff its topology is completely distributive and coarser than or equal to the Scott topology;

(4)

A topological T₁ space is a discrete space iff its topology is completely distributive.

These results generalize the relevant results obtained by J.D. Lawson for dcpos.     

The Fréchet-Urysohn property of Pixley-Roy hyperspaces

Let F[X] be the Pixley-Roy hyperspace of a regular space X. In this paper, we prove the following theorem.

Theorem. For a space X, the following are equivalent:

(1)

F[X]is a k-space;

(2)

F[X]is sequential;

(3)

F[X]is Fréchet-Urysohn;

(4)

Every finite power of X is Fréchet-Urysohn for finite sets;

(5)

Every finite power ofF[X]is Fréchet-Urysohn for finite sets.

As an application, we improve a metrization theorem onF[X].     

Norm continuity of weakly continuous mappings into Banach spaces

Let T be the class of Banach spaces E for which every weakly continuous mapping from an α-favorable space to E is norm continuous at the points of a dense subset. We show that:

•

T contains all weakly Lindelöf Banach spaces;

•

l^∞∉T, which brings clarity to a concern expressed by Haydon ([R. Haydon, Baire trees, bad norms and the Namioka property, Mathematika 42 (1995) 30-42], pp. 30-31) about the need of additional set-theoretical assumptions for this conclusion. Also, (l^∞/c₀)∉T.

•

T is stable under weak homeomorphisms;

•

E∈T iff every quasi-continuous mapping from a complete metric space to (E,weak) is densely norm continuous;

•

E∈T iff every quasi-continuous mapping from a complete metric space to (E,weak) is weakly continuous at some point.

     

Function spaces from core compact coherent spaces to continuous B-domains

It is proved in this paper that for a continuous B-domain L, the function space [X→L] is continuous for each core compact and coherent space X. Further, applications are given. It is proved that:

(1)

the function space from the unit interval to any bifinite domain which is not an L-domain is not Lawson compact;

(2)

the Isbell and Scott topologies on [X→L] agree for each continuous B-domain L and core compact coherent space X.

     

h-Vectors of matroids and logarithmic concavity

Let M be a matroid on E, representable over a field of characteristic zero. We show that h-vectors of the following simplicial complexes are log-concave:

1.

The matroid complex of independent subsets of E.