Ideal versions of the Bolzano-Weierstrass property

Limited Omniscience and the Bolzano-weierstrass Principle

The constructive study of metric spaces requires at first an examination of each classical proposition for numerical content. In classical mathematics it is a theorem that sequences in a compact space have convergent subsequences, but this is not constructively true. For compact intervals on the real line it has long been known that this theorem is nonconstructive because it implies the Limited...

The Complex Stone{weierstrass Property

The compact Hausdorff space X has the CSWP iff every subalgebra of C(X,C) which separates points and contains the constant functions is dense in C(X,C). Results of W. Rudin (1956), and Hoffman and Singer (1960), show that all scattered X have the CSWP and many non-scattered X fail the CSWP, but it was left open whether having the CSWP is just equivalent to being scattered. Here, we prove some g...

Strong Versions of the Theorems of Weierstrass, Montel and Hurwitz

In classical complex analysis, the theorems of Weierstrass, Montel and Hurwitz are of great use in very many contexts. The main goal of the present paper is to relax their strong hypotheses via the concept of A-statistical convergence, where A is a nonnegative regular summability matrix. The A-statistical convergence method is defined in the following way. Let A := [ajn] (j, n ∈ N := {1, 2, 3, ...

The Bolzano-Weierstrass Theorem is the Jump of Weak König's Lemma

We classify the computational content of the Bolzano-Weierstraß Theorem and variants thereof in the Weihrauch lattice. For this purpose we first introduce the concept of a derivative or jump in this lattice and we show that it has some properties similar to the Turing jump. Using this concept we prove that the derivative of closed choice of a computable metric space is the cluster point problem...

Strong Versions of the Dfr Property