Maximal $\alpha $-r.e. sets

Some results on maximal open sets

In this paper, the notion of maximal m-open set is introduced and itsproperties are investigated. Some results about existence of maximal m-open setsare given. Moreover, the relations between maximal m-open sets in an m-spaceand maximal open sets in the corresponding generated topology are considered.Our results are supported by examples and counterexamples.

ATOMLESS r-MAXIMAL SETS

We focus on L(A), the filter of supersets of A in the structure of the computably enumerable sets under the inclusion relation, where A is an atomless r-maximal set. We answer a long standing question by showing that there are infinitely many pairwise non-isomorphic filters of this type.

Concerning Maximal Sets

Maximal Equilateral Sets

A subset of a normed space X is called equilateral if the distance between any two points is the same. Let m(X) be the smallest possible size of an equilateral subset of X maximal with respect to inclusion. We first observe that Petty’s construction of a d-dimensional X of any finite dimension d ≥ 4 with m(X) = 4 can be generalised to show that m(X⊕1 R) = 4 for any X of dimension at least 2 whi...

D-maximal Sets

Soare [20] proved that the maximal sets form an orbit in E. We consider here D-maximal sets, generalizations of maximal sets introduced by Herrmann and Kummer [12]. Some orbits of D-maximal sets are well understood, e.g., hemimaximal sets [8], but many are not. The goal of this paper is to define new invariants on computably enumerable sets and to use them to give a complete nontrivial classifi...