We demonstrate that many properties of topological spaces connected with the notion of resolvability are preserved by the relation of similarity between topologies. Moreover, many of them can be characterised by the properties of the algebra of sets with nowhere dense boundary and the ideal of nowhere dense sets. We use these results to investigate whether a given pair of an algebra and an ideal is topological.2010 Mathematics subject classification: primary 54A10; secondary 54A05, 11B05, 28A05.

The connection between classical notions of abstract density topologies, semi-open sets and the relation of semi-correspondence, introduced by Levine, Crossley and Hildebrand, is demonstrated.

Arithmetic in the divisor class group of a hyperelliptic curve is a fundamental component of algebraic geometry packages implemented in computer algebra systems such as Magma and Sage. In this paper, we present an adaptation of Shanks' NUCOMP algorithm for split model hyperelliptic curves of arbitrary genus that uses balanced divisors and includes a number of enhancements to optimize its efficiency in that setting. Our version of NUCOMP offers better performance than Cantor's algorithm in the balanced divisor setting. Compared with Magma's builtin arithmetic, our Magma implementation shows significant speed-ups for curves of all but the smallest genera, with the improvement increasing as the genus grows.

We consider a special kind of structure resolvability and irresolvability for measurable spaces and discuss analogues of the criteria for topological resolvability and irresolvability.

In the paper we concentrate on lower, almost-lower and semi-lower density operators on measurable spaces. The existence of maximal element in the families of such operators is investigated. Moreover, we consider topologies generated by the above operators. Among others the existence of the greatest of such topologies (with respect to the inclusion) is studied.