The problem with which this paper is concerned is that of investigating a class of topological properties commonly called separation properties. A topological space which satisfies only the definition may be very limited in open sets. By use of the separation properties, specific families of open sets can be guaranteed.

A new class of functions, called somewhat e-continuous functions, has been defined and studied by making use of e-open sets. Characterizations and properties of somewhat e-continuous functions are presented.

No / In this paper we investigate the use of general topological spaces in connection with a generalised variant of membrane systems. We provide an approach which produces a fine grain description of local operations occurring simultaneously in sets of compartments of the system by restricting the interactions between objects. This restriction is given by open sets of a topology and multisets of objects associated with them, which dynamically change during the functioning of the system and which together define a notion of vicinity for the objects taking part in the interactions. / The first, the second, and the third authors were partially supported under the Royal Society International Exchanges Scheme (ref. IE110369). The second author was also partially supported by the project MuVet, Romanian National Authority for Scientific Research (CNCS – UEFISCDI) grant number PNII- ID-PCE-2011-3-0688. This work was partially completed whilst the third author was a visiting fellow at the Isaac Newton Institute for the Mathematical Sciences in the programme ‘Semantics & Syntax: A Legacy of Alan Turing’. The work of the first and the fourth author was also supported in part by the Hungarian Scientific Research Fund (OTKA), Grant no. K75952. The fourth author was supported by the European Union through the T´AMOP-4.2.2.C-11/1/KONV-2012-0001 project which is co-financed by the European Social Fund.

A function is dened to be hardly e-open provided that theinverse image of each e-dense subset of the codomain that is contained in a proper open set is e-dense in the domain. Characterizations and properties of hardly e-open functionsare presented.

The concept of (b, s)-continuous functions in topological spaces is introduced and studied. Some of their characteristic properties are considered. Also we investigate the relationships between these classes of functions and other classes of functions.

In this paper we introduce two classes of functions called weakly δ -open and weakly δ -closed functions. We obtain their characterizations, their basic properties and their relationships with other types of functions between topological spaces.