In General Topology, continuous functions occur naturally as more general objects than spaces. In fact, it is evident that every topological space can be trivially identified with a continuous function defined on it and with values in a space consisting of a single point. Since the early 1950s, this simple observation has suggested the idea of studying particular properties of continuous functions which, – reflecting in some way known properties for spaces – could generalize them. In 1975 Uljanov, studying first numerable compactifications of topological spaces, gave an impetus to this theme by introducing the notion of Hausdorff function, but it was not until 1984 that Pasynkov dealt in a broad and systematic way with the generalization of notions and properties of spaces to continuous functions, thus laying the foundations of the Fibrewise General Topology for Fibre (briefly FGT).
FGT is a relatively new area of research whose main goal is to generalize to continuous functions notions and properties originally defined for topological spaces. From a categorical point of view, this means replacing the study of the category \mathrm{Top} whose objects are topological spaces and morphisms continuous functions, with the study of the category \mathrm{Top}_Y in which the objects are all continuous functions on a fixed topological space Y and the morphisms are the continuous functions that make the triangular diagram of two objects commute.
Because a property \mathcal{P}_Y of \mathrm{Top}_Y can be considered a generalization of a corresponding property \mathcal{P} of \mathrm{Top} it must coincide with \mathcal{P} whenever the space Y reduces to a point, that is, when every object f : X\to Y of \mathrm{Top}_Y can be identified with the space X i.e. with an object of \mathrm{Top}.
Sometimes, the analogous property \mathcal{P}_Y in \mathrm{Top}_Y of a property \mathcal{P} of \mathrm{Top} can be obtained rather easily, being -- for example -- well known that the analogue in \mathrm{Top}_Y of compact spaces are perfect functions. In other cases (e.g., for the separation axioms T_0 and T_1), the property \mathcal{P}_Y can be obtained simply by requiring that the corresponding property \mathcal{P} be satisfied on every fiber of the functions, but in most cases new and more complex definitions must be given. The advantage of this kind of approach lies mainly in the fact that almost all the main statements of Classical Topology can be easily obtained from simple and elegant FGT propositions in the case where the space Y reduces to a singleton.
Investigation in this area of research has involved several properties of continuous functions such as several axioms of separations and compactness as well as notions such as compactness and metrizability of a function. In particular, in the paper "On completion of metric mapping" the notion of Cauchy net tied to a point has been introduced in order to provide a characterization of complete functions (introduced by Nordo and Pasynkov in 2002) and to describe a general method for the construction of complete extensions of any continuous function. The paper "On trivially semi-metrizable and D-completely regular mappings" is also in the area of fibrewise general topology. In it, through the introduction of the notions of function T-metrizable and T-semi-metrizable, they are characterized in terms of parallelism to symmetrizable, semi-metrizable and D-completely regular spaces, respectively, and it is also proved that the subfunctions of D-completely regular functions (i.e., the subfunctions of a fiber product of developable functions) coincide (barring homemorphisms) with the subfunctions of fiber products of semi-metrizable functions.

Digital Geometry studies the combinatorial and topological properties of discrete sets (digital objects) such as computer generated images, digitized images, or digital models in 2D or 3D space. The study of this discipline, introduced by Rosenfeld and Kong in the 1980s for binary images, has been reformulated by Kovalevsky, Eckhardt and Latecki in a more general context. Digital Geometry has useful applications in areas such as computer graphics and image analysis, and some of the main lines of research involve the study of properties such as digital convexity, digital straightness and digital flatness. Digital Topology studies the properties of digitized objects (sets of pixels or voxels) that correspond to the topological properties of the original objects. Originating from the early fundamental work of Kopperman, Herman, Khalimksy and Kovalevsky, this branch of Digital Geometry provides the theoretical basis for operations such as digitization, searching and counting connected components, frontier extraction, closed area filling and thinning of 2, 3 or multi-dimensional digital images. Among the different approaches described in the literature to define a topology on an n-dimensional digital space are the cellular complexes introduced by Kovalevsky or, equivalently, the product of Khalimsky topologies on a grid of integers. Many computational applications in computer graphics or image analysis need to represent and manipulate discrete subsets of a digital space, and since the size of the input can become very large, the need arises to represent these sets in a compact form. On the other hand, many topological notions (such as, for example, that of gap) have a rather complex formulation in the context of cellular complexes that makes them unsuitable for translating theoretical results into algorithms for concrete computational implementations. For this reason, the need has recently arisen to represent the notions and the main properties of digital spaces through the use of an algebraic-topological framework that also allows to describe the data structures for digital objects and the operations (algorithms) applicable to them. Through the use of combinatorial, topological and computational techniques, the local research unit aims to complete the study - already undertaken - of the relationships between the notions of gap and dimension of a digital object as well as to define an organic set of relationships and algebraic-topological tools that allow to represent, in a single framework, the notions, properties, elementary operations and algorithms of digital objects and their underlying data structures in computational applications.
Research activity in Digital Geometry initially focused on the study of gaps in a two-dimensional binary image. This number characterizes the structure of a digital image and is of considerable interest in various computational and image analysis applications. In this case, after providing a classification of the types of gaps (or tunnels) of a 2D digital object, a combinatorial approach based on the notion of free vertex has been followed which has allowed to obtain a general formula able to provide the number of gaps by means of some intrinsic parameters of the object in a manner analogous to what happens for the classical Euler formula. The methods and tools used in this investigation appeared suitable for generalizations in digital spaces of arbitrary dimension.

Soft topological spaces are particular topological structures defined on soft sets, a notion introduced in 1999 by Molodtsov, as a generalization of fuzzy sets, as a useful tool for modeling problems in computer science, physics, economics, social sciences, medical sciences, etc. Soft sets are defined as a parameterized family of subsets of a universe in which each element is considered as an approximate set of elements.
In this area, the research activity has focused mainly on the study of soft bi-topological spaces and of some properties of the neutrosophic soft topological spaces, in particular of some separation axioms. Recently, I have defined the notion of soft N-topological space as generalization both of the concepts of soft topological space given by Shabir and Naz and that of N-topological space as introduced by Tawfiq and Majeed and, independently, by Khan and I have investigated some basic properties of such class of spaces with particular regard to its subspaces and to the parameterized families of crisp topologies generated by the soft N-space, also introducing some new separation axioms called N-wise soft T_0, N-wise soft T_1, and N-wise soft T_2.

The collaboration with Prof. Hovik A. Matevossian of the Russian Academy of Sciences in Moscow and the fruitful exchange of methods, techniques and knowledge between different research areas (in this case, General Topology and Theory of Partial Derivative Differential Equations) allowed to study some properties of the solutions of Steklov-type boundary problems for biharmonic equations that can find some practical applications in Engineering and Medicine.

The notion of neutrosophic set was introduced in 1999 by Smarandache as a generalization of both the notions of fuzzy set introduced by Zadeh in 1965 and intuitionistic fuzzy set introduced by Atanassov in 1983.
In 2012, Salama and Alblowi introduced the notion of neutrosophic topological space which generalizes both fuzzy topological spaces given by Chang and that of intuitionistic fuzzy topological spaces given by Coker.
Further contributions to Neutrosophic Sets Theory, which also involve many fields of theoretical and applied mathematics were recently given by numerous authors and, at the moment, the various areas of research derived from the Neutrosophic Theory of Smarandache know a particularly lively scientific production.
Research activity in this area has focused on the study of single valued filters and on some properties of the neutrosophic soft topological spaces.