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Consider a set M equipped with a structure ∗. We call a natural topology T∗, on (M, ∗), the topology induced by ∗. For example, a natural topology for a metric space (X, d) is a topology Td induced by the metric d, and for a linearly ordered set (X

Why is the manifold topology in a spacetime taken for granted? Why do we prefer to use Riemann open balls as basic-open sets, while there also exists a Lorentz metric? Which topology is a best candidate for a spacetime: a topology sufficient for the description of spacetime singularities or a topology which incorporates the causal structure? Or both? Is it more preferable to consider a topology with as many physical properties as possible, whose description might be complicated and counterintuitive, or a topology which can be described via a countable basis but misses some important information? These are just a few from the questions that we ask in this chapter, which serves as a critical review of the terrain and contains a survey with remarks, corrections and open questions.

The motion planning problem is a fundamental problem in robotics, so that every autonomous robot should be able to deal with it. A number of solutions have been proposed and a probabilistic one seems to be quite reasonable. However, here we propose a more adoptive solution that uses fuzzy set theory and we expose this solution next to a sort survey on the recent theory of soft robots, for a future qualitative comparison between the two.

This chapter is divided into two parts. The first part is a survey of some recent results on nests and the orderability problem. The second part consists of results, partial results and open questions, all viewed in the light of nests. From connected LOTS, to products of LOTS and function spaces, up to the order relation in the Fermat Real Line.

The classical Linear Regression is an approximation for the creation of a model which connects a dependent variable y with one or more independent variables X, and it is subjected to some assumptions. The violation of these assumptions can influence negatively the power of the use of statistical regression and its quality. Nowadays, to exceed this problem, a new method has been introduced, and is in use, that is called fuzzy regression. Fuzzy regression is considered to be possibilistic, with the distribution function of the possibility to be connected with the membership function of fuzzy numbers. In this article, we examine three models of fuzzy regression, for confidence level h=0, for the case of crisp input values, fuzzy output values and fuzzy regression parameters, with an application to Hydrology, in two rainfall stations in Northern Greece.

The class of LOTS (linearly ordered topological spaces, i.e. spaces equipped with a topology generated by a linear order) contains many important spaces, like the set of real numbers, the set of rational numbers and the ordinals. Such spaces have rich topological properties, which are not necessarily hereditary. The Orderability Problem, a very important question on whether a topological space admits a linear order which generates a topology equal to the topology of the space, was given a general solution by van Dalen and Wattel (Gen. Topol. Appl. 3:347–354, 1973). In this article we first examine the role of the interval topology in van Dalen’s and Wattel’s characterization of LOTS, and we then discuss ways to extend this model to transitive relations that are not necessarily linear orders.