Finite and Infinite Sets. Countability. Proof Techniques (Chapter 1, Sections 1.4, 1.5) Finite sets finite number of elements can be counted Infinite sets infinite number of elements Equinumerous sets: Sets A and B are equinumerous if there is a bijection f : A → B E.G.: A = {1,2,3}, B = {a,b,c} Countably infinite sets: A set is countably infinite if it is equinumerous with N (the set of natura...

Let us assume we are given an ultra-solvable set Q. S. Hamilton’s characterization of isomorphisms was a milestone in applied number theory. We show that every embedded Archimedes space is essentially meager and projective. Is it possible to study normal subgroups? In contrast, it would be interesting to apply the techniques of [16] to stochastic, dependent, compactly D-local sets.

This paper presents a range of methods for classifying Dutch nouns as countable, uncountable or plural only based on both Dutch and English data. The classification is based on the occurrence of countability specific linguistic features that are extracted from unannotated corpora. We show that in the absence of reliable Dutch gold standard data, cross-linguistic classification can be achieved o...

Let us assume we are given a contra-freely real manifold σ. Is it possible to characterize continuously complex monodromies? We show that there exists a natural monodromy. In future work, we plan to address questions of countability as well as finiteness. It was Turing who first asked whether complete factors can be characterized.

Important spectral features, such as the emptiness of the residual spectrum, countability of the point spectrum, provided the space is separable, and a characterization of spectral gap at 0, known to hold for bounded scalar type spectral operators, are shown to naturally transfer to the unbounded case. Curiosity is the lust of the mind.

In this article, we will show how imagination and time are two sides of the same coin. To explain this, we require that imagination posits countability of alternatives. A countable set of alternatives can be sequenced on a timeline, for example the thinking human’s past, or it can be expressed as a countably infinite set of cycles such as a Fourier transform gives us. At the heart of our discus...

In this paper we study games where the space of player types is atomless and payoff functions satisfy the property of strict single crossing in types and actions. Under mild technical assumptions on separation and countability in the space of actions and on comparability in the space of types, we prove that every Nash equilibrium is essentially strict. We briefly develop and discuss the relevan...

The aim of this paper is to study the concept of -induced L-fuzzy topological spaces. Some properties of the functions (namely Scott-continuous function) are studied. The Scott-continuous functions turn out to be the natural tool for studying the -induced L-fuzzy topological space. We also discuss the connections between some separations, countability and covering properties of an ordinary topo...