Related papers: An Elementary Expository Study: From Metric Spaces…
When one considers the collection $\mathcal{H}(\mathbb{R}^n)$ of all compact subsets of $\mathbb{R}^n$ and equip it with a topology, many questions can be asked about the topological space one ends up with. This is an example of a…
A self-contained introduction is presented of the notion of the (abstract) differentiable manifold and its tangent vector fields. The way in which elementary topological ideas stimulated the passage from Euclidean (vector) spaces and linear…
Metric embeddings are central to metric theory and its applications. Here we consider embeddings of a different sort: maps from a set to subsets of a metric space so that distances between points are approximated by minimal distances…
We introduced the concept of a metric value set (MVS) in an earlier paper \cite{GM}. In this paper we study the algebraic structure of MVSs. For an MVS $M$ we define the concept of $M$-metrizability of a topological space and prove some…
In various areas of modern physics and in particular in quantum gravity or foundational space-time physics it is of great importance to be in the possession of a systematic procedure by which a macroscopic or continuum limit can be…
Learning faithful graph representations as sets of vertex embeddings has become a fundamental intermediary step in a wide range of machine learning applications. The quality of the embeddings is usually determined by how well the geometry…
In this work we provide a way to introduce a probability measure on the space of minimal fillings of finite additive metric spaces as well as an algorithm for its computation. The values of probability, got from the analytical solution,…
There are versions of "calculus" in many settings, with various mixtures of algebra and analysis. In these informal notes we consider a few examples that suggest a lot of interesting questions.
Combining gravity with quantum theory is still work in progress. On the one hand, classical gravity, is the geometry of space-time determined by the energy-momentum tensor of matter and the resulting nonlinear equations; on the other hand,…
In this dissertation, we explore models based on the idea that there are two metrics in spacetime: One describes the standard gravity, and the other provides a geometry in which matter fields propagate. In order to do that, we provide the…
Using the wedge sum of metric spaces, for all compact metrizable spaces, we construct a topological embedding of the compact metrizable space into the set of all metric trees in the Gromov--Hausdorff space with finite prescribed values. As…
Associated to any finite metric space are a large number of objects and quantities which provide some degree of structural or geometric information about the space. In this paper we show that in the setting of subsets of weighted Hamming…
The enumeration degrees of sets of natural numbers can be identified with the degrees of difficulty of enumerating neighborhood bases of points in a universal second-countable $T_0$-space (e.g. the $\omega$-power of the Sierpi\'nski space).…
Learning Spaces are certain set systems that are applied in the mathematical modeling of education. We propose a suitable compression (without loss of information) of such set systems to facilitate their logical and statistical analysis.…
Lecture notes as per the title. In the first part, the concepts of a measurable space, measurable maps between measurable spaces and that of a measure on a measurable space are introduced, after which the fundamentals of the theory of…
In this paper we will study the representations of isomorphisms between bases of topological spaces. It turns out that the perfect setting for this study is that of regular open subsets of complete metric spaces, but we have achieved some…
General relativity does not allow one to specify the topology of space, leaving the possibility that space is multiply rather than simply connected. We review the main mathematical properties of multiply connected spaces, and the different…
We study the properties of topological spaces $(X,\tau)$, where $X$ is a definable set in an o-minimal structure and the topology $\tau$ on $X$ has a basis that is (uniformly) definable. Examples of such spaces include the canonical…
We present a general way to define a topology on orthomodular lattices. We show that in the case of a Hilbert lattice, this topology is equivalent to that induced by the metrics of the corresponding Hilbert space. Moreover, we show that in…
The first aim of this study is to define soft sequential compact metric spaces and to investigate some important theorems on soft sequential compact metric space. Second is to introduce net and totally bounded soft metric space and study…