On the Mathematical Structure for Discrete and Continuous Metric Point Sets

Abstract

We have studied fundamental properties of continuous and discrete metric point sets. Our focus is pure geometrical objects. We show how geodesic lines and angles can be constructed from an imposed metric even in discrete spaces. Lines in discrete and continuous metric point sets are constructed and compared with Euclid’s five axioms. The angle between two lines is defined. Euclid’s axioms E1 and E2 are sufficient to achieve local angles and to define an infinite space. Axiom E3 is sufficient to define a space with more than one dimension. Axiom E4 is666 J. F. Moxnes and K. Hausken sufficient to define a homogenous space. Axiom E5 is sufficient to define a flat space. We study how the concepts of vector spaces could appear from the metric point set. We have constructed arrows from each point in the metric point set. These arrows can be conceived as lines with a direction. The sum of arrows from each point is constructed algebraically without parallel transport. A method is presented for constructing coordinates. We have constructed coordinates in a metric point space by assuming that the arrow from a specific point o in the metric point space defines a vector space at each point p. We comment on the force concept. Different parallel transports are constructed geometrically. The concepts of tensors and tensor fields are briefly addressed