Do all examples of metric spaces have infinite number of points?
A non-trivial metric space is a mathematical concept used to describe a set of points where the distance between any two points is defined by a function called a metric. This function satisfies certain properties, such as being positive and symmetric, and allows for the measurement of distance between points in the space.
A non-trivial metric space has an infinite number of points, whereas a trivial metric space only has a finite number of points. In a non-trivial metric space, the distance between any two points can be infinitely small, while in a trivial metric space, the distance between any two points is always a non-zero constant value.
Infinite points by convention refers to the idea that in non-trivial metric spaces, the number of points is considered to be infinite, even if the actual number of points in the space is finite. This allows for the measurement of distances between points that may be infinitely small, and allows for a more flexible and abstract understanding of metric spaces.
Some examples of non-trivial metric spaces include Euclidean space, which is the set of all points in three-dimensional space, and the set of real numbers with the metric defined as the absolute value of the difference between two numbers. Other examples include graphs, networks, and other abstract mathematical structures.
Non-trivial metric spaces are used in various fields of science, such as physics, engineering, and computer science. They are used to model and analyze complex systems, such as the behavior of particles in quantum mechanics or the flow of information in computer networks. Non-trivial metric spaces are also used in data analysis and machine learning to measure similarities and differences between data points.