- #1
vandanak
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how is discrete metric space given by d((x1,x2,...xn)(y1,y2,...yn))=0 if xi=yi else 1
disc
is complete
disc
is complete
A discrete metric is a mathematical concept used in the field of topology to measure the distance between two points in a set. It is defined as the number of steps it takes to get from one point to another, where each step is of equal distance.
Unlike other metrics, the discrete metric only considers the distance between two points to be either 0 or 1. This means that there are no fractional or decimal distances, and all points are either adjacent (distance of 1) or the same point (distance of 0).
The discrete metric is used in various fields, including computer science, biology, and physics. In computer science, it is used to measure the similarity between two data points, while in biology, it is used to analyze genetic sequences. In physics, it is used to study the behavior of particles in discrete spaces.
Topology is the study of geometric properties that are preserved under continuous deformations, and the discrete metric is used to define a topology on a set. It helps to identify topological properties such as connectedness, compactness, and continuity.
Yes, the concept of discrete metric can be extended to higher dimensions, but it becomes more complex as the number of dimensions increases. In higher dimensions, the distance between two points is measured by the number of steps it takes to go from one point to another, where each step is of equal length in each dimension.