Metric space

Manhattan distance

Metric space is a fundamental concept in the field of mathematics, specifically within the branch of topology. A metric space is defined as a set together with a metric on the set, which is a function that defines a distance between any two elements of the set. The concept of a metric space generalizes the notion of Euclidean distance to more abstract sets, allowing for the rigorous study of the geometry and topology of these sets.

Definition

A metric space is an ordered pair (M, d) where M is a set and d is a metric on M. The metric d is a function d: M × M → R (where R denotes the set of real numbers) that satisfies the following conditions for any elements x, y, and z in M:

1. d(x, y) ≥ 0 (non-negativity)
2. d(x, y) = 0 if and only if x = y (identity of indiscernibles)
3. d(x, y) = d(y, x) (symmetry)
4. d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality)

These properties ensure that the distance between any two points is a non-negative number, the distance is zero if and only if the two points are the same, the distance is the same in both directions, and the direct distance between two points is always less than or equal to the distance via a third point.

Examples

1. The set of real numbers R with the distance function d(x, y) = |x - y| is a metric space. This is the standard Euclidean metric. 2. The set of n-dimensional real vectors (R^n) with the Euclidean distance function defined as d(x, y) = sqrt((x_1 - y_1)^2 + ... + (x_n - y_n)^2) is also a metric space. 3. The set of continuous functions on a closed interval [a, b] with the distance function defined by the maximum absolute difference over the interval is another example of a metric space.

Properties

Metric spaces have several important properties that are studied in topology and related fields:

• Open and closed sets: In a metric space, one can define open balls (the set of points within a certain distance from a center point), which are used to define open and closed sets.
• Convergence and limits: A sequence in a metric space is said to converge to a limit if the distances from the sequence elements to the limit become arbitrarily small.
• Continuity: A function between two metric spaces is continuous if it preserves the convergence of sequences.
• Completeness: A metric space is complete if every Cauchy sequence (a sequence where the distance between successive elements becomes arbitrarily small) converges to a limit within the space.

Applications

Metric spaces are used in various areas of mathematics and applied sciences. They are fundamental in the study of geometry, analysis, and topology. In computer science, metric spaces are used in algorithms for searching and optimizing, particularly in spaces where the notion of distance is defined but does not necessarily follow the standard Euclidean metric. In physics, metric spaces can model the physical space under certain conditions, and in economics, they can represent distances in commodity spaces.

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