Metric space

Hierarchy of mathematical spaces. Inner product spaces are a subset of normed vector spaces, which are a subset of metric spaces, which in turn are a subset of topological spaces. The plane (a set of points) can be equipped with different metrics. In the taxicab metric the red, yellow and blue paths have the same length (12), and are all shortest paths. In the Euclidean metric, the green path has length Q on a sphere.

To see the utility of different notions of distance, consider the surface of the Earth as a set of points. One notion of distance (a metric) between two such points is the length of the shortest path along the surface, or "as the crow flies"; this is particularly useful for shipping and aviation. The straight-line distance between two points through the Earth's interior is another metric; this notion of distance is, for example, natural in seismology, since it roughly corresponds to the length of time it takes for seismic waves to travel between those two points.

The notion of distance encoded by the metric space axioms has relatively few requirements. This generality gives metric spaces a lot of flexibility. At the same time, the notion is strong enough to encode many intuitive facts about what distance means. This means that general results about metric spaces can be applied in many different contexts.

Like many fundamental mathematical concepts, the metric on a metric space can be interpreted in many different ways. A particular metric may not be best thought of as measuring physical distance, but, instead, as the cost of changing from one state to another (as with Wasserstein metrics on spaces of measures) or the degree of difference between two objects (for example, the Hamming distance between two strings of characters, or the Gromov–Hausdorff distance between metric spaces themselves).

Formally, a metric space is an ordered pair d is a metric on x,y,z∈M:

1. The distance from a point to itself is zero: x to x: z from M as an abbreviation for d(x,y)=|y−x| given by the absolute difference form a metric space. Many properties of metric spaces and functions between them are generalizations of concepts in real analysis and coincide with those concepts when applied to the real line.

   Metrics on Euclidean spaces

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   Comparison of Chebyshev, Euclidean and taxicab distances for the hypotenuse of a 3-4-5 triangle on a chessboard

   The Euclidean plane d1((x1,y1),(x2,y2))=|x2−x1|+|y2−y1| and can be thought of as the distance you need to travel along horizontal and vertical lines to get from one point to the other, as illustrated at the top of the article.

   The maximum, d∞(p,q)≤d2(p,q)≤d1(p,q)≤2d∞(p,q), which holds for every pair of points α∈(1,∞) that interpolates between these three distances. It is defined byℝn as well as A⊆M, we can consider A is a function S as a subset of S described above. Two more useful examples are the open interval x in a metric space r around x: N⊆M is a neighborhood of r around M. In other words, the open sets of d2, and ∞</sub>"}},"i":0}}]}'/>d∞ defined above all induce the same topology on (0, 1) with the induced metric are homeomorphic but have very different metric properties.

   Conversely, not every topological space can be given a metric. Topological spaces which are compatible with a metric are called metrizable and are particularly well-behaved in many ways: in particular, they are paracompact Hausdorff spaces (hence normal) and first-countable. The Nagata–Smirnov metrization theorem gives a characterization of metrizability in terms of other topological properties, without reference to metrics.

   Convergence

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   Convergence of sequences in Euclidean space is defined as follows:

   A sequence n</sub>'')"}},"i":0}}]}'/>(xn) converges to a point N such that for all ''N''"}},"i":0}}]}'/>n > N, n</sub>'', ''x'') < ε"}},"i":0}}]}'/>d(xn, x) < ε.

   Convergence of sequences in a topological space is defined as follows:

   A sequence n</sub>'')"}},"i":0}}]}'/>(xn) converges to a point x there is an integer xn∈U.

   In metric spaces, both of these definitions make sense and they are equivalent. This is a general pattern for topological properties of metric spaces: while they can be defined in a purely topological way, there is often a way that uses the metric which is easier to state or more familiar from real analysis.

   Completeness

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   Informally, a metric space is complete if it has no "missing points": every sequence that looks like it should converge to something actually converges.

   To make this precise: a sequence n</sub>'')"}},"i":0}}]}'/>(xn) in a metric space N such that for all ''N''"}},"i":0}}]}'/>m, n > N, m</sub>'', ''x_n'') < ε"}},"i":0}}]}'/>d(xm, xn) < ε. By the triangle inequality, any convergent sequence is a Cauchy sequence: if m</sub>"}},"i":0}}]}'/>xm and n</sub>"}},"i":0}}]}'/>xn are both less than M converges—then (0, 1) and the rationals, each with the metric induced from ℝ (for example, its successive decimal approximations). These examples show that completeness is not a topological property, since [0, 1] is the completion of