17 Facts About Metric space

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points.

Every metric space is a topological space, and some metric properties can be rephrased without reference to distance in the language of topology; that is, they are really topological properties.

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

Metric space is bounded if there is an such that no pair of points in is more than distance apart.

The converse does not hold: an example of a metric space that is bounded but not totally bounded is with the discrete metric.

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For example, a curve in a metric space is rectifiable if and only if it has a Lipschitz reparametrization.

Normed vector Metric space is a vector Metric space equipped with a norm, which is a function that measures the length of vectors.

Curve in a metric space is a continuous function.

Geodesic metric space is a metric space which admits a geodesic between any two of its points.

An example of a length Metric space which is not geodesic is the Euclidean plane minus the origin: the points and can be joined by paths of length arbitrarily close to 2, but not by a path of length 2.

An example of a metric space which is not a length space is given by the straight-line metric on the sphere: the straight line between two points through the center of the earth is shorter than any path along the surface.

Riemannian manifold is a space equipped with a Riemannian metric tensor, which determines lengths of tangent vectors at every point.

Formally, a metric measure space is a metric space equipped with a Borel regular measure such that every ball has positive measure.

For example, not every finite metric space can be isometrically embedded in a Euclidean space or in Hilbert space.

Any undirected connected graph, the set of vertices of can be turned into a metric space by defining the distance between vertices and to be the length of the shortest edge path connecting them.

Topological space is sequential if and only if it is a quotient of a metric space.

Every premetric space is a topological space, and in fact a sequential space.