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Formally, a metric space is an ordered pair (M, d) where M is a set and d is a metric on M, i.e., ... One example of a compact space is the closed interval [0, 1].
In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. The metric captures all the geometric and causal structure of spacetime , being used to define notions such as time, distance, volume, curvature, angle, and separation of the future and the past.
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior . Intuitively speaking, a neighbourhood of a point is a set of points containing that point where one can move some amount in any direction away from ...
The metric space is the real line . is a set of real numbers whose absolute value is at most .Then, there is an external covering of ⌈ ⌉ intervals of length , covering the interval [,].
If is viewed as a metric space, its open balls are the open bounded intervals (c + r, c − r), and its closed balls are the closed bounded intervals [c + r, c − r]. In particular, the metric and order topologies in the real line coincide, which is the standard topology of the real line.
A spacetime diagram is a graphical illustration of locations in space at various times, especially in the special theory of relativity.Spacetime diagrams can show the geometry underlying phenomena like time dilation and length contraction without mathematical equations.
The unit interval is a complete metric space, homeomorphic to the extended real number line. As a topological space, it is compact, contractible, path connected and locally path connected. The Hilbert cube is obtained by taking a topological product of countably many copies of the unit interval.
A metric tensor field g on M assigns to each point p of M a metric tensor g p in the tangent space at p in a way that varies smoothly with p. More precisely, given any open subset U of manifold M and any (smooth) vector fields X and Y on U, the real function (,) = (,) is a smooth function of p.