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In mathematics, a space is a set (sometimes known as a universe) with a definition ( structure) of relationships among the elements of the set. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself.
In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. [1] Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry .
A point in three-dimensional Euclidean space can be located by three coordinates. Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer ...
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the σ -algebra) and the method that is used for measuring (the measure ). One important example of a measure space is a probability space .
Complete metric space. In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M . Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because ...
Plane (mathematics) In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. When working exclusively in two-dimensional Euclidean space, the definite article is used, so the Euclidean ...
Hilbert space. The state of a vibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinct overtones is given by the projection of the point onto the coordinate axes in the space. In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear ...
In probability theory, the sample space (also called sample description space, [1] possibility space, [2] or outcome space [3]) of an experiment or random trial is the set of all possible outcomes or results of that experiment. [4] A sample space is usually denoted using set notation, and the possible ordered outcomes, or sample points, [5] are ...