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A three-dimensional Euclidean space is a special case of a Euclidean space. In Bourbaki's terms, [2] the species of three-dimensional Euclidean space is richer than the species of Euclidean space. Likewise, the species of compact topological space is richer than the species of topological space. Fig. 3: Example relations between species of spaces
Unlike in a geodesic metric space, the infimum does not have to be attained. An example of a length space which is not geodesic is the Euclidean plane minus the origin: the points (1, 0) and (-1, 0) 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 ...
Most non-Euclidean geometries can be modeled by a manifold, and embedded in a Euclidean space of higher dimension. For example, an elliptic space can be modeled by an ellipsoid. It is common to represent in a Euclidean space mathematical objects that are a priori not of a geometrical nature. An example among many is the usual representation of ...
Cartesian coordinates identify points of the Euclidean plane with pairs of real numbers. In mathematics, the real coordinate space or real coordinate n-space, of dimension n, denoted R n or , is the set of all ordered n-tuples of real numbers, that is the set of all sequences of n real numbers, also known as coordinate vectors.
An example of a quotient space of a manifold that is also a manifold is the real projective space, identified as a quotient space of the corresponding sphere. One method of identifying points (gluing them together) is through a right (or left) action of a group , which acts on the manifold.
In the counter-example, the proposition "all unmarried men are bachelors" is true by virtue of each term's meaning. Further, space is a priori because it is the form of our receptive abilities to receive information about the external world. For example, someone without sight can still perceive spatial attributes via touch, hearing, and smell.
A simple example is a volume (how big an object occupies a space) as a measure. In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and ...
This page will attempt to list examples in mathematics. To qualify for inclusion, an article should be about a mathematical object with a fair amount of concreteness. Usually a definition of an abstract concept, a theorem, or a proof would not be an "example" as the term should be understood here (an elegant proof of an isolated but particularly striking fact, as opposed to a proof of a ...