Definition:Euclidean Space/Euclidean Topology/Real Number Plane
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Definition
Let $\R^n$ be an $n$-dimensional real vector space.
Let $M = \struct {\R^2, d}$ be a real Euclidean space of $2$ dimensions.
The topology $\tau_d$ induced by the Euclidean metric $d$ is called the Euclidean topology.
The space $\struct {\R^2, \tau_d}$ is known as the (real) Euclidean plane.
Also known as
The Euclidean topology, when applied to a real Cartesian space, is often referred to as the usual topology.
Also see
- Results about Euclidean spaces can be found here.
Source of Name
This entry was named for Euclid.
Historical Note
Euclid himself did not in fact conceive of the Euclidean metric and its associated Euclidean space, Euclidean topology and Euclidean norm.
They bear that name because the geometric space which it gives rise to is Euclidean in the sense that it is consistent with Euclid's fifth postulate.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Cartesian product
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Cartesian product