The space which satisfies all five of the postulates (aka axioms) upon which Euclid built his ideas of geometry.

Euclidean spaces are nowadays defined in terms of Cartesian coordinates, but they have various definitions.

E as a topological space

The one-dimensional Euclidean space E is defined on the set of real numbers taken in their natural order.

Given some real numbers r, e, consider the set {n| r-e < n < r+e}.

Let us construct the collection S of all such subsets for any r and any e. The Euclidean topology is the topology for which S is a base. (That is, any open interval or any combination of open intervals is considered "open" in this topology).

The N-dimensional Euclidean space EN is the product space of N copies of E. For example, a set in E2 is open if it is the union of open rectangles, and a set in E3 is open if it is the union of open rectanglular prisms.

Several metric spaces induce the topology EN. (examples here)

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