A Euclidean space is a zerocurve, infinitely large, real metric space that follows Euclidean geometry, and thus, all of Euclid's Postulates.
Euclidean spaces are constructed from repeated Cartesian products of a real line , and can be modelled with a real coordinate space (often, Euclidean spaces are denoted with , though can be used). This means that coordinates in a Euclidean space have to be an ordered tuple of real numbers, the number of numbers is the dimensionality of the space.
Euclidean spaces are typically known as "flat" spaces due to the fact that they have zero curvature.
Distances in a Euclidean space can be determined with the Pythagorean theorem.
Euclidean spaces can be generalised with Hilbert spaces, which can allow spaces to pertain an infinite dimensionality.
Spaces that don't follow Euclidean geometry do not follow one of the postulates, and may have curvature, imaginary dimensions, a different geometry.
See Also
Real dimensionality  0  1  2  3  ... 

Real space

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Real projective space

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Complex space

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Complex projective space

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Quaternionic space

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Quaternionic projective space

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Octonionic space

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Octonionic projective space

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