Definition: 1.05: Surface
And a surface is that which has length and breadth only.
Modern Definition
A surface (or a curved surface) \(s\) is a connected subset of a Euclidean metric space \(A\in\mathbb R^n\), \(n\ge 2\) such that taking away a line \(g\subset s\), and given any point \(A\in g\) on the line \(g\) will disconnect a sufficiently small neighborhood of \(A\) in \(s\).
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Definitions: 1 2 3
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References
Adapted from CC BYSA 3.0 Sources:
 Callahan, Daniel: "Euclidâ€™s 'Elements' Redux" 2014
Adapted from (Public Domain)
 Casey, John: "The First Six Books of the Elements of Euclid"
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"