Notes on Bernard Cache, Euclidean geometry and topology

"Einstein's theory certainly teaches us that space(-time) is actually "curved" (i.e. not Euclidean) in the presence of a gravitational field, but generally, one perceives this curvature only in the case of bodies moving at speeds close to that of the light."
"Klein would go so far as to define the various geometries by the group of movements which transform geometrical figures without affecting distances nor angles in these figures. This group of movements defines what is called: "metric geometry". Now, if we forget the distances and concentrate on the "shape" of the figures defined by the angles between elements, we come upon a new transformation which is the scaling. Translation, rotation, symmetry and scaling form a wider group of transformations, the group of similitudes which defines Euclidean geometry."
"And finally, if we do away with position properties, and only look at the continuity of the figures and at the order in which their elements are linked together, just as if figures were made of an elastic material which can be stretched and deformed, but not torn, we encounter another group of transformations: the homeographies which define topology."
"Euclidean geoemtry requires more axioms and more structured properties. Projective geometry and topology can be more general only to the extent that they deal with looser transformations and objects. As such topology enables one to focus on fundamental properties from which our Euclidean intuition is distracted by the metric appearances. Because topology doesn't register any difference between a cube and a sphere, it focuses on what is left, order and continuity..."
"One single topological structure has an infinity of Euclidean incarnations, the variations of which are not relevant for topology, about which topology has nothing to say ... Topology cannot be said to be curved because it precedes any assignment of metrical curvature. Because topological structures are often represented with in some ways indefinite curved surfaces, one might think that topology brings free curvature to architecture, but this is a misunderstanding. When mathematicians draw those kind of free surfaces, they mean to indicate that they do not care about the actual shape in which topology can be incarnated ... And, of course, as soon as it comes to actually making a geometrical figure out of a topological structure, we enter into Euclidean geometry; that is, the design of complex curvature is essentially Euclidean."