A converse to this theorem is given by Alexandrov's uniqueness theorem, according to which a metric space that is locally Euclidean except for a finite number of points of positive angular defect, adding to 4?, can be realized in a unique way as the surface of a convex polyhedron.
22.
Calculus has enormous numbers of applications throughout the physical sciences and elsewhere, point-set topology and general measure theory don't ( they have applications, certainly, but not to the same extent ) . 4 ) A manifold is a topological space that is locally Euclidean, nothing else is a manifold.
23.
It states that if a metric space ( " X ", " d " ) is geodesic, homeomorphic to a sphere, and locally Euclidean except for a finite number of cone points of positive angular defect summing to 4, then ( " X ", " d " ) can be represented as the development of a convex polyhedron.
24.
For example, if a space is locally Euclidean at a point you can define its tangent space at that point ( which is, itself, a useful thing to do for all kinds of purposes ), you can't define the tangent space to a figure of 8 at that central point ( it has two tangents there, so you would end up with the union of two lines, which isn't a vector space ) . 3 ) A manifold is a space that is locally Euclidean everywhere ( possibly with some extra conditions, depending on who you ask ), that is the minimal condition ( I'm not really sure what you mean by " some sort " of manifold, there are generalisations of manifolds, but they are really manifolds any more even if the word may appear in their name, you could argue that " topological space " is a generalisation of " manifold ", but that doesn't mean much ) . 4 ) No, the only neighbourhood of any point in an indiscrete space is the whole space, which can't be homeomorphic to any Euclidean space because no Euclidean space ( beyond "'R "'0, I guess ) is indiscrete .-- talk ) 00 : 04, 12 January 2009 ( UTC)
25.
For example, if a space is locally Euclidean at a point you can define its tangent space at that point ( which is, itself, a useful thing to do for all kinds of purposes ), you can't define the tangent space to a figure of 8 at that central point ( it has two tangents there, so you would end up with the union of two lines, which isn't a vector space ) . 3 ) A manifold is a space that is locally Euclidean everywhere ( possibly with some extra conditions, depending on who you ask ), that is the minimal condition ( I'm not really sure what you mean by " some sort " of manifold, there are generalisations of manifolds, but they are really manifolds any more even if the word may appear in their name, you could argue that " topological space " is a generalisation of " manifold ", but that doesn't mean much ) . 4 ) No, the only neighbourhood of any point in an indiscrete space is the whole space, which can't be homeomorphic to any Euclidean space because no Euclidean space ( beyond "'R "'0, I guess ) is indiscrete .-- talk ) 00 : 04, 12 January 2009 ( UTC)
