However, this is just the most likely reason as to why people don't study non-locally Euclidean topological spaces.
12.
The Hausdorff property is not a local one; so even though Euclidean space is Hausdorff, a locally Euclidean space need not be.
13.
The theorem has a number of equivalent statements, one of which is that the topology induced by the Carnot Carath�odory metric is equivalent to the intrinsic ( locally Euclidean ) topology of the manifold.
14.
Quasi-projective varieties are " locally affine " in the sense that a manifold is locally Euclidean & mdash; every point of a quasiprojective variety has a neighborhood given by an affine variety.
15.
But anyhow, differenciation involves the notion of'direction'and arbitrary topological space don't have this notion unless you have a local vector space structure defined on them ( for instance locally Euclidean ).
16.
The polyhedron can be thought of as being folded from a sheet of paper ( a homeomorphic ( topologically equivalent ) to a sphere, and locally Euclidean except for a finite number of cone points whose angular defect sums to 4.
17.
Where the latter equals to zero, the metric structure is locally Euclidean ( it means that at least some open set in the coordinate space is isometric to a piece of Euclidean space ), no matter whether coordinates are affine or curvilinear.
18.
The " unrestricted " version of Hilbert's fifth problem, closer to Hilbert's original formulation, considers both a locally Euclidean group " G " and another manifold " M " on which " G " has a continuous action.
19.
A topological manifold is a locally Euclidean Hausdorff space . ( In Wikipedia, a manifold need not be paracompact or second-countable . ) A " C k " manifold is a differentiable manifold whose chart overlap functions are " k " times continuously differentiable.
20.
A space " M " is locally Euclidean if and only if it can be atlas "'on " M " . ( The terminology comes from an analogy with cartography whereby a spherical globe can be described by an atlas of flat maps or charts ).
