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... Let us recall the definition of a topological group; this is a group (G, ., e) together with a topology on G such that (x, y) 7→ xy −1 is continuous, i.e., from G × G into G. Note also that G × G is regarded as a topological space defined by the product topology. Definition 0.1. Consider G to be a t ...
... Let us recall the definition of a topological group; this is a group (G, ., e) together with a topology on G such that (x, y) 7→ xy −1 is continuous, i.e., from G × G into G. Note also that G × G is regarded as a topological space defined by the product topology. Definition 0.1. Consider G to be a t ...
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... The following theorem holds in geometries in which isosceles triangle can be defined and in which SSS, AAS, and SAS are all valid. Specifically, it holds in Euclidean geometry and hyperbolic geometry (and therefore in neutral geometry). Theorem 1 (Isosceles Triangle Theorem). Let 4ABC be an isoscele ...
... The following theorem holds in geometries in which isosceles triangle can be defined and in which SSS, AAS, and SAS are all valid. Specifically, it holds in Euclidean geometry and hyperbolic geometry (and therefore in neutral geometry). Theorem 1 (Isosceles Triangle Theorem). Let 4ABC be an isoscele ...
HW 3 - Solutions to selected exercises
... Ex 1. See http://www.math.toronto.edu/dalvit/courses/mat402/sphere.pdf Ex 2. The same construction as in Euclidean geometry works. Ex 3. Consider the three great circles defined as intersections of the coordinate planes (xy-plane, xz-plane, yz-plane) with the sphere. Each of the eight regions define ...
... Ex 1. See http://www.math.toronto.edu/dalvit/courses/mat402/sphere.pdf Ex 2. The same construction as in Euclidean geometry works. Ex 3. Consider the three great circles defined as intersections of the coordinate planes (xy-plane, xz-plane, yz-plane) with the sphere. Each of the eight regions define ...
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. Intuitively, a 3-manifold can be thought of as a possible shape of the universe. Just like a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.