June 2008
... Do three of the problems from section A and three questions from section B. If you work more than the required number of problems, make sure that you clearly mark which problems you want to have counted. If you have doubts about the wording of a problem or about what results may be assumed without p ...
... Do three of the problems from section A and three questions from section B. If you work more than the required number of problems, make sure that you clearly mark which problems you want to have counted. If you have doubts about the wording of a problem or about what results may be assumed without p ...
A1 Partitions of unity
... Theorem in A1.7 (Partitions of unity) Let X be a second countable finite-dimensional topological manifold, and let W be an open cover of X. Then there exists a locally finite refinement U of W, and a continuous partition of unity subordinate to U. Moreover, if X is a C ∞ manifold, the partition of u ...
... Theorem in A1.7 (Partitions of unity) Let X be a second countable finite-dimensional topological manifold, and let W be an open cover of X. Then there exists a locally finite refinement U of W, and a continuous partition of unity subordinate to U. Moreover, if X is a C ∞ manifold, the partition of u ...
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.