§2.1. Topological Spaces Let X be a set. A family T of subsets of X is
... (b) Let X be a set. Let T1 be the family of all subsets of X . Then T1 is a topology for X . It is called the discrete topology. It is the largest topology for X in the sense that if T is a topology for X , then T ⊂ T1 . (c) Let X be a metric space. Then the family of open subsets (defined in terms ...
... (b) Let X be a set. Let T1 be the family of all subsets of X . Then T1 is a topology for X . It is called the discrete topology. It is the largest topology for X in the sense that if T is a topology for X , then T ⊂ T1 . (c) Let X be a metric space. Then the family of open subsets (defined in terms ...
The Hyperbolic Plane
... In order for the geodesic path to be non-empty, we require a2 + 2b = r2 > 0, and so geodesics in H correspond to upper semicircles in U centered on the x-axis. Again one can determine the parameterization using the condition g(u̇, u̇) = 1. Note that geodesics in H (of both types) are what we have al ...
... In order for the geodesic path to be non-empty, we require a2 + 2b = r2 > 0, and so geodesics in H correspond to upper semicircles in U centered on the x-axis. Again one can determine the parameterization using the condition g(u̇, u̇) = 1. Note that geodesics in H (of both types) are what we have al ...
Embeddings vs. Homeomorphisms (Lecture 13)
... Rn to itself: Theorem 1 is equivalent to the assertion that the inclusion between these topological spaces is a weak homotopy equivalence. Remark 3. We can also define simplicial sets which parametrize PL embeddings and PL homeomorphisms from Rn to itself. Theorem 1 continues to hold in this case, u ...
... Rn to itself: Theorem 1 is equivalent to the assertion that the inclusion between these topological spaces is a weak homotopy equivalence. Remark 3. We can also define simplicial sets which parametrize PL embeddings and PL homeomorphisms from Rn to itself. Theorem 1 continues to hold in this case, 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.