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An Introduction to Non-Euclidean Geometry
... An Introduction to Non-Euclidean Geometry Nate Black ...
... An Introduction to Non-Euclidean Geometry Nate Black ...
local and global convexity for maps
... 13. Lemma (Global properties imply convexity). If Ψ : X → Rn is a convex map, then its image, Ψ(X), is convex, and its level sets, Ψ−1 (w), for w ∈ Ψ(X), are connected. Proof. Take any two points in Ψ(X); write them as Ψ(x0 ) and Ψ(x1 ) where x0 and x1 are in X. Because the map Ψ is convex, there ex ...
... 13. Lemma (Global properties imply convexity). If Ψ : X → Rn is a convex map, then its image, Ψ(X), is convex, and its level sets, Ψ−1 (w), for w ∈ Ψ(X), are connected. Proof. Take any two points in Ψ(X); write them as Ψ(x0 ) and Ψ(x1 ) where x0 and x1 are in X. Because the map Ψ is convex, there ex ...
strongly connected spaces - National University of Singapore
... PROPOSITION 2.10: For any space X, the following statements are equivalent: (1) X is locally connected. (2) The components of every open subspace of X are open. (3) The connected open sets of X form a basis of the topology of X. Proof: (1) ⇒ (2). Let X be a locally connected space and let U be an o ...
... PROPOSITION 2.10: For any space X, the following statements are equivalent: (1) X is locally connected. (2) The components of every open subspace of X are open. (3) The connected open sets of X form a basis of the topology of X. Proof: (1) ⇒ (2). Let X be a locally connected space and let U be an o ...
Homology stratifications and intersection homology Geometry & Topology Monographs Colin Rourke Brian Sanderson
... The purpose of this section is to generalise proposition 2.2 to homology manifolds. Proposition 4.1 If M is a homology manifold then the natural map φ: Hiπ (M ) → Hi (M ) is an isomorphism. The proof is very similar to the proof of 2.2. However the key point in the proof (the application of PL gener ...
... The purpose of this section is to generalise proposition 2.2 to homology manifolds. Proposition 4.1 If M is a homology manifold then the natural map φ: Hiπ (M ) → Hi (M ) is an isomorphism. The proof is very similar to the proof of 2.2. However the key point in the proof (the application of PL gener ...
3-manifold
![](https://commons.wikimedia.org/wiki/Special:FilePath/3-Manifold_3-Torus.png?width=300)
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.