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Math 535 - General Topology Fall 2012 Homework 7 Solutions
... Solution. For any x ∈ X, the subset {x, p} is open and contains x, hence is a neighborhood of x. Moreover {x, p} is finite, hence compact. b. Show that X is compact if and only if X is finite. Solution. (⇐) Every finite space is compact. S (⇒) Consider the open cover X = x∈X {x, p}. Since X is compa ...
... Solution. For any x ∈ X, the subset {x, p} is open and contains x, hence is a neighborhood of x. Moreover {x, p} is finite, hence compact. b. Show that X is compact if and only if X is finite. Solution. (⇐) Every finite space is compact. S (⇒) Consider the open cover X = x∈X {x, p}. Since X is compa ...
The Word Geometry
... and by his ability to solve extremely complicated mathematical operations Some of his teachers were Gauss,Jacobi, Dirichlet, and Steiner Riemannian geometry ...
... and by his ability to solve extremely complicated mathematical operations Some of his teachers were Gauss,Jacobi, Dirichlet, and Steiner Riemannian geometry ...
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.