![Sufficient Conditions for Paracompactness of](http://s1.studyres.com/store/data/001351290_1-044bba308d9d73907a01c970709cd89d-300x300.png)
1.1 Patterns and Inductive Reasoning
... 2 can be written as the sum of two primes. This is called Goldbach’s Conjecture. No one has ever proven this conjecture is true or found a counterexample to show that it is false. As of the writing of this text, it is unknown if this conjecture is true or false. It is known; however, that all even n ...
... 2 can be written as the sum of two primes. This is called Goldbach’s Conjecture. No one has ever proven this conjecture is true or found a counterexample to show that it is false. As of the writing of this text, it is unknown if this conjecture is true or false. It is known; however, that all even n ...
Functional Analysis Exercise Class
... d) Given a set M ⊂ X , what is its closure/interior/boundary? Solution: a) We have ∅ = (+∞, +∞), and R = (−∞, +∞). Given ai , i ∈ I, where I is any index set, we have ∪i∈I (ai , +∞) = (inf i ai , +∞) ∈ τhl . If I is finite then ∩i∈I (ai , +∞) = (maxi ai , +∞) ∈ τhl . Hence, τhl contains the empty se ...
... d) Given a set M ⊂ X , what is its closure/interior/boundary? Solution: a) We have ∅ = (+∞, +∞), and R = (−∞, +∞). Given ai , i ∈ I, where I is any index set, we have ∪i∈I (ai , +∞) = (inf i ai , +∞) ∈ τhl . If I is finite then ∩i∈I (ai , +∞) = (maxi ai , +∞) ∈ τhl . Hence, τhl contains the empty se ...
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.