![Chapter 6: Hyperbolic Analytic Geometry](http://s1.studyres.com/store/data/005456980_1-b19a8c3f76e7ecabdb6c2c9650804373-300x300.png)
FUNDAMENTAL GROUPS - University of Chicago Math Department
... g(2s), for s ∈ [0, 21 ] h(s) = f (2s − 1), for s ∈ [ 12 , 1]. This product on paths induces an operation on equivalence classes of paths defined by the equation [g] ∗ [f ] = [g ∗ f ]. 2.2. The Fundamental Group. The set of all path homotopy equivalence classes of paths in a space is not a group unde ...
... g(2s), for s ∈ [0, 21 ] h(s) = f (2s − 1), for s ∈ [ 12 , 1]. This product on paths induces an operation on equivalence classes of paths defined by the equation [g] ∗ [f ] = [g ∗ f ]. 2.2. The Fundamental Group. The set of all path homotopy equivalence classes of paths in a space is not a group unde ...
Mathematical Preliminaries
... of a set A, denoted P (A), is the collection of all subsets of A. Notice that if the cardinality (see below for definition) of the set A is finite (and equal to a), then the number of subsets of A, i.e. the cardinality of the power set of A, is 2a . Next, we (intuitively) define a map from one sour ...
... of a set A, denoted P (A), is the collection of all subsets of A. Notice that if the cardinality (see below for definition) of the set A is finite (and equal to a), then the number of subsets of A, i.e. the cardinality of the power set of A, is 2a . Next, we (intuitively) define a map from one sour ...
ON SOMEWHAT β-CONTINUITY, SOMEWHAT β
... Theorem 3.4: A function f : (X, τ) → (Y, σ) is hardly β-open if and only if intβ(f–1(A)) = 0/ for each set A ⊂ Y having the property that intβ(A) = 0/ and A containing a nonempty closed set. Proof: Assume f is hardly β-open. Let A ⊂ Y such that intβ(A) = 0/ and let F be a nonempty closed set contain ...
... Theorem 3.4: A function f : (X, τ) → (Y, σ) is hardly β-open if and only if intβ(f–1(A)) = 0/ for each set A ⊂ Y having the property that intβ(A) = 0/ and A containing a nonempty closed set. Proof: Assume f is hardly β-open. Let A ⊂ Y such that intβ(A) = 0/ and let F be a nonempty closed set contain ...
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.