![Recall : A topology on a set X is a collection Τ of subsets of X having](http://s1.studyres.com/store/data/000786013_1-b1879787d5dfeec4d9bbc9d14174bf50-300x300.png)
High School Essential Curriculum
... e. Analyze the properties and relationships of geometric solids with bases other than rectangles, triangles, or circles. f. Analyze the properties and relationships of truncated three-dimensional solids. ...
... e. Analyze the properties and relationships of geometric solids with bases other than rectangles, triangles, or circles. f. Analyze the properties and relationships of truncated three-dimensional solids. ...
Outline - Durham University
... a Cartesian plane: {(x, y) | x, y ∈ R} with the distance d(A1 , A2 ) = (x1 − x2 )2 + (y1 − y2 )2 ; a Gaussian plane: {z | z ∈ C}, with the distance d(u, v) = |u − v|. Definition 1.1. A Euclidean isometry is a distance-preserving transformation of E2 , i.e. a map f : E2 → E2 satisfying d(f (A), f (B) ...
... a Cartesian plane: {(x, y) | x, y ∈ R} with the distance d(A1 , A2 ) = (x1 − x2 )2 + (y1 − y2 )2 ; a Gaussian plane: {z | z ∈ C}, with the distance d(u, v) = |u − v|. Definition 1.1. A Euclidean isometry is a distance-preserving transformation of E2 , i.e. a map f : E2 → E2 satisfying d(f (A), f (B) ...
Topology (Part 2) - Department of Mathematics, University of Toronto
... Corollary: If f:[a,b]→ → R, then f assumes all values between f(a) & f(b) (Intermediate Value Theorem) Proof: [a,b] is connected (in its relative topology) – this really requires proof. By above theorem, f([a,b]) (image) is connected. Suppose f(a) < f(b) (Etc. if other way as usual). Suppose f(a)
... Corollary: If f:[a,b]→ → R, then f assumes all values between f(a) & f(b) (Intermediate Value Theorem) Proof: [a,b] is connected (in its relative topology) – this really requires proof. By above theorem, f([a,b]) (image) is connected. Suppose f(a) < f(b) (Etc. if other way as usual). Suppose f(a)
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.