ALgebraic TOpolology [ALTO-102B]

Prerequisites: The major portion of the course in devoted to algebraic topology, but some glimpses of the smooth manifolds theory will be given along the way. This course will assume previous knowledge of general topology, though, a brief overview of the contents required for this course will be covered.

 

Contents:

Ch-1- General Topology

Ch-2- Differential Manifolds

Ch-3- Fundamental Group

Ch-4- Homology Theory

Ch-5- Cohomology

Ch-6- Products and Duality

Ch-7- Homotopy Theory

 

Books:

Topology and Geometry (Graduate texts in mathematics) by G.E. Bredon

Questions to be Answered

(1)

Let ( \mathbb{R}^n, d_{l^2}) be a metric space and x_{0},y_{0} \in X be two distinct points. Prove that there exists no radius r_1,r_2 \in \mathbb{R} s.t. B_{r_{1}} (x_{0})= B_{r_{2}} (y_{0})  i.e. ball of radius r_1 around x_0 is equal to ball of  r_{2} around y_0.

Hints: \epsilon- balls in \mathbb{R}^n are open sets, you may take cases when r_{1}=r_{2}, r_{1} > r_{2} , r_{1} < r_{2}. The latter two cases are symmetrical. In fact, first prove this easy result that

(1-A) Prove that in particular, B_{r} (x_{0}) \neq B_{r} (y_{0})  whenever x_{0} \neq y_{0}.

 

Advertisements