# 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

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 $x_{0} \neq y_{0}$.