Metric spaces

Concepts

• Metric space 10 exercisesA metric space is a set X together with a distance d between its points: a nonnegative number for each pair, symmetric, zero only…
• Open ball 10 exercisesThe open ball B(x,r) with center x and radius r>0 in a metric space (X,d) is the set of points of X whose distance to x is less t…
• Open set in a metric space 10 exercisesA subset U of a metric space (X,d) is open if every point x∈ U is the center of some open ball B(x,r) contained in U.
• Convergence in a metric space 10 exercisesA sequence (x_n) in a metric space (X,d) converges to x∈ X if, for every ε>0, there is an index N from which x_n lies within dist…
• Complete metric space 10 exercisesA metric space X is complete if every Cauchy sequence (x_n) in X converges to a point x of X. A sequence is Cauchy if its terms e…