Proofs 1

Question 1

Let (X, d) be a metric space. Show that X is Hausdorff.

Answer 1

Let x,y ∈ X be such that x != y and define two open balls with radius d(x,y)/2. Because those sets do not intersect, X is Hausdorff.

Question 2

Any finite set in a topological space is compact.

Answer 2

Let M be a finite set with an open cover {Ui}. Then for every x ∈ M there is an element i(x) such that x ∈ Ui(x) and {Ui(x)} is a finite subcover of M.

Question 3

Show that a compact set in a Hausdorff space is closed.

Answer 3

Let X be a Hausdorff space and K ⊂ X be compact.

Question 4

Show that e(x, y):= |arctan(x) − arctan(y)| induces the same topology as d(x, y):= |x − y|, i.e, the two metrics are equivalent. Conclude that completeness is not a topological property.

Answer 4

It suffices to show that every e-ball contains a d-ball and every d-ball contains an e-ball. So, fix the radius and find a radius for the other metric, such that the ball is contained. This can be done by solving for d(x,x+-ε)