Metric spaces as topological spaces

Show that the discrete topology for a non-empty set $X$ is a metric topology.

Verify that the definition of an open set for a metric space satisfies the requirements for a topology.

Let $d$ denote the usual Euclidean metric in the plane and let $d'((x, y), (u, v)) = \left|x - u\right| + \left|y - v\right|$. Verify that $d'$ is a metric.

The points in a space are continuous functions defined on the interval $[0, 1]$. Define the distance between two such functions to be

Show that this definition of the distance between two functions does satisfy the three properties required of a metric. This depends on the fact, which you may assume in your argument, that the integral of a nonnegative continuous function is positive unless the function is identically $0$.

If $x, y \in \mathbb {R}^{2}$, let $d(x, y) = \left|x_1 - y_1\right|$. Which of the three properties of a metric does $d$ satisfy?

On $\mathbb {R}^{2} \setminus \left\{ 0\right\}$, write each point in polar coordinates as $(r, \theta )$, where $0 \leq \theta < 2\pi$. Define $d((r_1, \theta_1), (r_2, \theta_2)) = \left|r_1 - r_2\right| + \left|\theta_1 - \theta_2\right|$. Show that this gives a metric on $\mathbb {R}^{2} \setminus \left\{ 0\right\}$ but the topology formed is not the usual topology.