### Bases and Metrics

##### due by Monday, Aug 23, 2021

This assignment is based on material in lectures 3 and 4.

1. Let $X$ be a space with discrete topology and $B$ a collection of subsets of $X$. Prove or disprove that $B$ is a basis if and only if every singleton set $\{x\}$, with $x\in X$, is in $B$.

2. Which of the following form a basis for some topology on $\mathbb{R}^2$? In each case prove or disprove.

1. All open discs centred at the origin.
2. All open discs centred at some point $(x, 0)$ on the $x$-axis.
3. All open discs centred at some point $(n,m)$ with $n, m\in\mathbb{Z}$.
3. Let $X$ be a set. Show that the collection $$\mathcal{F}= \{A \subset X: \textrm{A is countable}\}\cup\{X\}$$ is the collection of closed sets for some topology on $X$.

4. For each of the following functions $d: \mathbb{R}\times\mathbb{R}\to\mathbb{R}$, determine whether $d$ is a metric. Prove or disprove in each case.

1. $d(x, y) = (x - y)^2$.
2. $d(x, y) = \left|x - y\right|^{1/2}$.
5. For each of the following functions $d: \mathbb{R^2}\times\mathbb{R^2}\to\mathbb{R}$, determine whether $d$ is a metric. Prove or disprove in each case.

1. $d((x_1, y_1), (x_2, y_2)) = (x_1 - x_2)^2$.
2. $d((x_1, y_1), (x_2, y_2)) = \min(\left|x_1 - x_2\right|, \left|y_1 - y_2\right|)$.