Bases and Metrics
due by Monday, Aug 23, 2021
This assignment is based on material in lectures 3 and 4.

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$B$
. 
Which of the following form a basis for some topology on $\mathbb{R}^2$? In each case prove or disprove.
 All open discs centred at the origin.
 All open discs centred at some point $(x, 0)$ on the $x$axis.
 All open discs centred at some point $(n,m)$ with $n, m\in\mathbb{Z}$.

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$X$
. 
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.$d(x, y) = (x  y)^2$
.$d(x, y) = \leftx  y\right^{1/2}$
.

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.$d((x_1, y_1), (x_2, y_2)) = (x_1  x_2)^2$
.$d((x_1, y_1), (x_2, y_2)) = \min(\leftx_1  x_2\right, \lefty_1  y_2\right)$
.